The local Burkholder functional, quasiconvexity and Geometric Function Theory

Theorem 1.3. Let $p \ge 2$ . For any $A \in \mathbb{R}^{2 \times 2}$ and any $f \in A + W_0^{1,2}(\Omega, \mathbb{R}^2)$ such that $\mathbf{B}_p(\mathrm{D}f) \le 0$ a.e. in $\Omega$ , we have <span id="page-3-1"></span>(1.7) $$\mathbf{B}_{p}(A) \leqslant \int_{\Omega} \mathbf{B}_{p}(\mathrm{D}f(z)) \,\mathrm{d}m(z).$$ On the other hand, it was shown in [37] that the positive part $\mathbf{B}_p^+ \equiv \max\{\mathbf{B}_p, 0\}$ is polyconvex. In particular, it follows that (1.7) persists for maps $f \in A + W_0^{1,2}(\Omega, \mathbb{R}^2)$ satisfying $\mathbf{B}_p(\mathrm{D}f) \geqslant 0$ almost everywhere in $\Omega$ . Thus combined, these two results provide strong evidence towards the full quasiconvexity of $\mathbf{B}_p$ . Theorem 1.3 was established in the special case $A = \operatorname{Id}$ in [5]. Also, as we will explain later, the result entails a sharp integrability statement for quasiconformal maps. At the moment let us emphasize that it comprises delicate cancellation properties: for any K-quasiregular map f and for $p = p_K$ , we have $\mathbf{B}_{p_K}(\mathrm{D}f(z)) \in \mathrm{L}^1_{\mathrm{loc}}$ , even if in general the map $f \in \mathrm{W}^{1,s}_{\mathrm{loc}}$ only for $s < p_K$ [2]. In order to establish the existence of minimizers for the induced Burkholder energy we actually need a stronger form of quasiconvexity. To this end, it is convenient to introduce the local Burkholder functional, defined by <span id="page-3-3"></span>(1.8) $$\mathcal{B}_K(A) \equiv \begin{cases} \mathbf{B}_{p_K}(A), & \text{if } |A|^2 \leqslant K \det A, \\ +\infty & \text{otherwise.} \end{cases}$$ Thus $\mathcal{B}_K$ equals the Burkholder functional $\mathbf{B}_p$ with the largest p for which $\mathbf{B}_p(\mathrm{D}f) \leq 0$ for every K-quasiconformal map f, and it becomes defined in all of $\mathbb{R}^{2\times 2}$ at the cost of admitting the value $+\infty$ outside the K-quasiconformal cone. However, since $\mathcal{B}_K$ assumes the value $+\infty$ , the notion of quasiconvexity needs to be strengthened to closed quasiconvexity [65, 76]: briefly, one requires the Jensen inequality, that is (1.7), to hold not just for maps but also for gradient Young measures, cf. Section 2 for the precise definition. With this terminology, we obtain the following stronger version of Theorem 1.3.

Theorem 1.4. Let $K \ge 1$ and $p > \frac{2K}{K+1}$ . Then the local Burkholder functional $\mathcal{B}_K \colon \mathbb{R}^{2 \times 2} \to \mathbb{R} \cup \{+\infty\}$ is closed W^1,p-quasiconvex. By combining Theorem 1.4 with standard results from the theory of Young measures and the Direct Method of the Calculus of Variations we obtain the existence of minimizers:

Corollary 1.5. Let $K \ge 1$ and $2 \le p < \frac{2K}{K-1}$ . Then for any K-quasiregular map $g: \mathbb{C} \to \mathbb{C}$ , the problem $$\inf \left\{ \int_{\Omega} \mathbf{B}_{p}(\mathrm{D}f) \, \mathrm{d}m(z) : f \in g + \mathrm{W}_{0}^{1,p}(\Omega) \text{ is } K\text{-}quasiregular \right\}$$ admits a minimizer $f \in g + W_0^{1,p}(\Omega)$ . Note that here we do not need to require the maps to be homeomorphisms. 1.2. The Burkholder area inequality. We next turn to a further refinement of Theorem 1.3, of independent interest, but also one of the key points in the study of the associated weak lower semicontinuity and minimization problems, see Section 12. Namely, given a $W_{loc}^{1,2}(\mathbb{C})$ -homeomorphism f, analytic outside $\mathbb{D}$ , we say that f is a principal map if it has the Laurent expansion <span id="page-4-2"></span>(1.9) $$f(z) = z + \frac{b_1}{z} + \sum_{j=2}^{\infty} \frac{b_j}{z^j}, \qquad |z| > 1.$$ It follows from the classical Grönwall–Bieberbach area formula that in this expansion $|b_1| < 1$ , see Section 3. This allows us to interpret the first two terms of the above series, i.e. the main asymptotics of f, in terms of the invertible linear map <span id="page-4-4"></span>(1.10) $$A_f(z) = z + b_1 \bar{z},$$ equivalently $A_f \equiv \int_{\mathbb{D}} \mathrm{D}f(w) \, \mathrm{d}m(w).$ As we will see, in many respects $A_f$ plays the role which the linear boundary values have in the standard definition of quasiconvexity. For instance, with this notation the classical area formula, cf. (3.3), asserts that <span id="page-4-1"></span>(1.11) $$\int_{\mathbb{D}} \left[ -\det \mathbf{D}f + \det A_f \right] \mathrm{d}m(z) = \sum_{j=2}^{\infty} j |b_j|^2,$$ and one can think of this identity as a sharpening of the well-known null-Lagrangian property of the Jacobian determinant. In the same spirit one can consider also the Burkholder functional, recalling that $\mathbf{B}_2(A) = -\det(A)$ . In fact, with Theorem 1.3 we find an $\mathbf{L}^p$ version of (1.11):

Theorem 1.6. (Burkholder Area Inequality). Let f be a K-quasiconformal principal map as in (1.9). Then, for any $2 \le p \le p_K$ , we have $$\int_{\mathbb{D}} \left[ \mathbf{B}_p(\mathbf{D}f) - \mathbf{B}_p(A_f) \right] dm(z) \geqslant \gamma_p(A_f) \sum_{j=2}^{\infty} j |b_j|^2$$ where $$\gamma_p(A_f) \equiv \frac{p}{2} \frac{\mathbf{B}_p(A_f)}{\mathbf{B}_2(A_f)} > 0$$ . Note that $\gamma_2 = 1$ . Theorem 1.6 sharpens the main result in [5], since $\mathbf{B}_p(A_f) \geq \mathbf{B}_p(\mathrm{Id})$ for any principal map f as above (note, however, the different sign-convention for the Burkholder functional in [5]). Since $A_f = \int_{\mathbb{D}} \mathrm{D}f(z) \, \mathrm{d}m(z)$ , a surprising feature of Theorem 1.6 is that it establishes a Jensen inequality even without requiring that the map takes affine boundary values, as in the definition of quasiconvexity! In fact, the result shows that, if $$\int_{\mathbb{D}} \mathbf{B}_p(\mathrm{D}f) \, \mathrm{d}m(z) = \int_{\mathbb{D}} \mathbf{B}_p(A_f) \, \mathrm{d}m(z),$$ then $f(z) = z + \frac{b_1}{z}$ for |z| > 1; in particular, $f|_{\mathbb{S}^1}$ is linear. 1.3. Functionals for Nonlinear Elasticity, as $p \to 2$ . As observed in [45] the theory of Burkholder functionals has very interesting consequences at the limit when the index p goes to 2. Namely, we have $\mathbf{B}_2(A) = -\det(A)$ and for the next order of approximation $$\lim_{p\to 2} \frac{p}{p-2} \left[ \mathbf{B}_p(A) - \mathbf{B}_2(A) \right] = \mathscr{F}(A),$$ where the functional $$\mathscr{F}(A) \equiv |A|^2 - (1 + \log |A|^2) \det(A),$$ is rank-one convex in $\mathbb{R}^{2\times 2}$ , but not polyconvex. On the other hand, the local quasiconvexity of the Burkholder functional, as formulated in Theorem 1.3, allows us to show that $\mathscr{F}$ satisfies the quasiconvexity inequality (1.4) for all W^1,2-homeomorphisms with linear boundary values $A \in \mathbb{R}_+^{2\times 2}$ , see Corollary 8.1 for the precise statement. Notice that this for instance implies a sharp and quantitative version of the celebrated L log L-higher integrability properties for the derivatives of W^1,2-homeomorphisms. Another consequence of these relations concerns the quasiconvexity of the functional <span id="page-5-1"></span>(1.12) $$\mathscr{W}(A) \equiv \frac{|A|^2}{\det A} - \log\left(\frac{|A|^2}{\det A}\right) + \log \det A, \quad A \in \mathbb{R}_+^{2 \times 2}.$$ This is an example of a rank-one convex but non-polyconvex functional, which diverges as the $det(A) \rightarrow 0$ . It was introduced in [5] and further studied in the recent works [86, 87], where its quasiconvexity remained undecided. In fact, $\mathcal{W}$ arises from $\mathcal{F}$ by applying the Shield transform [83]. Therefore Theorem 1.3 leads us to the following: <span id="page-6-0"></span>Corollary 1.7. The functional $\mathcal{W}: \mathbb{R}^{2\times 2}_+ \to \mathbb{R}$ is quasiconvex. In Section 12 we investigate weak lower semicontinuity properties of $\mathcal{W}$ and establish that some features similar to closed quasiconvexity hold also for $\mathcal{W}$ ; for precise formulations see Proposition 12.5. Originating from $\mathbf{B}_p$ , the initial functional (1.12) assumes all real values, and even tends to $-\infty$ along suitable directions when the determinant goes to zero. However, it allows easy modifications creating quasiconvex and non-polyconvex functionals that satisfy (1.2). The following is perhaps the easiest example: <span id="page-6-2"></span>(1.13) $$\widetilde{\mathscr{W}}(A) \equiv \frac{|A|^2}{\det A} - \log\left(\frac{|A|^2}{\det A}\right) + |\log \det A|,$$ see Remark 12.9. The interest in [86, 87] on $\mathcal{W}$ originates from the fact that it spans the only non-polyconvex extreme ray in a class of functionals satisfying the additive volumetric-isochoric split, see also [36] for further information on extremal functionals. Thus, as a consequence of Corollary 1.7, we obtain a solution to Morrey's problem in a class of elastic functionals:

Theorem 1.8. Let $\mathbf{E} \colon \mathbb{R}^{2\times 2}_+ \to \mathbb{R}$ be a functional of the form $$\mathbf{E}(A) = g(\det A) + h(K_A), \qquad K_A \equiv \frac{|A|^2}{\det A},$$ where $h: [1, +\infty) \to \mathbb{R}$ is convex and $g: (0, +\infty) \to \mathbb{R}$ . Then $\mathbf{E}$ is rank-one convex $\iff$ $\mathbf{E}$ is quasiconvex. The additive volumetric-isochoric split goes back at least to the work of Flory [31] and since then it has been used extensively to model slightly compressible materials, see for instance [41, 75] and the references therein. 1.4. Lower semicontinuity and existence of minimizers. As discussed above the notion of quasiconvexity was introduced by Morrey to characterize sequential weak lower semicontinuity for integral functionals in the vectorial calculus of variations. On the other hand, for functionals $\mathbf{E} \colon \mathbb{R}^{2\times 2}_+ \to \mathbb{R}$ the condition (1.2) expresses the intuitive and natural requirement of hyperelasticity that an infinite amount of energy is required to compress a finite volume of material into zero volume [10, 24]. However, for such functionals with (1.2) it is not clear if quasiconvexity suffices for lower semicontinuity results. The stronger notion of polyconvexity does suffice, allowing a wealth of interesting models for hyperelastic materials [8]. In fact, Theorem 1.8 already provides a natural class of functionals that have been considered before in the engineering literature [24], but their weak lower semicontinuity and minimization properties had not been established. In addition to Theorem 1.8 or Corollary 1.7 we shall also address this point here. In this connection, the extended Stoilow factorization due to Iwaniec and Šverák [53] suggested to us that Jensen inequality with respect to principal maps might be sufficient for lower semicontinuity. Indeed, this was our original indication that a theorem like the Burkholder area inequality might be true. Applying this line of thought leads us to the following Jensen inequality for principal maps:

Theorem 1.9. Let $f \in W^{1,1}_{loc}(\mathbb{C})$ be a homeomorphism, and a principal map with integrable distortion $K_f \in L^1(\mathbb{D})$ . Then $$\int_{\mathbb{D}} \left[ \mathscr{W} \left( \mathrm{D} f(z) \right) - \mathscr{W} \left( A_f \right) \right] \mathrm{d} m(z) \ge 0.$$ Remark 1.10. Notice that the above inequality implies sharp bounds on the integrability of $\log(1/J_f)$ in terms of those of $K_f$ . Here the assumption $K_f \in L^1$ is optimal mathematically. Moreover, notice that in the study of incompressible neo-Hookean materials, the first invariant of the isochoric part of the Cauchy-Green tensor of the deformation f is $\hat{I}_1 = K_f + 1/K_f$ , see [75]. Since it is unclear how to measure experimentally the response of materials as the determinant tends to zero [24], the condition $K_f \in L^1$ might in fact be the right postulate within our current knowledge. In addition, recall that the norm $||K_f||_{L^1}$ equals the W^1,2-Sobolev norm of $f^{-1}$ ; thus it is plausible that the condition $K_f \in L^1$ is the right regularity requirement in order to have a flexible lower semicontinuity theory. It turns out that, for general functionals with the blow-up (1.2), proving lower semicontinuity requires two properties: the Jensen inequality for principal maps, allowing analysis via gradient Young measures, and secondly, control of concentration, typically via suitable equiintegrability. With these properties available we easily obtain the following lower semicontinuity result.

Theorem 1.11. Let $g \in W^{1,2}_{loc}(\mathbb{C})$ be a homeomorphism. Consider a sequence $(f_j)$ in $g + W^{1,2}_0(\Omega)$ such that $f_j \rightharpoonup f$ in $W^{1,2}(\Omega)$ and for some q > 1 we have $||K_{f_j}||_{L^q(\Omega)} \leq C < \infty$ . Then $$\liminf_{j\to\infty} \int_{\Omega} \mathscr{W}(\mathrm{D} f_j(z)) \,\mathrm{d} m(z) \geqslant \int_{\Omega} \mathscr{W}(\mathrm{D} f(z)) \,\mathrm{d} m(z).$$ Similar weak lower semicontinuity holds for the rank-one convex functionals from Theorem [1.8](#page-6-1) with the appropriate volumetric-isochoric split, see Corollary [12.6.](#page-60-0) Notice that as in the Burkholder setting the endpoint result, which in this context is K^f P L 1 , is missing. This amounts to study possible concentration effects and we will investigate it in a future work. Building on ^W or on ^W^Ă we obtain in Subsection [12.3](#page-60-1) a number of quasiconvex non-polyconvex functionals for which the direct method of the Calculus of Variation gives existence of minimizers, see Corollary [12.7](#page-60-2) and Example [12.8.](#page-62-1) Morever since, by the work of Koskela and Onninen [\[60\]](#page-65-9), the norm }K^f }L^q controls } log J^f }L^q , many of these functionals allow the blow-up condition [\(1.2\)](#page-0-1). This, in particular, sheds light on the problem of existence of minimizers in hyperelasticity, see [\[12\]](#page-63-7) and in particular Problem 1 there. On the other hand, this class also contains functionals which degenerate in various ways when the determinant vanishes, like the Burkholder functional itself. Therefore its interest is not restricted to the elasticity ecosystem but is relevant also to the geometric function theory interpretation of our work. Outline. Finally, we conclude the introduction with a description of the organization of the paper. Section [2](#page-10-0) revisits the standard notions of the vectorial calculus of variations, with an emphasis on the special care needed to treat extended-real valued functionals. Section [3](#page-21-0) reviews relevant parts of the basic quasiconformal theory, with special focus on principal maps. Section [4](#page-24-0) adapts the theory of quasiregular Young measures, initiated in [\[3\]](#page-63-8) and applied for instance in [\[30\]](#page-64-10), to the case of principal maps. It also extends this theory to maps of integrable distortion. Section [5](#page-30-0) provides the preliminary results needed in the proof of the local quasiconvexity of the Burkholder functional, in particular an extremality argument in the spirit of [\[5\]](#page-63-4). Section [6](#page-34-0) presents the proof of Theorems [1.3](#page-3-2) and [1.4.](#page-4-0) Section [7](#page-37-0) contains the proof of Theorem [1.6.](#page-5-0) Section [8](#page-42-1) studies the functional ^F, which is the derivative of ^B^p at ^p " 2, and is closely related to the higher integrability of the Jacobian. Section [9](#page-44-0) revisits the classical Shield transformation, which uses inverses to define new integral functionals and, in particular, presents W as a transformation of F, which leads to the proof of Corollary [1.7.](#page-6-0) For the sake of completeness we also investigate for which class of test functions the quasiconvexity inequality for W can be verified. Section [10](#page-48-0) proves Theorem [1.9,](#page-7-0) which gives quasiconvexity of W in the class of principal maps. Similar inequalities are established for F. Section 11 proves the quasiconvexity of functionals with volumetric isochoric split after that of $\mathcal{W}$ , streamlining the arguments in [86] and proving Theorem 1.8. In Section 12 we are then in position to apply a version of the direct method of calculus of variations to prove lower semicontinuity theorems and existence of minimizers for a quite large family of functionals, see in particular Theorem 12.7. In this section we also prove Corollary 1.5 and Theorem 1.11. Notation. We denote by $\mathbb{D} \subset \mathbb{C}$ the unit disk and by $\mathbb{A}(r,R)$ the annulus $\{z: r < |z| < R\}$ . Unless explicitly stated otherwise, $\Omega \subset \mathbb{R}^n$ is a bounded domain such that $\mathscr{L}^n(\partial\Omega) = 0$ ; in most of the paper we will take n = 2. Given a map $g \colon \mathbb{R}^n \to \mathbb{R}^n$ , we sometimes use the notation $W_g^{1,p}(\Omega) \equiv g + W_0^{1,p}(\Omega,\mathbb{R}^n)$ ; in particular, this space is well-defined even if $\partial\Omega$ is irregular. We also use the standard notation <span id="page-9-1"></span> $$\oint_{\Omega} \varphi(x) \, \mathrm{d}x \equiv \frac{1}{|\Omega|} \int_{\Omega} \varphi(x) \, \mathrm{d}x.$$ A matrix $A \in \mathbb{R}^{2\times 2}$ is naturally identified with a linear map $A \colon \mathbb{C} \to \mathbb{C}$ . It will also be useful to use conformal coordinates, whereby we identify $A \in \mathbb{R}^{2\times 2}$ with a pair $(a_+, a_-) \in \mathbb{C}^2$ according to the rule $$(1.14) A(z) = a_+ z + a_- \bar{z}.$$ Here, on the left-hand side we see z as an element of $\mathbb{R}^2$ , while on the right-hand side $z \in \mathbb{C}$ . In these coordinates we have <span id="page-9-0"></span>(1.15) $$\det A = |a_+|^2 - |a_-|^2, \qquad |A| = |a_+| + |a_-|,$$ where $|A| \equiv \max_{z \in \mathbb{S}^1} |A(z)|$ denotes the operator norm. Given a map $f \in W^{1,1}_{loc}(\Omega)$ , we write $J_f \equiv \det Df$ for its Jacobian. If $J_f > 0$ a.e. in $\Omega$ then there is a measurable function $K \colon \Omega \to [1, +\infty]$ such that $K < \infty$ a.e. and $$|\mathrm{D}f(z)|^2 \leqslant K(z)J_f(z),$$ a.e. in $\Omega$ . The distortion function of f, which we denote by $K_f$ , is the smallest such function K. Acknowledgments. D.F, K.A, A.K acknowledge the financial support of QUAMAP, the ERC Advanced Grant 834728, and of the Severo Ochoa Programme CEX2019-000904-S. A.G. was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. D.F and A.K were partially supported by CM and UAM, and A.K by Academy of Finland CoE Randomness and Structures, and Academy Fellowship Grant 355840. D.F acknowledge financial support by PI2021-124-195NB-C32. K.A, D.F, A.G, A.K acknowledge the hospitality and financial support of the Institute of Advanced studies during various periods in 2021-2022 and the discussions there with C. De Lellis, V.Šverák and L.Székelyhidi Jr, on topics related to the paper. K.A, A.G, J.K acknowledge the hospitality of Universidad Autónoma de Madrid and ICMAT during the autumn of 2022. D.F also wants to acknowledge the hospitality of the Mathematical Institute of Oxford during the summer of 2023.

Lemma 2.1. Assume $F: \mathbb{R} \to \overline{\mathbb{R}}$ is convex and that for some $x_0 \in \mathbb{R}$ we have $F(x_0) = -\infty$ . Then there exist $\alpha, \beta \in \overline{\mathbb{R}}$ with $\alpha \leq x_0 \leq \beta$ such that $$F = \begin{cases} -\infty & in (\alpha, \beta), \\ +\infty & on \mathbb{R} \setminus [\alpha, \beta]. \end{cases}$$ Furthermore, when $\alpha \in \mathbb{R}$ or $\beta \in \mathbb{R}$ , the values of F there are unrestricted. Let $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \overline{\mathbb{R}}$ be an extended real-valued function (henceforth we refer to functions defined on $\mathbb{R}^{m \times n}$ as functionals). The effective domain of $\mathbf{E}$ is the subset <span id="page-10-2"></span>(2.1) $$\operatorname{dom}(\mathbf{E}) \equiv \left\{ A \in \mathbb{R}^{m \times n} : \mathbf{E}(A) < +\infty \right\}$$ of $\mathbb{R}^{m \times n}$ , where we emphasize that the value $-\infty$ is allowed for E on dom(E).

Lemma 2.3. Assume $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \overline{\mathbb{R}}$ is a rank-one convex functional and that for some $A_0 \in \operatorname{core}(\operatorname{dom}(\mathbf{E}))$ we have $\mathbf{E}(A_0) \in \mathbb{R}$ . Then $\mathbf{E} > -\infty$ on all lines through $A_0$ that are parallel to a rank one matrix.

Theorem 2.8. Let $p \in [1, \infty]$ and let $(\varphi_j)$ be a bounded sequence in $W^{1,p}(\Omega, \mathbb{R}^m)$ . Then there is a subsequence, which we do not relabel, such that $(D\varphi_j)$ generates a $W^{1,p}$ -gradient Young measure $\nu = (\nu_x)_{x \in \Omega}$ . However, even if the original sequence $(\varphi_j)$ converges weakly in $W^{1,p}(\Omega, \mathbb{R}^m)$ , the gradient Young measures its subsequences generate need not be unique. On the other hand, since the center of mass or barycenter is the weak limit of (2.6), this is of course the same for all Young measures generated by the subsequences of a given weakly converging sequence. In view of (2.4) the gradient Young measures provide flexible methods to compute weak limits of nonlinear quantities. Hence the notion is very useful also in our setting. For later purposes we list here some of their basic and well-known properties. For instance, given a Young measure one can often modify or improve the generating sequence.

Theorem 2.9. [77, Theorem 8.15 and Lemma 6.3] Suppose $\nu = (\nu_x)_{x \in \Omega}$ is a W^1,p-gradient Young measure in $\Omega$ and suppose $\langle \nu_x, \operatorname{Id} \rangle = \operatorname{D}g(x)$ a.e. in $\Omega$ , where $g \in \operatorname{W}^{1,p}(\mathbb{R}^n, \mathbb{R}^m)$ . (1) Then there is a bounded sequence $(\varphi_j)$ in $g + W_0^{1,p}(\Omega, \mathbb{R}^m)$ whose gradients generate $\nu$ and for which (2.7) $$(|D\varphi_i|^p)$$ is equiintegrable on $\Omega$ . (2) If $(\varphi_j)$ is a bounded sequence in $W^{1,p}(\Omega, \mathbb{R}^m)$ that generates $\nu$ , and $(\psi_j)$ is another sequence for which $D\psi_j - D\varphi_j \to 0$ in $L^p(\Omega)$ , then also $(D\psi_j)$ generates $\nu$ . The fact that the generating sequence can be chosen so that $(|D\varphi_j|^p)$ is equiintegrable will be important for us and is related to the so-called Decomposition Lemma [32, 63, 64]. <span id="page-14-0"></span>Remark 2.10. On the other hand, the limit (2.4) exists even for a general continuous $\mathbf{E} \in \mathrm{C}(\mathbb{R}^{m \times n})$ which does not vanish at $\infty$ , but where, for the generating sequence, $(\mathbf{E}(\mathrm{D}\varphi_i))$ is equiintegrable, see [77, Theorem 6.2]. The literature on Gradient Young measures is by now quite extensive and for further properties we refer to the monographs [66, 73, 77, 79]. <span id="page-14-4"></span>Remark 2.11. It is not difficult to extend the convergence (2.4) also to lower semicontinuous functionals $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \mathbb{R} \cup \{+\infty\}$ which are continuous on dom( $\mathbf{E}$ ), provided that ( $\mathbf{E}(\mathbf{D}\varphi_j)$ ) is equiintegrable. This can be proved by approximating $\mathbf{E}$ with the sequence of truncations $\mathbf{E}_k \equiv \min{\{\mathbf{E}, k\}}$ and applying the standard lower semicontinuity result <span id="page-14-1"></span>(2.8) $$\liminf_{j \to \infty} \int_{\Omega} \eta(x) \mathbf{E}_k(\mathrm{D}\varphi_j) \, \mathrm{d}x \geqslant \int_{\Omega} \eta(x) \int_{\mathbb{R}^{m \times n}} \mathbf{E}_k(A) \, \mathrm{d}\nu_x(A) \, \mathrm{d}x,$$ where $\eta \in L^{\infty}(\Omega)$ is arbitrary, cf. [73, Corollary 3.3]; the opposite direction simply follows from the equi-integrability assumption. We note that (2.8) can also be proved by approximating lower semicontinuous functionals with continuous ones, using the Scorza–Dragoni theorem. In this paper, only in Section 12 we really use gradient Young measures in the full generality of Definition 2.6. Instead, most of the time we will be content to work with homogeneous gradient Young measures:

Proposition 2.14. (The Localization Principle [56]) Given a W^1,p-gradient Young measure $\nu = (\nu_x)_{x \in \Omega}$ , we have $\nu_x \in \mathcal{M}_{qc}^p$ for a.e. $x \in \Omega$ . <span id="page-15-1"></span>Remark 2.15. In fact, if $\nu$ is generated by a sequence $(\varphi_j)$ then, for a.e. $x_0$ , the measure $\nu_{x_0}$ is generated by a diagonal subsequence of the sequence $\psi_{j,\lambda}(x) \equiv \lambda^{-1}(\varphi_j(x_0 + \lambda x) - \varphi(x_0))$ , as $j \to \infty$ and $\lambda \to 0$ . Note that $$\mathrm{D}\psi_{j,\lambda}(x) = \mathrm{D}\varphi_j(x_0 + \lambda x).$$ Since $\lambda \to 0$ , we can assume that the maps $\psi_{j,\lambda}$ are defined on any bounded open set $\Omega$ for which $\mathcal{L}^n(\partial\Omega) = 0$ . Also, the basic invariance properties of $\mathcal{M}_{qc}^p$ follow quickly: For t > 0, let $$\langle \nu_t, \mathbf{E} \rangle \equiv \langle \nu, \mathbf{E}(t \cdot) \rangle = \int_{\mathbb{D}^{n \times n}} \mathbf{E}(tA) \, \mathrm{d}\nu(A),$$ and similarly, if m = n and $Q, R \in SO(n)$ , we define $$\langle \nu_{Q,R}, \mathbf{E} \rangle \equiv \int_{\mathbb{R}^{n \times n}} \mathbf{E}(QAR) \, \mathrm{d}\nu(A).$$

Lemma 2.16. Fix $p \in [1, \infty]$ . - <span id="page-16-2"></span>(1) For any t > 0, the map $\nu \mapsto \nu_t$ , maps $\mathcal{M}_{qc}^p$ bijectively onto itself. - (2) Similarly, if m = n and $Q, R \in SO(n)$ , the map $\nu \mapsto \nu_{Q,R}$ , maps $\mathcal{M}_{qc}^p$ bijectively onto itself.

Theorem 2.17. Fix $p \in [1, \infty)$ and let $\nu$ be a Borel probability measure on $\mathbb{R}^{m \times n}$ . Here and throughout we denote its center of mass by $$\langle \nu, \mathrm{Id} \rangle \equiv \int_{\mathbb{R}^{m \times n}} A \, \mathrm{d}\nu(A),$$ so that in particular $\langle \nu, \operatorname{Id} \rangle \in \mathbb{R}^{m \times n}$ . <span id="page-16-3"></span>Then we have $\nu \in \mathcal{M}_{qc}^p$ if and only if $\nu$ satisfies the following two conditions: (1) For all quasiconvex $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \mathbb{R}$ with $\sup_{A \in \mathbb{R}^{m \times n}} \frac{|\mathbf{E}(A)|}{1 + |A|^p} < \infty$ , (no condition if $p = \infty$ ), the Jensen inequality $$\mathbf{E}(\langle \nu, \mathrm{Id} \rangle) \leqslant \int_{\mathbb{R}^{m \times n}} \mathbf{E}(A) \, \mathrm{d}\nu(A)$$ holds; <span id="page-16-4"></span>(2) $\nu$ has finite p-th moment, in the sense (2.5). In this connection we also recall that $$\mathcal{M}_{qc}^p = \mathcal{M}_{qc}^1 \cap \{\nu : \nu \text{ has a finite } p\text{-th moment } \}$$ for $p \in (1, \infty]$ , see [64, Corollary 1.8] for $p < \infty$ and [89] for $p = \infty$ . <span id="page-16-5"></span>Remark 2.18. Note also that if $p \in (1, \infty)$ , $\nu \in \mathscr{M}_{qc}^p(\mathbb{R}^{m \times n})$ and $\varphi$ is the weak limit of a sequence $(\varphi_i)$ generating $\nu$ , then by (2.6) $$D\varphi(x) \equiv \langle \nu, \mathrm{Id} \rangle$$ and therefore $\varphi$ is affine. As a last remark, Theorem 2.17 also motivates the definition of the class of measures which satisfy Jensen's inequality with respect to rank-one convex instead of quasiconvex functionals.

Lemma 2.26. Let $\mathbf{E} \colon \mathbb{R}^{2 \times 2} \to \overline{\mathbb{R}}$ be a rank-one convex functional such that $\{t \text{ Id} : t > 0\}$ is in the interior of $\operatorname{dom}(\mathbf{E})$ . Then, for a radial map $\phi \colon \mathbb{D} \to \mathbb{C}$ such that $\operatorname{D} \phi \in \operatorname{int}(\operatorname{dom}(\mathbf{E}))$ a.e., we have $$\mathbf{E}(\mathrm{Id}) \leqslant \int_{\mathbb{D}}^{*} \mathbf{E}(\mathrm{D}\phi(z)) \, \mathrm{d}m(z).$$

Theorem 4.1. Let $$\frac{2K}{K+1} < s < \frac{2K}{K-1}$$ and let $\nu \in \mathcal{M}_{qc}^s(Q_2(K))$ be such that (4.1) $\langle \nu, \operatorname{Id} \rangle = A, \qquad Az = z + a\bar{z}.$ Then there is a sequence of K-quasiconformal principal maps which generates in $\mathbb{D}$ the Young measure $\nu$ . Here recall that any matrix in $Q_2(K)$ is a scalar multiple of $Az = z + a\bar{z}$ where $|a| \leq \frac{K-1}{K+1}$ . Thus the choice (4.1) is merely a normalisation. Moreover, recall that if $f \in W^{1,2}_{loc}(\mathbb{C})$ is the weak limit of the generating sequence given by Theorem 4.1, then f is K-quasiconformal and principal; this follows from the compactness of the family $\mathscr{F}_k$ in (3.12). In addition, (2.6) shows that $$Df(x) = \langle \nu, Id \rangle$$ a.e. $x \in \mathbb{D}$ . Thus also the center of mass $\langle \nu, \mathrm{Id} \rangle$ is a K-quasiconformal matrix and for some $|a| \leqslant \frac{K-1}{K+1}$ , $$f(z) = \begin{cases} z + a\bar{z} & \text{if } |z| \leq 1, \\ z + \frac{a}{z} & \text{if } |z| > 1. \end{cases}$$ For the proof of Theorem 4.1 we borrow an auxiliary result from [3, Lemma 4.1]:

Lemma 4.2. Suppose $\frac{2K}{K+1} < s < \frac{2K}{K-1}$ and $\nu \in \mathcal{M}^s_{qc}(Q_2(K))$ is generated by a sequence $(\phi_j)$ such that $(|D\phi_j|^s)$ is equiintegrable in $\mathbb{D}$ . Then there are measurable functions $\mu_j : \mathbb{D} \to \mathbb{C}$ , with $\|\mu_j\|_{\infty} \leqslant \frac{K-1}{K+1}$ , such that $$\lim_{j \to \infty} \|\partial_{\bar{z}} \phi_j - \mu_j \partial_z \phi_j\|_{L^s(\mathbb{D})} = 0.$$ Proof of Theorem 4.1. Let $(\phi_j) \subset W^{1,s}(\mathbb{D},\mathbb{C})$ be a sequence generating $\nu$ , given by Theorem 2.9, for which $(|D\phi_j|^s)$ is equiintegrable. We then set $$\eta_i \equiv (\partial_{\bar{z}}\phi_i - \mu_i\partial_z\phi_i)\chi_{\mathbb{D}} \in L^s(\mathbb{C}),$$ where the $\mu_j$ are given by Lemma 4.2. As in (3.6), via the Cauchy transform we find global solutions $\omega_i$ to $$\partial_{\bar{z}}\omega_i - \mu_i \partial_z \omega_i = \eta_i$$ with $|D\omega_i| \in L^s(\mathbb{C})$ , simply by letting $$\partial_{\bar{z}}\omega_j = (I - \mu_j \mathbf{S})^{-1} \eta_j, \quad \omega_j = \mathbf{C}(\partial_{\bar{z}}\omega_j).$$ In particular, $\|D\omega_j\|_{L^s(\mathbb{C})} \leq C_s(K)\|\eta_j\|_{L^s(\mathbb{C})} \to 0$ as $j \to \infty$ . By Theorem 2.9, also the sequence of maps $$\psi_j \equiv \phi_j - \omega_j \in \mathbf{W}^{1,s}(\mathbb{D})$$ generates the given Young measure $\nu$ , since $\|\mathbf{D}\psi_j - \mathbf{D}\phi_j\|_{\mathbf{L}^s(\mathbb{D})} \to 0$ . Further, the maps $\psi_j$ are all K-quasiregular, $\partial_{\bar{z}}\psi_j - \mu_j\partial_z\psi_j = 0$ in $\mathbb{D}$ , but they need not be homeomorphisms. On the other hand, in Section 3 we saw that there is a principal solution $f_j \in W^{1,p}_{loc}(\mathbb{C},\mathbb{C})$ to the Beltrami equation $$\partial_{\bar{z}} f_j - \chi_{\mathbb{D}} \mu_j \partial_z f_j = 0.$$ Then by Stoilow's factorization [4, Theorem 5.5.1] we have $$\psi_j = h_j \circ f_j,$$ where the maps $h_j$ are holomorphic in $f_j(\mathbb{D})$ . We can next estimate as in [3, (4.11)]. Namely, as $J_{f_j}(z) \leq |Df_j(z)|^2$ , by the change of variables formula and (3.10) we have $$(4.2) \qquad \int_{f_{j}(\mathbb{D})} |h'_{j}(w)| \, \mathrm{d}m(w) = \int_{\mathbb{D}} |h'_{j}(f_{j}(z))| J_{f_{j}}(z) \, \mathrm{d}m(z)$$ $$\leq \left( \int_{\mathbb{D}} |h'_{j}(f_{j}(z))|^{s} |\mathrm{D}f_{j}(z)|^{s} \right)^{1/s} \left( \int_{\mathbb{D}} |\mathrm{D}f_{j}(z)|^{t} \right)^{1/t}$$ $$\leq C_{t}(K) \left( \int_{\mathbb{D}} |\mathrm{D}\psi_{j}(z)|^{s} \right)^{1/s},$$ where t, the Hölder conjugate of s, also satisfies (3.11). Thus the norms $||h'_j||_{L^1(f_j(\mathbb{D}))}$ of the derivatives of the holomorphic factors are uniformly bounded. Choosing a point $x_0 \in \mathbb{D}$ and adding a constant to elements of the generating sequence $\phi_j$ , we can assume that $\psi_j(x_0) = x_0$ , i.e. that $h_j$ takes $f_j(x_0)$ to $x_0$ . Choosing then a subsequence such that $f_j(x_0)$ converge, we see that the holomorphic functions $h_j: f_j(\mathbb{D}) \to \mathbb{C}$ form a normal family. All in all, taking subsequences we can assume that $f_j \to f \in \mathscr{F}_k$ uniformly on $\mathbb{C}$ and $h_j \to h$ locally uniformly on $f(\mathbb{D})$ , where h is analytic on $f(\mathbb{D})$ . Further, we have the weak convergence (4.3) $$\mathrm{D}\psi_j = h'_i(f_i(z))\mathrm{D}f_i \to \mathrm{D}(h \circ f) \quad \text{in } \mathrm{L}^s(\mathbb{D}).$$ Since $(\psi_j)$ generate the given Young measure $\nu$ , we see from (4.1) and Remark 2.18 that $\mathrm{D}(h\circ f)(z)=A$ for a.e. $z\in\mathbb{D}$ . In particular, this means that f has the complex dilatation $\mu=a\chi_{\mathbb{D}}$ . But given any compactly supported dilation with $\|\mu\|_{\infty}<1$ , this time $\mu=a\chi_{\mathbb{D}}$ , there is a unique $W_{\mathrm{loc}}^{1,2}$ -principal mapping with this dilatation [4, Theorem 5.1.2]. So f must be equal to $$f(z) = \begin{cases} A(z) = z + a\overline{z}, & \text{if } |z| \leq 1, \\ z + \frac{a}{z}, & \text{if } |z| \geq 1. \end{cases}$$ It follows that Dh = Id in $f(\mathbb{D})$ and so we see that $h'_j(f_j(z)) \to 1$ locally uniformly on $\mathbb{D}$ as $j \to \infty$ . Thus with (3.10), for any r < 1, $$\|\mathrm{D}\psi_j - \mathrm{D}f_j\|_{\mathrm{L}^s(\mathbb{D}(0,r))} = \|(h_j' \circ f_j - 1)\mathrm{D}f_j\|_{\mathrm{L}^s(\mathbb{D}(0,r))} \to 0.$$ Therefore, again by Theorem 2.9, $(f_j)$ and $(\psi_j)$ generate the same (homogeneous) Young measure $\nu$ in every disc compactly contained in $\mathbb{D}$ . Since the sequence $(|\mathcal{D}f_j|^s)$ is equiintegrable over $\mathbb{D}$ , and as $\mathscr{L}^2(\mathbb{S}^1) = 0$ , this finally shows that the sequence of principal maps $(f_j)$ generates $\nu$ in $\mathbb{D}$ . Combining Theorem 4.1 with (3.10) we obtain the following consequence, cf. [3, Corollary 1.6]: <span id="page-26-1"></span>Corollary 4.3. If $$\nu \in \mathcal{M}_{qc}^s(Q_2(K))$$ for some $\frac{2K}{K+1} < s$ , then $\nu \in \mathcal{M}_{qc}^p(Q_2(K))$ for all $p < \frac{2K}{K+1}$ . We conclude this section with a version of Theorem 4.1 for gradient Young measures which are generated by sequences with integrable distortion.

Theorem 4.4. Let $\nu \in \mathcal{M}_{qc}^2(\mathbb{R}_+^{2\times 2})$ be such that, for some |a| < 1, (4.4) $$\langle \nu, \operatorname{Id} \rangle = A, \qquad A(z) = z + a\bar{z}.$$ Assume that $\nu$ is generated by a bounded sequence $\{\psi_j\} \subset W^{1,2}(\mathbb{D})$ of homeomorphisms such that for some q > 1, <span id="page-26-0"></span> $$||K_{\psi_j}||_{\mathbf{L}^q(\mathbb{D})} \leqslant C.$$ Then there is a sequence of maps $f_i : \mathbb{C} \to \mathbb{C}$ such that: - (1) $f_i$ are principal maps; - (2) For each r < 1, the sequence $(f_j|_{\mathbb{D}(0,r)}) \subset W^{1,2}(\mathbb{D}(0,r))$ is bounded and generates $\nu$ ; - (3) $\psi_j = h_j \circ f_j$ for some conformal maps $h_j : f_j(\mathbb{D}) \to \psi_j(\mathbb{D})$ . The proof strategy is similar to that of Theorem 4.1. However, since we are in a setting where the Beltrami equations are degenerate elliptic the argument is more subtle. For instance, one does not know if the sequence $(f_j)$ is bounded in $W^{1,2}(\mathbb{D})$ , which causes technical problems. On the other hand, for the applications we have in mind it suffices to consider generating sequences which consist of homeomorphisms. The argument relies crucially on the Stoilow factorization for maps with integrable distortion, due to Iwaniec and Šverák [53].

Theorem 5.1. [5, Theorem 3.5] Suppose $f: \mathbb{C} \to \mathbb{C}$ is a principal solution to the Beltrami equation <span id="page-31-4"></span> $$(5.4) f_{\overline{z}} = \mu f_z, \quad |\mu(z)| \leqslant k \chi_{\mathbb{D}}(z), \quad 0 \leqslant k < 1,$$ Then for all exponents $2 \le p \le 1 + 1/k$ , we have <span id="page-31-2"></span>(5.5) $$f_{\mathbb{D}} \left( 1 - \frac{p|\mu(z)|}{1 + |\mu(z)|} \right) (|f_z(z)| + |f_{\overline{z}}(z)|)^p \, \mathrm{d}m(z) \leqslant 1.$$ Finally, comparing now (5.3) and (5.5), we may rewrite Theorem 5.1 as follows

Theorem 5.2. [5, Theorem 1.3] Suppose $f: \mathbb{C} \to \mathbb{C}$ is a principal solution to the Beltrami equation (5.4). Then for all $p \in [2, 1 + 1/k]$ we have <span id="page-31-5"></span>(5.6) $$\mathbf{B}_p(\mathrm{Id}) \leqslant \int_{\mathbb{D}} \mathbf{B}_p(\mathrm{D}f(z)) \, \mathrm{d}m(z).$$ In particular, if a quasiconformal map f on $\mathbb{D}$ has identity boundary values, it extends trivially to a principal map of $\mathbb{C}$ . Thus the above result shows that the Burkholder functional $\mathbf{B}_p$ is quasiconvex at identity when tested with K-quasiconformal maps, under the condition that $2 \leq p \leq \frac{2K}{K-1}$ . Furthermore, the equality in (5.6) occurs for a large class of radial mappings, see for instance Subsection 5.2 below. Remark 5.3. Note that holomorphic deformations of solutions f to (5.4), such as $\lambda \mapsto f_{\mu_{\lambda}}(z)$ above, in general change the boundary values of the mapping, even if the original map f has identity boundary values on $\partial \mathbb{D}$ . For this reason, in particular, the principal mappings and their integral bounds are indispensable for the quasiconvexity estimates (5.6). On the other hand, to prove quasiconvexity bounds such as (5.6) for quasiconformal maps on $\mathbb{D}$ with linear boundary values $A \neq \mathrm{Id}$ , different methods appear necessary. Here we will make extensive use of the gradient Young measures discussed in Sections 2 and 4. However, even for these Theorem 5.2 is required as their basis and the starting point for the estimates they provide, see e.g. Proposition 6.1. Remark 5.4. Note that, in general, at the endpoint exponent p=1+1/k we have $\mathrm{D}f \notin \mathrm{L}^p_{\mathrm{loc}}$ for the solutions (5.4): simple examples are obtained, for instance, by considering radial stretchings. Nonetheless, we always have $\mathrm{D}f \in \mathrm{weak}\text{-}\mathrm{L}^p_{\mathrm{loc}}$ [4, Theorem 13.2.1]. A surprising feature of Theorem 5.2 is that, even if $\mathbf{B}_p$ has p-growth, we are able to test the quasiconvexity inequality (5.6) with maps which are not in $\mathrm{W}^{1,p}_{\mathrm{loc}}$ , and in particular $\mathbf{B}_p$ is integrable along such maps. One can regarded this as an extension to the planar quasiconformal setting of the results of [52], where it is shown that $\det \mathrm{D}f \in L^1_{\mathrm{loc}}$ if $f \in \mathrm{weak}\text{-}W^{1,n}_{\mathrm{loc}}$ is orientation-preserving. <span id="page-32-0"></span>5.2. Burkholder functional and radial mappings. In Lemma 2.26 we saw that rank-one convex functionals are quasiconvex along radial maps. The Burkholder functional is special, as it is quasiaffine on a large class of radial stretchings:

Lemma 5.5. Let $\phi(z) = \rho(r) \frac{z}{r}$ be a radial stretching satisfying the condition $$|\dot{\rho}(r)| \leqslant \frac{\rho(r)}{r}.$$ The functional $\mathbf{B}_p$ is quasiaffine along such radial stretchings: $$-1 = \mathbf{B}_p(\mathrm{Id}) = \int_{\mathbb{D}} \mathbf{B}_p(\mathrm{D}\phi) \,\mathrm{d}m(z).$$

Theorem 5.6. Let $\mathbf{E} \colon \mathbb{R}^{2 \times 2} \to \overline{\mathbb{R}}$ be an functional such that: - (1) for some $K \ge 1$ , $Q_2(K) \subset \operatorname{int}(\operatorname{dom}(\mathbf{E}))$ ; - (2) E is rank-one convex and $\mathbf{E}(\mathrm{Id}) = -1$ ; - <span id="page-33-2"></span>(3) E is positively p-homogeneous, for some $p \ge 2$ ; - (4) E is isotropic, that is $\mathbf{E}(QAR) = \mathbf{E}(A)$ for all $A \in \mathbb{R}^{2 \times 2}$ and all $Q, R \in \mathrm{SO}(2)$ . Then $\mathbf{E}(A) \geqslant \mathbf{B}_p(A)$ for all $A \in Q_2(K)$ .

Proposition 6.1. Let K > 1 and fix $p \in (2, \frac{2K}{K-1})$ . Then the functional $\mathbf{B}_{K,p} \colon \mathbb{R}^{2 \times 2} \to \mathbb{R} \cup \{+\infty\}$ is closed $W^{1,p}$ -quasiconvex at Id.

Proposition 6.2. Let K > 1. Then for each $2 , the local Burkholder functional <math>\mathbf{B}_{K,p} \colon \mathbb{R}^{2 \times 2} \to \mathbb{R} \cup \{+\infty\}$ is closed $\mathbf{W}^{1,p}$ -quasiconvex.

Theorem 7.1. Let f be a K-quasiconformal principal map, conformal outside $\mathbb{D}$ with expansion <span id="page-37-1"></span>(7.1) $$f(z) = z + \frac{b_1}{z} + \sum_{j=2}^{\infty} \frac{b_j}{z^j} \equiv z + \frac{b_1}{z} + \phi(z), \quad |z| > 1.$$ Then with the linear asymptotics $A_f(z) \equiv z + b_1 \bar{z}$ , we have <span id="page-37-4"></span>(7.2) $$\int_{\mathbb{D}} \left( \mathbf{B}_p(\mathrm{D}f) - \mathbf{B}_p(A_f) \right) \mathrm{d}m(z) \ge -\frac{p}{2} \frac{\mathbf{B}_p(A_f)}{\det(A_f)} \int_{\mathbb{C}\setminus\mathbb{D}} |\phi'(z)|^2 \, \mathrm{d}m(z),$$ provided that $2 \leqslant p \leqslant \frac{2K}{K-1}$ . We begin with a rather general lemma which improves the asymptotics of the map in a controlled manner, while keeping the map unchanged in the disk.

Lemma 7.2. Suppose $f \in W^{1,1}_{loc}(\mathbb{C})$ is a principal map with expansion (7.1). Then the map $\tilde{f} : \mathbb{C} \to \mathbb{C}$ defined by $$\tilde{f} \equiv \begin{cases} f & \text{in } \mathbb{D}, \\ h \circ A_f & \text{in } \mathbb{C} \backslash \mathbb{D}, \end{cases}$$ is a $W^{1,1}_{loc}$ -homeomorphism, where $h: A_f(\mathbb{C}\backslash \mathbb{D}) \to f(\mathbb{C}\backslash \mathbb{D})$ is a conformal map defined by $$h \equiv f \circ R^{-1}, \qquad R(z) \equiv z + b_1/z.$$ Moreover, $$h(z) = z + \mathcal{O}(z^{-2})$$ as $|z| \to \infty$ . Finally, if f is K-quasiconformal for some $K \ge 1$ , then so is $\tilde{f}$ .

Lemma 7.3. Let f be as in Theorem 7.1 and define $\tilde{f}$ as in Lemma 7.2. Then $$\int_{\mathbb{C}} \left( \mathbf{B}_p(\mathrm{D}\tilde{f}) - \mathbf{B}_p(A_f) \right) \mathrm{d}m(z) \ge 0.$$

Lemma 7.4. Let $G_p(w) = |1 + w|^p - 1$ . Then for any 2 , $$G_p(w) \geqslant p \operatorname{Re} w + \frac{p}{2}|w|^2, \qquad w \in \mathbb{C},$$ where the coefficient in front of $|w|^2$ is the largest possible.

Proposition 9.2. Let $\mathbf{E} \colon \mathbb{R}_{+}^{2 \times 2} \to \mathbb{R}$ be a $W^{1,2}$ -quasiconvex functional, in the sense that $\mathbf{E}$ satisfies (1.4) for any $A \in \mathbb{R}_{+}^{2 \times 2}$ and any homeomorphism $f \in A + W_0^{1,2}(\Omega)$ . Then the Shield transformation $\hat{\mathbf{E}} \colon \mathbb{R}_{+}^{2 \times 2} \to \mathbb{R}$ satisfies $$\hat{\mathbf{E}}(A) \leqslant \oint_{\Omega} \hat{\mathbf{E}}(\mathrm{D}f(z)) \,\mathrm{d}m(z)$$ for all $A \in \mathbb{R}^{2 \times 2}_+$ and all homeomorphisms $f \in A + W_0^{1,1}(\Omega)$ with $K_f \in L^1(\Omega)$ .

Proposition 9.7. Let $A \in \mathbb{R}_+^{2 \times 2}$ and let $f \in A + W_0^{1,2}(\Omega)$ be monotone with $\mathcal{W}(\mathrm{D}f) \in \mathrm{L}^1(\mathbb{D})$ . Then $$\mathcal{W}(A) \leqslant \int_{\mathbb{D}} \mathcal{W}(\mathrm{D}f(z)) \, \mathrm{d}m(z).$$

Theorem 10.2. Let $f \in W^{1,1}_{loc}(\mathbb{C})$ be a homeomorphism with $K_f \in L^1(\mathbb{D})$ and suppose that f is conformal outside $\mathbb{D}$ with expansion (10.3) $$f(z) = z + \frac{b_1}{z} + \sum_{j=2}^{\infty} \frac{b_j}{z^j}, \qquad |z| > 1.$$ Letting $A_f \in \mathbb{R}^{2 \times 2}$ be given by $A_f(z) = z + b_1 \overline{z}$ , we have <span id="page-50-0"></span> $$\mathcal{W}(A_f) \leqslant \int_{\mathbb{D}} \mathcal{W}(\mathrm{D}f(z)) \,\mathrm{d}m(z).$$ Here recall from (3.4) that for any principal homeomorphism $\det(A_f) > 0$ . In fact, under the normalisation (10.3) we have $1 \leq \mathcal{W}(A_f) < +\infty$ . The proof of the above Theorem follows the same broad strategy as for the Burkholder area inequality in Theorem 7.1, but some of the details are quite different. As before, the first step is to establish a quasiconvexity inequality over the full space, as in Lemma 7.3.

Lemma 10.3. Given f as in Theorem 10.2, define the auxiliary function $\tilde{f}$ as in Lemma 7.2. Then we have $$0 \leqslant \int_{\mathbb{C}} \left( \mathcal{W}(D\tilde{f}(z)) - \mathcal{W}(A_f) \right) dm(z).$$

Lemma 10.4. Let $\mathscr{E} \subset \mathbb{C}$ be an ellipse and suppose $u \in L^1(\mathbb{C} \backslash \mathscr{E})$ is holomorphic in $\mathbb{C} \backslash \mathscr{E}$ . Then $$\int_{\mathbb{C}\setminus\mathscr{E}} u(z) \, \mathrm{d}m(z) = 0.$$

Theorem 11.1. Let $\mathbf{E} \colon \mathbb{R}_{+}^{2 \times 2} \to \mathbb{R}$ be a rank-one convex functional of the form (11.1), where $\mathbf{G} \colon (0, \infty) \to \mathbb{R}$ and $\mathbf{H} \colon [1, +\infty) \to \mathbb{R}$ is convex. Then there is a polyconvex functional $\mathbf{F} \colon \mathbb{R}_{+}^{2 \times 2} \to \mathbb{R}$ and a constant $c \geqslant 0$ such that $$\mathbf{E} = \mathbf{F} + c \mathcal{W}$$ . We recall that a functional $\mathbf{F} \colon \mathbb{R}^{2 \times 2} \to \overline{\mathbb{R}}$ is said to be polyconvex if there is a convex function $\tilde{\mathbf{F}} \colon \mathbb{R}^5 \to \overline{\mathbb{R}}$ such that $\mathbf{F}(A) = \tilde{\mathbf{F}}(A, \det(A))$ , see also [26] for further details. Since the determinant is a null Lagrangian, Jensen's inequality easily implies that polyconvex functionals are quasiconvex. Theorem 11.1 was proved implicitly in [86] and in this section we give a short, direct proof. Combining Corollary 9.5 and Theorem 11.1, we obtain:

Corollary 11.2. Any functional $\mathbf{E} \colon \mathbb{R}^{2 \times 2} \to \mathbb{R}$ as in Theorem 11.1 is quasiconvex: if $A \in \mathbb{R}^{2 \times 2}_+$ and if $f \in A + W_0^{1,1}(\Omega, \mathbb{R}^2)$ is a homeomorphism such that $K_f \in L^1(\Omega)$ , then $$\mathbf{E}(A) \leqslant \int_{\Omega}^{*} \mathbf{E}(\mathrm{D}f(z)) \, \mathrm{d}m(z).$$ The proof of Theorem 11.1 relies on the classical Baker–Ericksen inequality. Given $A \in \mathbb{R}^{2\times 2}$ , we write $\lambda(A) \equiv (\lambda_1(A), \lambda_2(A))$ for the vector of singular values of A, which is are the eigenvalues of the positive-definite matrix $\sqrt{A^T A}$ . The Baker–Ericksen inequality read as follows:

Lemma 11.3. Let $\mathbf{E} \colon \mathbb{R}_+^{2 \times 2} \to \mathbb{R}$ be an isotropic rank-one convex functional: thus there is a symmetric function $\Phi \colon (0, \infty)^2 \to \mathbb{R}$ such that $$\mathbf{E}(A) = \Phi(\lambda_1(A), \lambda_2(A)).$$ If $\Phi$ is $C^1$ and $\lambda_1 \neq \lambda_2$ then $$\frac{\lambda_1 \partial_1 \Phi(\lambda) - \lambda_2 \partial_2 \Phi(\lambda)}{\lambda_1 - \lambda_2} \geqslant 0$$ for all $\lambda = (\lambda_1, \lambda_2) \in \mathbb{R}^2$ such that $\lambda_1, \lambda_2 > 0$ . Lemma 11.3 is well-known and the interested reader can find a short proof for instance in [37, Proposition 3.2]. We will also require the following result:

Lemma 11.4. Let $\mathbf{E} \colon \mathbb{R}_{+}^{2 \times 2} \to \mathbb{R}$ be a rank-one convex functional with the representation (11.1). - <span id="page-54-2"></span>(1) If $\mathbf{H} = 0$ then $\mathbf{G} : (0, \infty) \to \mathbb{R}$ is convex and $\mathbf{E}$ is polyconvex. - (2) If $\mathbf{G} = 0$ then $\mathbf{H} \colon [1, \infty) \to \mathbb{R}$ is convex and non-decreasing, and $\mathbf{E}$ is polyconvex. The first claim in Lemma 11.4 is classical, see e.g. [26, Theorem 5.46]. The result in Lemma 11.4(2) is not difficult to obtain, see e.g. [68]. Here we present a short proof for the sake of completeness. The crucial point is the easily-checked fact that $A \mapsto K_A$ is a polyconvex functional. We refer the reader to [49] for a systematic study of polyconvexity properties of distortion functions in higher dimensions. Proof of Lemma 11.4(2). Since E is rank-one convex, for $\lambda_1 \ge 1$ we have that $\lambda_1 \mapsto \mathbf{E}(\operatorname{diag}(\lambda_1, 1)) = \mathbf{H}(\lambda_1)$ is convex. To prove the monotonicity, fix $1 \le s < t$ and let $\theta \in (0, 1)$ be such that $\theta t + (1 - \theta)t^{-1} = s$ . Thus, by rank-one convexity, $$\mathbf{H}(s) = \mathbf{E}(\operatorname{diag}(s, 1))$$ $$\leq \theta \mathbf{E}(\operatorname{diag}(t, 1)) + (1 - \theta) \mathbf{E}(\operatorname{diag}(t^{-1}, 1))$$ $$= \theta \mathbf{H}(t) + (1 - \theta) \mathbf{H}(t) = \mathbf{H}(t)$$ and hence H has the claimed properties. Since H is non-decreasing and convex, and $K_A$ is polyconvex, it follows that $\mathbf{E}(A) = \mathbf{H}(K_A)$ is polyconvex as well. Proof of Theorem 11.1. Since polyconvexity and rank-one convexity are preserved under pointwise limits, there is no loss of generality in assuming that both $\mathbf{G} \colon (0,\infty) \to \mathbb{R}$ and $\mathbf{H} \colon (1,\infty) \to \mathbb{R}$ are smooth. Note, however, that we do not assume that $\mathbf{H}$ is smooth up to t=1. We consider arbitrary x > y > 0. Since E is rank-one convex, a simple calculation yields $$0 \le x^2 \partial_{xx} \mathbf{E}(\operatorname{diag}(x,y)) = (xy)^2 \mathbf{G}''(xy) + \left(\frac{x}{y}\right)^2 \mathbf{H}''(x/y).$$ By changing variables t = xy, s = x/y, we deduce the inequality $$\inf_{t>0} t^2 \mathbf{G}''(t) + \inf_{s>1} s^2 \mathbf{H}''(s) \equiv G_0 + H_0 \geqslant 0.$$ Similarly, with $\Phi(x,y) = \mathbf{E}(\operatorname{diag}(x,y))$ as in Lemma 11.3, we calculate $$\frac{x\partial_x \Phi(x,y) - y\partial_y \Phi(x,y)}{x - y} = \frac{2x}{y} \frac{\mathbf{H}'(x/y)}{x - y}$$ and thus the Baker-Ericksen inequality implies the condition $$\mathbf{H}'(t) \geqslant 0$$ for $t > 1$ . By assumption $H_0 \ge 0$ . Suppose that $G_0 \ge 0$ as well; in this case, both H and G are convex and Lemma 11.4 shows that E, being the sum of two polyconvex functionals, is itself polyconvex, so we may take c = 0. Hence we now assume that $G_0 \le 0$ and we take $c \equiv -G_0$ . We claim that $F \equiv \mathbf{E} - c \mathcal{W}$ is polyconvex. In fact, F can be written as $$\mathbf{F}(A) = \left[\mathbf{G}(\det A) - c\log(\det A)\right] + \left[\mathbf{H}(K_A) - c(K_A - \log K_A)\right]$$ $$\equiv \widetilde{\mathbf{G}}(\det A) + \widetilde{\mathbf{H}}(K_A)$$ and we claim that both terms are polyconvex functionals. This will follow from Lemma 11.4. That $\tilde{\mathbf{G}}$ is convex follows from the definition of c: $$\widetilde{\mathbf{G}}''(t) = \mathbf{G}''(t) + c/t^2 \geqslant 0.$$ Again from the definition of c, we have $$\widetilde{\mathbf{H}}''(t) = \mathbf{H}''(t) - c/t^2 \geqslant (H_0 - c)/t^2 \geqslant 0,$$ so $\widetilde{\mathbf{H}}$ is convex. Suppose now that $\widetilde{\mathbf{H}}$ is not non-decreasing, so in particular there is $t_0 > 1$ such that $\widetilde{\mathbf{H}}'(t_0) < 0$ . For t > 1, since $\mathbf{H}'(t) \ge 0$ , $$\widetilde{\mathbf{H}}'(t) = \mathbf{H}'(t) + c(1 - 1/t) \ge c(1 - 1/t).$$ The right-hand side vanishes in the limit $t \to 1$ ; so, by choosing t sufficiently close to 1, we may suppose that $t < t_0$ and that $\widetilde{\mathbf{H}}'(t) \geqslant \widetilde{\mathbf{H}}'(t)/2 > \widetilde{\mathbf{H}}'(t_0)$ . This contradicts the fact that $\widetilde{\mathbf{H}}'$ is non-decreasing in $(1, \infty)$ , since $\widetilde{\mathbf{H}}$ is convex in the same interval.

Proposition 12.1. Let $K \ge 1$ and fix $2 \le p \le \frac{2K}{K-1}$ . Given a sequence $(f_j) \subset W^{1,p}(\Omega)$ of K-quasiregular maps such that $f_j \to f$ in $W^{1,p}(\Omega)$ and $(\mathbf{B}_p(\mathrm{D}f_j))$ is equiintegrable, we have $$\liminf_{j\to\infty} \int_{\Omega} \mathbf{B}_p(\mathrm{D}f_j(z)) \,\mathrm{d}m(z) \geqslant \int_{\Omega} \mathbf{B}_p(\mathrm{D}f(z)) \,\mathrm{d}m(z).$$

Proposition 12.2. Let $g \in W^{1,1}_{loc}(\mathbb{C})$ be a homeomorphism with $K_g \in L^q_{loc}$ for $q \geq 1$ . If $f \in g + W^{1,1}_0(\Omega)$ is a homeomorphism such that $K_f \in L^q(\Omega)$ , then $$\int_{\Omega} \log^q \left( e + \frac{1}{J_f(z)} \right) dm(z) \leqslant C(q, g, \Omega) \left( 1 + \int_{\Omega} K_f(z)^q dm(z) \right).$$

Lemma 12.3. Let $(f_j) \subset W^{1,1}(\Omega)$ be a sequence of homeomorphisms that $\sup_j \|K_{f_j}\|_{L^q(\Omega)} < \infty$ for some q > 1. Suppose that either - <span id="page-58-2"></span><span id="page-58-1"></span>(1) $f_j = g$ on $\partial \Omega$ for a homeomorphism $g \in W^{1,1}_{loc}(\mathbb{C})$ with $K_g \in L^q_{loc}$ , or - (2) $\Omega = \mathbb{D}$ and $f_j$ are principal maps. Then $(\mathcal{W}(Df_i))$ is equiintegrable.

Proposition 12.5. Let $\nu \in \mathscr{M}^2_{qc}(\mathbb{R}^{2\times 2}_+)$ be a gradient Young measure generated by a bounded sequence $(\psi_j) \subset W^{1,2}(\mathbb{D})$ of homeomorphisms such that $$||K_{\psi_i}||_{\mathbf{L}^q(\mathbb{D})} \leqslant C,$$ for some q > 1. Then $$\mathscr{W}(\langle \nu, \mathrm{Id} \rangle) \leqslant \int_{\mathbb{R}^{2 \times 2}} \mathscr{W}(A) \, \mathrm{d}\nu(A).$$

Corollary 12.6. Let $g \in W^{1,1}_{loc}(\mathbb{C})$ be a homeomorphism with $K_g \in L^q_{loc}(\mathbb{C})$ for some q > 1 and let $\mathbf{E} \colon \mathbb{R}_+^{2 \times 2} \to \mathbb{R}$ be a functional as in Theorem 11.1, $$\mathbf{E}(A) = \mathbf{G}(\det A) + \mathbf{H}(K_A),$$ which we assume to be rank-one convex. If $f_j \to f$ in $g + W_0^{1,2}(\Omega)$ and $\sup_j ||K_{f_j}||_{L^q(\Omega)} < \infty$ then $$\liminf_{j\to\infty} \int_{\Omega} \mathbf{E}(\mathrm{D}f_j(z)) \,\mathrm{d}m(z) \geqslant \int_{\Omega} \mathbf{E}(\mathrm{D}f(z)) \,\mathrm{d}m(z).$$ In conclusion, to promote the lower semicontinuity to the existence of minimizers requires now some form of coercivity, and this takes us to the following examples. <span id="page-60-2"></span>Corollary 12.7. Suppose $q > q_0 \ge 1$ with $p \ge 2$ , and let $$\mathbf{E}(A) \equiv \mathbf{G}(\det A) + \mathbf{H}(K_A) + |A|^p,$$ where we assume that for some C > 0, - (1) $A \mapsto \mathbf{G}(\det A) + \mathbf{H}(K_A)$ is rank-one convex; - (2) $|\mathbf{G}(t)| \leq C(1 + |\log(t)|^{q_0})$ ; - (3) H is convex and $\mathbf{H}(t) \geqslant t^q/C C$ . Then for any homeomorphism $g \in W^{1,1}_{loc}(\mathbb{C})$ with $K_g \in L^q_{loc}(\mathbb{C})$ there is a minimizer $f \in W^{1,p}(\Omega)$ of the problem $$\inf \left\{ \int_{\Omega} \mathbf{E}(\mathrm{D}h(z)) \, \mathrm{d}m(z) : h \in g + \mathrm{W}_{0}^{1,p}(\Omega) \right\}.$$ In addition, f is a homeomorphism such that $f^{-1} \in W^{1,2}(g(\Omega))$ .

Definition 1.1. Let $\mathbf{E} \colon \mathbb{R}_{+}^{n \times n} \to \mathbb{R}$ be locally bounded and Borel measurable. Then $\mathbf{E}$ is said to be quasiconvex at $A \in \mathbb{R}_{+}^{n \times n}$ if <span id="page-1-0"></span>(1.4) $$\mathbf{E}(A) \leqslant \oint_{\Omega} \mathbf{E}(\mathrm{D}f) \,\mathrm{d}x$$ whenever $\Omega \subset \mathbb{R}^n$ is a bounded domain and f a $C^1$ -diffeomorphism with f = A in $\partial \Omega$ ; we clarify that $f \in C^1(\overline{\Omega})$ and $f^{-1} \in C^1(\overline{A(\Omega)})$ are required. Remark 1.2. From the general point of view, it is important to note that in this paper we study functionals $\mathbf{E}$ defined on $\mathbb{R}^{2\times 2}_+$ and their quasiconvexity properties, so that the test functions are typically homeomorphisms. Some of these functionals satisfy (1.2), some not, and thus the applications of our results goes beyond non-linear elasticity. Another important remark is that for a given functional, inequality (1.4) typically holds for a much larger class of Sobolev functions. For instance, for all explicit functionals studied in this work we will prove (1.4) for homeomorphisms such that both $f \in W^{1,2}(\Omega, \Omega')$ and $f^{-1} \in W^{1,2}(\Omega', \Omega)$ , where $\Omega' = A(\Omega)$ . It is still a question of interest to study (1.4) when testing the inequality for general Lipschitz maps with a.e. $\det(\mathrm{D}f) > 0$ . Then, however, the question becomes a problem of approximation of Sobolev homeomorphisms, an active area of its own [44, 47]. See Section 9 for a discussion on this and for some positive partial results. Testing the quasiconvexity inequality with smooth approximations of a planar wave, i.e. a map which only takes two values, shows that that quasiconvexity implies convexity along rank-one directions, abbreviated as rank-one convexity. Whether conversely rank-one convexity implies quasiconvexity is a famous problem, going back to Morrey's work [69, 70]. The celebrated work of V. Šverák [85] gives a counterexample in dimensions $m \geq 3$ . His example consists of a map which is a superposition of three planar waves (see also [35] for a different example when $m \geq 8$ ). When m = 2 Šverák's example does not work [78] and indeed three waves cannot provide a counterexample [81], see also [38]. In fact, in two dimensions there are partial positive results [30, 40, 59, 72] which suggest that rank-one convexity might imply quasiconvexity. The purpose of this paper is to investigate whether for n=m=2 rankone convexity might imply quasiconvexity, at least for functionals with symmetries and additional structure, such as the Burkholder functionals. In the context of Definition 1.1 it is natural to set $\mathbf{E}(A) = +\infty$ when $\det(A) \leq 0$ . In this case quasiconvexity as defined above is no longer a sufficient condition for weak lower semicontinuity, cf. Section 2. Thus, in addition, one also needs an exploration towards the properties of the functionals on existence of minimizers. Here we realised that a stronger quasiconvexity inequality, one which for the Burkholder functional $\mathbf{B}_p(A)$ can be viewed as a version of the classical area formula, is then needed, see Theorem 1.6. Moreover, versions of such inequalities in the limit $p \to 2$ lead to new lower semicontinuity and existence theorems of interest in their own. 1.1. The Burkholder functional. The Burkholder functional is known to be rank-one convex, since the original work [20, 21]. For other approaches on this see [7, 84] or [45]. Our first theorem asserts that the Burkholder functional is quasiconvex when restricted to the set where it takes non-positive values. To interpret this setting, note that <span id="page-3-0"></span>(1.5) $$\mathbf{B}_p(A) \leqslant 0 \iff |A|^2 \leqslant \frac{p}{p-2} \det(A),$$ thus such a map $A \in \mathbb{R}^{2 \times 2}$ is K-quasiconformal with $K = \frac{p}{p-2}$ , equivalently $$(1.6) p = p_K \equiv \frac{2K}{K - 1}.$$ <span id="page-3-2"></span>In particular, if (1.5) holds then $A \in \mathbb{R}^{2\times 2}_+$ , unless A = 0.

Definition 2.2. A functional $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \overline{\mathbb{R}}$ is rank-one convex if, for all $A, X \in \mathbb{R}^{m \times n}$ with rank(X) = 1, the function $t \mapsto \mathbf{E}(A + tX)$ is convex. We emphasize that in the considered generality this notion of rank-one convexity is rather weak, see already Example 2.22 below. For the next result we need the notion of algebraic interior or core of a subset $S \subseteq \mathbb{R}^{m \times n}$ , namely $A \in \text{core}(S)$ provided for each $X \in \mathbb{R}^{m \times n}$ we can find $\delta > 0$ such that $A + tX \in S$ for all $t \in [0, \delta)$ .

Definition 2.6. Let $p \in [1, \infty]$ . A parametrized family of probability measures $\nu = (\nu_x)_{x \in \Omega}$ on $\mathbb{R}^{m \times n}$ is a W^1,p-gradient Young measure if there exists a weakly converging sequence $(\varphi_j)$ in W^1,p $(\Omega, \mathbb{R}^m)$ whose gradients generate $\nu$ , that is, (2.4) holds for all $\mathbf{E} \in C_0(\mathbb{R}^{m \times n})$ . Remark 2.7. We collect a few immediate consequences of the definition of $W^{1,p}$ gradient Young measure. First, as a limit of weak\-measurable functions also the family $(\nu_x)$ is weak\-measurable. This means that for every $\mathbf{E} \in C_0(\mathbb{R}^{m \times n})$ the function $$\Omega \ni x \mapsto \int_{\mathbb{R}^{m \times n}} \mathbf{E}(A) \, \mathrm{d}\nu_x(A)$$ is Lebesgue measurable. This property ensures that standard constructions involving $(\nu_x)$ result in measurable functions. Next, the fact that $(\nu_x)$ is generated by an L^p-bounded sequence implies that the measure satisfies the p-moment condition: For almost all $x \in \Omega$ , <span id="page-12-2"></span>(2.5) $$\int_{\Omega} \langle \nu_x, |\cdot|^p \rangle \, \mathrm{d}x < \infty \quad \text{when } p \in [1, \infty),$$ whereas when $p = \infty$ there exists R > 0 such that the supports $\sup(\nu_x) \subset B_R(0)$ for $\mathscr{L}^n$ almost all $x \in \Omega$ . The moment condition in particular means that for $\mathcal{L}^n$ almost all $x \in \Omega$ the probability measure $\nu_x$ has a centre of mass, and it is not difficult to see from (2.4) and Remark 2.10 that <span id="page-13-0"></span>(2.6) $$\langle \nu_x, \operatorname{Id} \rangle \equiv \int_{\mathbb{R}^{m \times n}} A \, d\nu_x(A) = \operatorname{D} \phi(x) \quad \text{for } a.e. \ x \in \Omega,$$ where $\phi$ is the weak limit of the generating sequence $(\phi_i)$ . Finally we note that since for any bounded open set $\Omega$ with $\mathcal{L}^n(\partial\Omega) = 0$ the inclusion $W^{1,s}(\Omega,\mathbb{R}^m) \subset W^{1,p}(\Omega,\mathbb{R}^m)$ holds if $p \leq s$ , every $W^{1,s}$ -gradient Young measure is a $W^{1,p}$ -gradient Young measure whenever $p \leq s$ . Any norm-bounded sequence of gradients admits a subsequence that generates a gradient Young measure [73, Theorem 3.1]:

Definition 2.12. A W^1,p-gradient Young measure $(\nu_x)_{x\in\Omega}$ is homogeneous if there is a probability measure $\nu$ on $\mathbb{R}^{m\times n}$ such that $\nu_x = \nu$ for a.e. $x \in \Omega$ . In this case we naturally identify $(\nu_x)_{x\in\Omega}$ with $\nu$ . We often denote by $\mathcal{M}_{\mathrm{qc}}^p$ the class of homogeneous W^1,p-gradient Young measures; the notation is motivated by Theorem 2.17 below. For a subset $\mathcal{U} \subset \mathbb{R}^{m \times n}$ we write $\mathcal{M}_{\mathrm{qc}}^p(\mathcal{U})$ for the set of measures in $\mathcal{M}_{\mathrm{qc}}^p$ whose support is contained in $\mathcal{U}$ . <span id="page-14-3"></span>Remark 2.13. There are many natural ways to construct homogeneous W^1,p-gradient Young measure. For instance, given $A \in \mathbb{R}^{m \times n}$ and $\phi \in W_0^{1,p}(\Omega, \mathbb{R}^m)$ we associate to them the measure $\nu_{A+\mathrm{D}\phi}$ defined by the rule <span id="page-14-2"></span>(2.9) $$\nu_{A+\mathrm{D}\phi}(\mathcal{S}) \equiv \frac{\mathscr{L}^n(\{x \in \Omega : A + \mathrm{D}\phi(x) \in \mathcal{S}\})}{\mathscr{L}^n(\Omega)},$$ where $\mathcal{S} \subset \mathbb{R}^{m \times n}$ is a Borel set. By inspection, $\nu_{A+\mathrm{D}\phi}$ is a Borel probability measure with a finite p-th moment $\langle \nu_{A+\mathrm{D}\phi}, |\cdot|^p \rangle < +\infty$ and centre of mass $\langle \nu_{A+\mathrm{D}\phi}, \mathrm{Id} \rangle = A$ . In particular, the measure $\nu_{A+\mathrm{D}\phi}$ describes the distribution of values of $A + \mathrm{D}\phi(x)$ in $\mathbb{R}^{m \times n}$ when x varies over $\Omega$ and we use the normalized volume $\mathscr{L}^n$ as weight. To represent $\nu_{A+\mathrm{D}\phi}$ as a homogeneous Young measure, since $\Omega$ is a bounded domain with $\mathcal{L}^n(\partial\Omega) = 0$ , we can realize this distribution on any other open bounded subset of $\mathbb{R}^n$ , and for later purposes we choose a realization on the open unit cube, $\mathbf{Q} \equiv \left(-\frac{1}{2}, \frac{1}{2}\right)^n$ . Indeed, a standard exhaustion argument allows us to write $\mathbf{Q}$ as a disjoint union of scaled and translated copies of $\Omega$ : $$\mathbf{Q} = N \cup \bigcup_{s \in \mathbb{N}} (x_s + r_s \Omega) \quad \text{(disjoint union!)}$$ where $\mathcal{L}^n(N) = 0$ . Next we import $\phi$ on $\mathbf{Q}$ by the definition $$\varphi(x) \equiv \begin{cases} r_s \phi\left(\frac{x - x_s}{r_s}\right) & \text{if } x \in x_s + r_s \Omega, s \in \mathbb{N} \\ 0 & \text{if } x \in N. \end{cases}$$ It is routine to check that hereby $\varphi \in W_0^{1,p}(\mathbf{Q}, \mathbb{R}^m)$ and that $\nu_{A+\mathrm{D}\varphi} = \nu_{A+\mathrm{D}\varphi}$ , if $\nu_{A+\mathrm{D}\varphi}$ is defined as in (2.9) with the obvious modifications. It is now easy to check that $\nu_{A+\mathrm{D}\varphi}$ is a homogeneous $W^{1,p}$ -gradient Young measure with centre of mass at A: Namely, extend $\varphi$ to $\mathbb{R}^n$ by $\mathbf{Q}$ -periodicity and define $u_j(x) \equiv Ax + \varphi(jx)/j$ , $x \in \Omega$ . Using the Riemann-Lebesgue lemma it follows that $u_j \to A$ in $W^{1,p}(\Omega, \mathbb{R}^m)$ and that $(\mathrm{D}u_j)$ generates the Young measure $(\nu_x)_{x\in\Omega}$ , where $\nu_x = \nu_{A+\mathrm{D}\varphi}$ for all $x \in \Omega$ . The examples provided in the next Subsection 2.3 show that $\mathcal{M}_{qc}^p$ , the set of homogeneous measures, contains probability measures that cannot be represented as $\nu_{A+D\phi}$ for any A, $\phi$ . On the other hand, using a variant of the above construction and a diagonalization argument it is not too difficult to see that $\mathcal{M}_{qc}^p$ can be defined as a suitable closure of the set $\{\nu_{A+D\phi}: \phi \in W_0^{1,p}(\Omega,\mathbb{R}^m), A \in \mathbb{R}^{m \times n}\}$ . However, we will not need this in the sequel. The usefulness of homogeneous gradient Young measures stems from the fact that a general gradient Young measure is essentially a collection of homogeneous gradient measures:

Definition 2.19. For $p \in [1, \infty]$ , $\mathscr{M}_{rc}^p$ is the set of those Borel probability measures $\nu$ in $\mathbb{R}^{m \times n}$ such that conditions (1) and (2) of Theorem 2.17 hold, with the word quasiconvex replaced by rank-one convex. The set $\mathcal{M}_{rc}^p$ agrees with the set of p-laminates, see e.g. [29, Definition 5.3], although we will not use laminates in the sequel. <span id="page-17-0"></span>2.3. Closed quasiconvexity and closed rank-one convexity. We are now ready to introduce what we believe are the correct notions of rank-one convexity and quasiconvexity for extended-real valued functionals.

Definition 2.20. A functional $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \overline{\mathbb{R}}$ is closed $W^{1,p}$ -quasiconvex (respectively closed p-rank-one convex) if for all $A \in \mathbb{R}^{m \times n}$ and all $\nu \in \mathcal{M}^p_{qc}$ (respectively all $\nu \in \mathcal{M}^p_{rc}$ ) with $\langle \nu, \operatorname{Id} \rangle = A$ we have (2.10) $$\mathbf{E}(A) = \mathbf{E}\left(\int_{\mathbb{R}^{m \times n}} \lambda \, \mathrm{d}\nu(\lambda)\right) \leqslant \int_{\mathbb{R}^{m \times n}}^{*} \mathbf{E}(\lambda) \, \mathrm{d}\nu(\lambda).$$ In particular we note that if $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \overline{\mathbb{R}}$ is closed $W^{1,p}$ -quasiconvex, then it is also closed $W^{1,s}$ -quasiconvex for every s > p. Furthermore, closed $W^{1,p}$ -quasiconvexity implies the standard $W^{1,p}$ -quasiconvexity. In order to verify this, let $\phi \in W_0^{1,p}(\Omega, \mathbb{R}^m)$ and $A \in \mathbb{R}^{m \times n}$ and recall Remark 2.13, where the probability measure $\nu_{A+\mathrm{D}\phi}$ was defined and shown to be a homogeneous $W^{1,p}$ -gradient Young measure with centre of mass A. Hence if $\mathbf{E}$ is a closed $W^{1,p}$ -quasiconvex functional, then $$\int_{\Omega}^{} \mathbf{E}(A + \mathrm{D}\phi(x)) \, \mathrm{d}x = \int_{\mathbb{R}^{m \times n}}^{} \mathbf{E} \, \mathrm{d}\nu_{A + \mathrm{D}\phi} \geqslant \mathbf{E}(A).$$ For other approaches see for instance [73, 77, 79]. However, the converse is not true, and it is instructive to exemplify this point here. Similarly to Example 2.5, the standard counterexamples are based on the following functional: Given a subset $\mathcal{U} \subset \mathbb{R}^{m \times n}$ we define its characteristic function, in the sense of convex analysis, by <span id="page-17-2"></span>(2.11) $$\chi_{\mathcal{U}}^{\infty}(A) \equiv \begin{cases} 0 & \text{if } A \in \mathcal{U}, \\ +\infty & \text{if } A \in \mathbb{R}^{2 \times 2} \backslash \mathcal{U}. \end{cases}$$ We then choose the set $\mathcal{U}$ to be a special finite set of matrices called a $T_N$ -configuration, see [33, Definition 2.1]. In particular such sets contain no rank-one connections, meaning that if $\mathcal{T}_N = \{A_1, \ldots, A_N\} \subset \mathbb{R}^{2\times 2}$ then <span id="page-17-1"></span>(2.12) $$\operatorname{rank}(A_i - A_j) > 1 \quad \text{ for all } i \neq j.$$ Kirchheim and Preiss [57, 58] provide the following optimal example concerning $T_N$ -configurations with $N \ge 5$ , see also [33]. Example 2.21 (five points). It is possible to construct a set $\mathcal{T}_5 = \{A_1, \dots, A_5\}$ of five matrices with the following property: <span id="page-18-1"></span>(2.13) There is $$A \notin \mathcal{T}_5$$ and $\varphi \in W_0^{1,\infty}(\Omega, \mathbb{R}^2)$ with $A + D\varphi \in \mathcal{T}_5$ a.e. Thus the functional $\chi_{T_5}^{\infty}$ is rank-one convex but not quasiconvex, it does not satisfy (2.3). Note that this in particular provides us with an example of a rank-one convex functional on symmetric 2-by-2 matrices which is not quasiconvex. It is an extended real-valued and lower semicontinuous functional and it is not clear if one could construct such an example which is continuous, let alone real-valued. The property (2.13) fails for all configurations of four matrices as proved by Chlebik and Kirchheim in [23]. This gives us the following example: <span id="page-18-0"></span>Example 2.22 (four points). Let $\mathcal{T}_4 = \{A_1, \dots, A_4\}$ be any set in $\mathbb{R}^{2\times 2}$ satisfying (2.12). Then (2.13) fails for the set. Thus the functional $\chi_{\mathcal{T}_4}^{\infty}$ is both quasiconvex and rank-one convex. Moreover, as observed in [65, Example 1.3] the corresponding variational integral $\int_{\Omega} \chi_{\mathcal{T}_4}^{\infty} (Du) dx$ is sequentially weakly lower semicontinuous on $W^{1,1}(\Omega, \mathbb{R}^2)$ . Nonetheless, if we assume the set $\mathcal{T}_4$ is a $\mathcal{T}_4$ configuration, then there exists nontrivial $\nu \in \mathscr{M}^{\infty}_{rc}(\mathcal{T}_4) \subseteq \mathscr{M}^{\infty}_{qc}(\mathcal{T}_4)$ (see for instance [73]), so $\chi_{\mathcal{T}_4}^{\infty}$ is neither closed $W^{1,p}$ -quasiconvex nor closed p-rank-one convex for any $p \in [1,\infty]$ . To summarize, the above examples with Example 2.5 show for the extended real-valued functionals in $\mathbb{R}^{m \times n}$ , with $m, n \geq 2$ , the following relations  and we emphasize that whether or not closed rank-one convexity implies either quasiconvexity or closed quasiconvexity is unknown if m = 2, $n \ge 2$ . If m > 2 then closed rank-one convexity is a strictly weaker notion [85]. As observed already by Morrey [69], quasiconvexity of a given functional $\mathbf{E} \colon \mathbb{R}^{m \times n} \to \mathbb{R} \cup \{+\infty\}$ is intimately tied to the sequential weak lower semicontinuity of the associated integral (compare also Remark 2.11, Proposition 2.14 and Theorem 2.17). In particular, see e.g. [26], for functionals with standard growth $|\mathbf{E}| \leq C(1+|\cdot|^p)$ , the W^1,p-quasiconvexity is equivalent to sequential weak lower semicontinuity in W^1,p. Here to be precise, for signed functionals such as the determinant which take also negative values, one must restrict to sequences with fixed boundary values, as otherwise concentration effects at the boundary can destroy the lower semicontinuity (see [26] for an example and [22] for some positive results). It is not known what are the weakest possible growth properties for functionals still guaranteing this equivalence. In any case, all growth conditions imply that $\mathbf{E}$ is real valued. Without going into too much detail, we remark that for non-negative functionals $\mathbf{E} \geqslant 0$ taking the value $+\infty$ and for $1 \leqslant p < \infty$ , we have, closed $$\mathbf{W}^{1,p}$$ -quasiconvex $\Longrightarrow$ $\mathbf{W}^{1,p}$ -seq. wlsc $\Longrightarrow$ $\mathbf{W}^{1,p}$ -quasiconvex To see that the implications above cannot be reversed, see Example 2.22 for the first and Example 2.5 for the second. Indeed, by [13, Example 3.5] the functional in Example 2.5 is quasiconvex but with planar waves, described in the introduction, one sees that the functional is not sequentially weak lower semicontinuous. The case of signed functionals that are allowed to assume also the value $+\infty$ , such as the local Burkholder functionals arising from our work, has not been considered before. Consequently, this requires a separate study of their lower semicontinuity properties, covered in the last Section 12. Finally, note that our approach here with pointwise definitions of the associated variational integrals, such as (1.1), is not the only possibility. Following an old tradition going back to H. Lebesgue, J. Serrin and introduced in the current context by P. Marcellini [67], a definition by relaxation from smooth maps is often more natural and desirable as it leads to variational integrals with better properties. We intend to return to this elsewhere. The characteristic function in (2.11) also leads us to a natural notion of quasiconvexity for sets:

Definition 2.23. A subset $\mathcal{U} \subset \mathbb{R}^{m \times n}$ is said to be $W^{1,p}$ -quasiconvex if $$\nu \in \mathscr{M}^p_{\mathrm{qc}}(\mathcal{U}) \quad \Longrightarrow \quad \langle \nu, \mathrm{Id} \rangle \in \mathcal{U}.$$ Thus a closed set $\mathcal{U}$ is $W^{1,p}$ -quasiconvex if and only if $\chi^{\infty}_{\mathcal{U}}$ is a closed $W^{1,p}$ -quasiconvex functional. We have been discussing $W^{1,p}$ -quasiconvex functionals, but we are yet to mention the role played by p. For later reference we record the following example, which displays the dependence of quasiconvexity on p. <span id="page-19-0"></span>Example 2.24. Let $K \ge 1$ . The K-quasiconformal cone $$Q_2(K) \equiv \{ A \in \mathbb{R}^{2 \times 2} : |A|^2 \leqslant K \det A \}$$ is a W^1,p-quasiconvex set if and only if $p \ge \frac{2K}{K+1}$ . The statement for $p > \frac{2K}{K+1}$ was shown in [3] and follows easily from Theorem 4.1 below, while the reader can find a proof of the case $p < \frac{2K}{K+1}$ in [29]. We also refer the reader to [88] where higher dimensional versions of Example 2.24 are discussed in detail. The borderline case $p = \frac{2K}{K+1}$ is more subtle and follows from a variant of Theorem 1.3, as will be shown elsewhere. 2.4. Rank-one convexity and radial maps. To conclude this section we relate rank-one convexity to radial maps. For simplicity and because it is our main focus we will only consider planar maps.

Definition 2.25. A radial map is a map $\phi \in W^{1,\infty}(\mathbb{D})$ of the form $$\phi(z) = \rho(r) \frac{z}{r}, \qquad r \equiv |z|,$$ where $\rho: [0,1] \to \mathbb{R}$ is a Lipschitz function such that $\rho(0) = 0$ . If $\rho: [0,1] \to [0,+\infty)$ we say that $\phi$ is a radial stretching. It is easy to verify that for a radial map $\phi$ we have a.e. in $\mathbb D$ the identities <span id="page-20-0"></span>(2.14) $$\partial_z \phi = \frac{1}{2} \left( \dot{\rho}(r) + \frac{\rho(r)}{r} \right), \qquad \partial_{\bar{z}} \phi = \frac{1}{2} \left( \dot{\rho}(r) - \frac{\rho(r)}{r} \right) \frac{z}{\bar{z}}.$$ Rank-one convex functionals are quasiconvex along radial maps, see e.g. [82] or [11]. Here we state a slightly more general version of this result in order to account for extended real-valued functionals.

Definition 3.1. A map $f: \mathbb{C} \to \mathbb{C}$ is said to be a principal map if: (1) f is a $\mathrm{W}^{1,1}_{\mathrm{loc}}(\mathbb{C})\text{-homeomorphism}$ and <span id="page-22-1"></span>(2) f is conformal outside $\mathbb{D}$ , with Laurent series (3.2) $$f(z) = z + \frac{b_1}{z} + \sum_{j=2}^{\infty} \frac{b_j}{z^j}, \qquad |z| > 1.$$ The classical area formula, a quick consequence of Green's theorem [4, Theorem 2.10.1] gives for any $W_{loc}^{1,2}(\mathbb{C})$ -principal map the identity <span id="page-22-0"></span>(3.3) $$\int_{\mathbb{D}} J_f(z) \, \mathrm{d}m(z) = \pi \left( 1 - \sum_{j=1}^{\infty} j |b_j|^2 \right),$$ which controls the size of the coefficients in (3.2). For instance, the Jacobian of a Sobolev homeomorphism does not change sign, thus $J_f(z) \ge 0$ for any principal map and hence $|b_1| \le 1$ in (3.2). Indeed, even if the principal map has only $W_{loc}^{1,1}$ -regularity we still have the bound $$\sum_{j=1}^{\infty} j|b_j|^2 \leqslant 1,$$ and in particular the condition $|b_1| \leq 1$ holds. But if in either case $|b_1| = 1$ , then the area formula forces all other coefficients to vanish, and that would force f affine and non-injective on the unit circle. Thus $|b_1| < 1$ for every principal homeomorphism as in Definition 3.1. We hence find that for any principal map as in (3.2), the associated linear operator <span id="page-22-3"></span> $$A_f(z) \equiv z + b_1 \bar{z}$$ is a homeomorphism with $$(3.4) det(A_f) > 0.$$ Principal maps are therefore, in a sense, close to having affine boundary values on the unit circle, but are yet flexible enough to allow deformations of maps. As another aspect of this view, the area formula (3.3) implies [4, Corollary 2.10.3] that <span id="page-22-2"></span>(3.5) $$f$$ is $K$ -quasiconformal $\Longrightarrow A_f$ is $K$ -quasiconformal. This is not immediate since the set of K-quasiconformal linear maps is not convex. The notion of a principal map is very natural also since for each coefficient $\mu$ supported in the unit disk with $\|\mu\|_{\infty} < 1$ , there is a unique $W_{loc}^{1,2}(\mathbb{C})$ principal solution $f = f_{\mu}$ to (3.1), cf. [4, Theorem 5.3.2]. This extends even to suitable degenerate Beltrami equations, see [53]. Indeed, a simple way to find principal solutions for the given coefficient $\mu$ is via the Cauchy transform <span id="page-23-3"></span> $$\mathbf{C}\varphi(z) = \frac{1}{\pi} \int_{\mathbb{C}} \frac{\varphi(\xi)}{z - \xi} \,\mathrm{d}\xi.$$ One now looks for a solution in the form (3.6) $$f(z) = z + (\mathbf{C}\omega)(z), \text{ with } \omega \in L^2(\mathbb{D}),$$ and the derivative $f_{\bar{z}} \equiv \omega$ is then found by a Neumann-series argument. Namely, if S is the Beurling–Ahlfors transform, i.e. a Calderón–Zygmund singular integral operator bounded in $L^s(\mathbb{C})$ for all $1 < s < \infty$ and defined by <span id="page-23-0"></span>(3.7) $$\mathbf{S}\varphi(z) \equiv -\frac{1}{\pi} \int_{\mathbb{C}} \frac{\varphi(\xi)}{(z-\xi)^2} d\xi,$$ then [4, (5.8)] shows that <span id="page-23-1"></span>(3.8) $$f_{\bar{z}} = (I - \mu \mathbf{S})^{-1} \mu, \quad f_z = 1 + (I - \mu \mathbf{S})^{-1} \mathbf{S} \mu.$$ Here $\|\mathbf{S}\|_{L^2(\mathbb{C})} = 1$ while by [4, Theorem 14.0.4], the operator $I - \mu \mathbf{S}$ is invertible on $L^s(\mathbb{C})$ whenever <span id="page-23-2"></span> $$(3.9) 1 + \|\mu\|_{\infty} < s < 1 + 1/\|\mu\|_{\infty},$$ with the operator-norm of the inverse $\|(I - \mu \mathbf{S})^{-1}\|_{L^s(\mathbb{C})}$ bounded by a constant that depends only on s and $\|\mu\|_{\infty}$ . Finally, to show that the mapping defined by (3.8) is a homeomorphism requires more work; for details see [4]. From (3.8) we obtain global higher integrability bounds for the derivatives of principal solutions to the Beltrami equation (3.1), in particular <span id="page-23-5"></span>(3.10) $$||f_{\bar{z}}||_{L^{s}(\mathbb{C})} + ||f_{z} - 1||_{L^{s}(\mathbb{C})} \leqslant C_{s}(K) < \infty,$$ whenever <span id="page-23-6"></span> $$(3.11) \frac{2K}{K+1} < s < \frac{2K}{K-1},$$ where this last condition comes from (3.9), recalling that $\|\mu\|_{\infty} \leqslant \frac{K-1}{K+1}$ . Note also that for exponents s > 2 and $\alpha < 1 - 2/s$ , the Cauchy operator $\mathbf{C} \colon \mathrm{L}^s(\mathbb{D}) \to \mathrm{C}^\alpha(\mathbb{C})$ is compact. Therefore from (3.6) and (3.8) we see that for each k < 1 the family <span id="page-23-4"></span>(3.12) $$\mathscr{F}_k \equiv \left\{ f \text{ is a principal solution to (3.1) with } \|\mu\|_{\infty} \leqslant k \right\}$$ is normal, i.e. every sequence of $\mathscr{F}_k$ contains a subsequence converging uniformly on $\mathbb{C}$ . The limit, too, belongs to $\mathscr{F}_k$ since any (non-constant) limit of a uniformly converging sequence of K-quasiconformal maps is K-quasiconformal.

Definition 9.1. Given $$\mathbf{E} \colon \mathbb{R}^{2 \times 2}_+ \to \mathbb{R}$$ we define $\hat{\mathbf{E}} \colon \mathbb{R}^{2 \times 2}_+ \to \mathbb{R}$ by $\hat{\mathbf{E}}(A) \equiv \mathbf{E}(A^{-1}) \det A$ . In the Elasticity literature, the involution $\hat{\cdot}$ is sometimes referred to as the Shield transformation, after Schield's work [83]. Since $\hat{1} = \det, \hat{\cdot}$ does not preserve convexity; yet it preserves the usual semi-convexity notions from the vectorial Calculus of Variations. Indeed, it is easy to verify that polyconvexity and rank-one convexity are preserved by $\hat{\cdot}$ , see [9, Theorem 2.6] for further details. The case of quasiconvexity, which is the one concerning us here, is more subtle. Indeed, as in Definition 1.1, in the case of deformations a crucial point is that one needs to address also the regularity of the inverse map when discussing quasiconvexity of $\hat{\mathbf{E}}$ , here see also Example 9.6. In fact, the inverse of a planar $W_{\text{loc}}^{1,1}$ -homeomorphism f is in $W_{\text{loc}}^{1,2}$ if and only if $K_f \in L_{\text{loc}}^1$ [42, Theorem 1.7], and hence this is the assumption that we shall make.