Univalent solutions of elliptic systems of Heinz-Lewy type

Theorem 1. — The first derivatives of u(x) are Hölder continuous in the interior of $\Omega$ with exponent $\mu$ , $u \in C^{1,\mu}_{loc}(\Omega)$ . Moreover, the Hölder norm on any compact subset $\Omega'$ of $\Omega$ can be estimated in the form $$||u||_{C^{1,\mu}(\Omega')} \leq C,$$ where C depends only on the parameters $\mu$ , $\lambda$ , $\Lambda$ , L, a, b, M, N and dist $(\Omega', \partial\Omega)$ . The proof of this result will be accomplished in four steps. First we establish a modulus of continuity for u(x) in the interior of $\Omega$ , essentially applying the Courant-Lebesque lemma [6], Lemma 3.1. In the following, we shall work only in the interior of $\Omega$ , particularly in discs $D_R = D_R(x_0)$ of radius R centered at $x_0 \in \Omega$ . Radii are always assumed to be less than min $\{1, \operatorname{dist}(x_0, \partial\Omega)\}$ . Constants named C are $\geq 1$ . They may change from line to line and may depend on all available parameters.

Lemma 1. — The modulus of continuity of u(x) can be estimated in the form $$\underset{\mathbf{D_R}}{\operatorname{osc}} u \leq 2 \sqrt{\frac{\pi \, \mathbf{N}}{\log 1/\mathbf{R}}}.$$

Lemma 2. – For all $\sigma$ , $0 < \sigma < 1$ , there is a radius $R_0$ , which depends only on the available parameters, such that for all R, $0 < R \le R_0$ , $$\int_{\mathbf{D}_{\mathbf{R}}} |\mathbf{D}u|^2 dx \leq \mathbf{C} \mathbf{R}^{2\sigma}.$$ The Dirichlet growth lemma therefore implies Corollary. $-u(x) \in C_{loc}^{\sigma}(\Omega)$ for all $\sigma$ , $0 < \sigma < 1$ . Proof of Lemma 2. - We freeze the coefficients u (x)) by rewriting the system (3) in the form (4) $$-D_{\alpha}(A^{\alpha\beta}(x_0)D_{\beta}u^k) = f^k + D_{\alpha}g^{\alpha k},$$ $$g^{\alpha k}(x) = (A^{\alpha\beta}(x) - A^{\alpha\beta}(x_0))D_{\beta}u^k.$$ By testing (4) with w = u - v, where v = (vl (x), v2 (x)) is the solution of the Dirichlet problem $$-D_{\alpha}(A^{\alpha\beta}(x_0)D_{\beta}v)=0$$ in $D_{\beta}$ , $v=u$ on $\partial D_{\beta}$ , we estimate (5) $$\int_{D_0} |Dw|^2 dx \le C \left\{ \int_{D_R} |w| |f| dx + \int_{D_R} |g - g_R|^2 dx \right\},$$ where gR is the mean of g over DR. Combined with the estimates (6) $$\int_{D_{n}} |Dv|^{2} dx \leq C(\lambda, \Lambda) \left(\frac{\rho}{R}\right)^{2} \int_{D_{R}} |Dv|^{2} dx,$$ which hold for all p, R, 0 P:5 R dist (xo, (5) in turn yields $$\int_{D_{\rho}} |Du|^2 dx \le C \left\{ \left( \frac{\rho}{R} \right)^2 \int_{D_{R}} |Du|^2 dx + \int_{D_{R}} |w| |f| dx + \int_{D_{R}} |g - g_{R}|^2 dx \right\}.$$ By the maximum principle, $$\sup_{\mathbf{D}_{\mathbf{P}}} |w| \leq \operatorname{osc} u$$ On the other hand, $$\operatorname{osc} A^{\alpha\beta}(x) \leq C((\operatorname{osc} u)^{\mu} + R^{\mu}).$$ Accordingly $$\int_{D_0} |Du|^2 dx \le C \left\{ \left( \left( \frac{\rho}{R} \right)^2 + \underset{D_R}{\operatorname{osc}} u + (\operatorname{osc} u)^{2\mu} + R^{2\mu} \right) \int_{D_R} |Du|^2 dx + R^2 \right\}.$$ By Lemma 1, if R is smaller than some Ro, Rodist(xo, which depends only A, J-l, L, a, b, M, then the quantity $$osc u + (osc u)^{2 \mu} + R^{2 \mu}$$ is so small that we can iterate the inequalities $$\int_{\mathbf{D}_{\mathbf{p}}} |\mathbf{D}u|^2 dx \leq \mathbf{C} \left\{ \left( \left( \frac{\mathbf{p}}{\mathbf{R}} \right)^2 + \varepsilon \right) \int_{\mathbf{D}_{\mathbf{R}}} |\mathbf{D}u|^2 dx + \mathbf{R}^{2\sigma} \right\}$$ for all $\sigma$ , $0 < \sigma < 1$ , to obtain, by the Calculus Lemma 2.1 of [5], $$\int_{D_{\mathbf{0}}} |Du|^2 dx \le C \left\{ \left( \frac{\rho}{R} \right)^{2\sigma} \int_{D_{\mathbf{R}}} |Du|^2 dx + \rho^{2\sigma} \right\} \le \frac{C}{R^{2\sigma}} \rho^{2\sigma}$$ for $0 < \rho \le R \le R_0$ as required.

Lemma 3. — For all $\sigma$ , $0 < \sigma < \min\{\mu, 1/2\}$ , there are radii $R_0 = R_0(\sigma)$ , such that for all R, $0 < R \le R_0$ , $$\int_{\mathbf{D}_{\mathbf{R}}} |\mathbf{D}u - (\mathbf{D}u)_{\mathbf{R}}|^2 dx \leq \mathbf{C}\mathbf{R}^{2+2\sigma}.$$ Corollary. $-u(x) \in C^{1,\sigma}_{loc}(\Omega)$ for all $\sigma$ , $0 < \sigma < \min\{\mu, 1/2\}$ . This follows from Campanato's characterization of the Hölder classes. In particular Du(x) is bounded. Proof of Lemma 3. — As in the proof of Lemma 2, using the inequalities $$\int_{D_{\rho}} |Dv - (Dv)_{\rho}|^2 dx \leq C \left(\frac{\rho}{R}\right)^4 \int_{D_{R}} |Dv - (Dv)_{R}|^2 dx$$ instead of (6), we have (7) $$\int_{D_{\rho}} |Du - (Du)_{\rho}|^{2} dx \leq C \left\{ \left( \frac{\rho}{R} \right)^{4} \int_{D_{R}} |Du - (Du)_{R}|^{2} dx + \int_{D_{R}} |w| |f| dx + \int_{D_{R}} |g - g_{R}|^{2} dx \right\}.$$ Incorporating Lemma 2 and its corollary, we estimate for all $\sigma < 1$ , $$\int_{D_{\rho}} |Du - (Du)_{\rho}|^2 dx \le C \left\{ \left( \frac{\rho}{R} \right)^4 \int_{D_{R}} |Du - (Du)_{R}|^2 dx + R^{(2+2\mu')\sigma} \right\},$$ whenever $R \le R_0 = R_0(\sigma)$ , where $\mu' = \min\{\mu, 1/2\}$ . The iteration argument then yields for all $\sigma$ , $0 < \sigma < \mu'$ , $$\int_{D_{\rho}} |Du - (Du)_{\rho}|^2 dx \le \frac{C}{R^{2+2\sigma}} \rho^{2+2\sigma}$$ for all $0 < \rho \le R \le R_0$ , which proves the lemma.

Lemma 4. – There exists a radius $R_0$ such that for all R, $0 < R \le R_0$ , $$\int_{D_R} |Du - (Du)_R|^2 dx \leq CR^{2+2\mu}.$$ Proof - Going back to (5), we use one of Poincare's inequalities to estimate for s > 0 $$\int_{D_{\mathbf{R}}} |Dw|^2 dx \le C \left\{ \int_{D_{\mathbf{R}}} |w| |f| dx + \int_{D_{\mathbf{R}}} |g - g_{\mathbf{R}}|^2 dx \right\}$$ $$\le C \left\{ \varepsilon \int_{D_{\mathbf{R}}} |Dw|^2 dx + \frac{1}{\varepsilon} R^{2+2\mu} \right\}.$$ The statement is then an immediate consequence of (7). Theorem 1 follows as a corollary to Lemma 4.

Theorem 2. - If cp (x) = o ( for some n then $$\lim_{\substack{|z| \to 0 \\ z \neq 0}} \frac{\varphi_z}{z^n} = \frac{1}{2} \lim_{\substack{|x| \to 0 \\ x \neq 0}} \frac{\varphi_{x^1} - i \varphi_{x^2}}{(x^1 + ix^2)^n}$$ exists. In order to prove this result, we first note that only differential inequalities of the form $$|D_{\alpha}(aD_{\alpha}\varphi)| \leq C(|\varphi| + |D\varphi|), \quad a \in C^{\mu}(\Omega),$$ have to be considered. We approximate a by a differentiable function a for x ~ 0, and then we modify Hartman-Wintner's proof of the case a (x) -1 [7], Theorem 1.

Lemma 5. - Let some w 0 ~. l, which satisfies $$\frac{0 < \lambda \leq a(x) \leq \Lambda < +\infty \qquad (x \in \Omega)}{\left| \frac{a(x') - a(x'')}{|x' - x''|^{\mu}} \right|} \leq L \qquad (x' \neq x'').$$ Then there exists a disc 0 Ro = Ro {u, l~, L) dist(O, ~), such that for each ~, 0 ~ ~ ~ Ro, there is a function (~ C 1 ( D B { 0 ~ ) such that $$\tilde{a}(0) = a(0), \qquad \tilde{a}(\zeta) = a(\zeta),$$ $$\frac{\lambda}{2} \leq \tilde{a}(x) \leq 2\Lambda \qquad (x \in D),$$ $$\frac{\left|\tilde{a}(x') - \tilde{a}(x'')\right|}{\left|x' - x''\right|^{\mu}} \leq C \qquad (x' \neq x''),$$ $$\left|D\tilde{a}(x)\right| \leq C\left|x\right|^{\mu - 1} \qquad (x \neq 0),$$ where C depends only on ~,, A, Land dist (0, aSZj. Proof - For E>O set $$\varphi_{\varepsilon}(x) = \int_{\Omega} \Phi_{\varepsilon}(y - x) \, a(y) \, dy$$ Here $$\Phi_{\varepsilon}(x) = \frac{1}{\varepsilon^2} \Phi\left(\frac{x}{\varepsilon}\right).$$ where O is a C~-function such that O (x) > 0, JO (x) dx = I, O (x) \*0 for ] x ( > I . Then let $$a_{\varepsilon}(x) = \varphi_{\varepsilon}(x) + \frac{(a(\zeta) - \varphi_{\varepsilon}(\zeta)) - (a(0) - \varphi_{\varepsilon}(0))}{|\zeta|^{\mu}} |x|^{\mu} + (a(0) - \varphi_{\varepsilon}(0))$$ If Ro and Eo are sufficiently small, depending only on J.1, Â., A, L and dist (0, then $$\tilde{a}(x) = a_{\epsilon_0}(x) \qquad (x \in D_{\mathbf{R}_0})$$ satisfies all the stated properties. Proof of Theorem 2. - By rewriting the coefficients a03B103B2 (x) as $$a^{\alpha\beta}(x) = a(x) A^{\alpha\beta}(x), \qquad a^2(x) = \det[a^{\alpha\beta}(x)]$$ we can solve the Cauchy-Riemann-Beltrami system $$D_1 \xi^1 = A^{2k} D_k \xi^2 D_2 \xi^1 = -A^{1k} D_k \xi^2$$ i. e., we can make a conformal change of variables $$x = (x^1, x^2) \to \xi = (\xi^1, \xi^2)$$ in a neighborhood of the origin in order to write the differential inequality (8) in the form (9). We change to complex notation as indicated in the statement and we assume that f or the sake of exposition. Let a, R o and ç be as in Lemma 5. Then $$-D_{\alpha}(\tilde{a}D_{\alpha}\varphi) = D_{\alpha}((a-\tilde{a})D_{\alpha}\varphi) + W.$$ Now $$4(\tilde{a}\varphi_z)_{\bar{z}} = D_{\alpha}(\tilde{a}D_{\alpha}\varphi) + i(D_1\varphi D_2\tilde{a} - D_2\varphi D_1\tilde{a})$$ and therefore $$-4(\tilde{a}\varphi_z)_{\bar{z}} = D_{\alpha}((a-\tilde{a})D_{\alpha}\varphi) + i(D_1\tilde{a}D_2\varphi - D_2\tilde{a}D_1\varphi) + W$$ = $\tilde{W}$ . Multiplying by $$h(z) = z^{-k}(z-\zeta)^{-1}, \quad k \in \{1, \ldots, n\},\$$ and integrating over DE (~)), it follows that $$2i\int_{\partial\Omega_{z}} \widetilde{a} \, \varphi_{z} \, h \, dz = \int_{\Omega_{z}} \widetilde{W} \, h \, dx.$$ If (10) $$\varphi_z = o(|z|^{k-1}) \qquad (|z| \to 0)$$ then we can to obtain (11) $$4\pi a (\zeta) \varphi_{z}(\zeta) \zeta^{-k} = -2i \int_{\partial D_{\mathbf{R}}} \tilde{a} \varphi_{z} h dz + \int_{D_{\mathbf{R}}} (\tilde{a} - a) D_{\alpha} \varphi D_{\alpha} h dx$$ $$+ \int_{\partial D_{\mathbf{R}}} (a - \tilde{a}) D_{\nu} \varphi h ds + i \int_{D_{\mathbf{R}}} (D_{1} \tilde{a} D_{2} \varphi - D_{2} \tilde{a} D_{1} \varphi) h dx + \int_{D_{\mathbf{R}}} W h dx.$$ For fixed n >\_ 1, it will be shown by induction over k, 1 \_ k ~ n, that (10) holds. First, (10) is true for k =1. Suppose now that (10) holds for a k, 1 k ~ n. By Lemma 5, all integrals are absolutely convergent. In particular, there is a constant C, which depends only on the data, such that (12) $$|D\varphi(\zeta)| |\zeta|^{-k} \leq C \left\{ \int_{\partial D_{\mathbf{R}}} |D\varphi| |z|^{-k} |z-\zeta|^{-1} ds + \int_{D_{\mathbf{R}}} |\varphi| |z|^{-k} |z-\zeta|^{-1} dx + \int_{D_{\mathbf{R}}} |D\varphi| (|z|^{-k-1+\mu} |z-\zeta|^{-1} + |z|^{-k} |z-\zeta|^{-2+\mu}) dx \right\}.$$ This inequality is multiplied by $|\zeta - z_0|^{-2+\mu}$ and then integrated over $D_R$ . By virtue of (13) $$|(z-\zeta)(\zeta-z_0)|^{-\gamma} \leq C(\gamma)|z-z_0|^{-\gamma} (|z-\zeta|^{-\gamma}+|\zeta-z_0|^{-\gamma})$$ for $\gamma > 0$ , we estimate after relabelling $$(1-CR^{\mu})\int_{D_{\mathbf{R}}} |D\phi| |z|^{-k} |z-\zeta|^{-2+\mu} dx$$ (14) $$\leq \operatorname{CR}^{\mu} \left\{ \int_{\partial D_{\mathbf{R}}} |D\phi| |z|^{-k} |z-\zeta|^{-1} ds + \int_{D_{\mathbf{R}}} |\phi| |z|^{-k} |z-\zeta|^{-1} dx + \int_{D_{\mathbf{R}}} |D\phi| |z|^{-k-1+\mu} |z-\zeta|^{-1} dx \right\}.$$ We now multiply (12) by $|\zeta|^{-1+\mu}|\zeta-z_0|^{-1}$ and integrate over $D_R$ to obtain similarly, but using (13) twice, $$(1 - CR^{\mu}) \int_{D_{\mathbf{R}}} |D\varphi| |z|^{-k-1+\mu} |z-\zeta|^{-1} dx$$ $$(15) \qquad \leq CR^{\mu} \left\{ \int_{\partial D_{\mathbf{R}}} |D\varphi| |z|^{-k} |z-\zeta|^{-1} ds + \int_{D_{\mathbf{R}}} |\varphi| |z|^{-k} |z-\zeta|^{-1} dx \right\}.$$ If R is so small that $CR^{\mu} \le 1/2$ , then we can combine (14) and (15) to arrive at $$\int_{\mathbf{D_R}} |\mathbf{D}\phi| (|z|^{-k-1+\mu} |z-\zeta|^{-1} + |z|^{-k} |z-\zeta|^{-2+\mu}) dx$$ (16) $$\leq \mathbf{CR}^{\mu} \left\{ \int_{\partial \mathbf{D_R}} |\mathbf{D}\phi| |z|^{-k} |z-\zeta|^{-1} ds + \int_{\mathbf{D_R}} |\phi| |z|^{-k} |z-\zeta|^{-1} dx \right\}.$$ The R.H.S. is O(1) as $|\zeta| \to 0$ , hence, by (12), $\varphi_z(\zeta)\zeta^{-k} = O(1)$ . In fact (17) $$\lim_{\substack{|\zeta| \to 0 \\ \zeta \neq 0}} \varphi_z(\zeta) \zeta^{-k}$$ exists by virtue of (11) and (12). If k < n, this limit is zero which completes the induction. Finally, by arguing as above and making use of (12) for k = n, we infer that the limit (17) also exists for k = n as required. By putting $\zeta = 0$ in (16), one can derive the well-known unique continuation principle, namely $\varphi(x) = o(|x|^n)$ ( $|x| \to 0$ ) for all $n \in \mathbb{N}$ implies $\varphi(x) \equiv 0$ . A consequence is the following extension of Hartman-Wintner's Theorem 2\* of [7]:

Corollary 1. - Let the assumptions of Theorem 2 be satisfied. T'hen either c~ (x) - 0 or there exists a non-negative integer m such that $$\lim_{\substack{|z|\to 0\\z\neq 0}}\frac{\varphi_z(z)}{z^m}\neq 0.$$ We shall need this theorem in the case n >\_ 1, where then obviously m > 1. The following corollaries are essentially identical with [15], Hilfssatz 1 and [11), Hilfssatz 2:

Corollary 2. - If n >\_ 1 and cp (x) ~ 0, then (p (z) has an asymptotic expansion of the form (18) $$\varphi(z) = \text{Re}(A z^{m+1}) + o(|z|^{m+1}) \qquad (|z| \to 0),$$ where A ~ 0 and m >\_ 1.

Corollary 3. - Let {03C6(k)(x) }~k=1 be a family of C1-solutions to the differential inequality (8), where C is independent of k. Assume that $$\Phi^{(k)}(x) \to \Phi(x), \qquad D\Phi^{(k)}(x) \to D\Phi(x)$$ uniformly in DR (k ~ oo). Let cp (x) = o ( I x I ) as x I --~ 0 and assume that (x) ~ 0 in DR for all Then cp (x) \_-- 0. Proof - If c~ (x) ~ 0, then by Corollary 1, $$\lim_{\substack{|z|\to 0\\z\neq 0}} \frac{\varphi_z(z)}{z^m} = A \neq 0.$$ Then there are k o, R o such that $$\left|\phi_z^{(k_0)}(z) - \psi(z)\right| < \left|\psi(z)\right|$$ for I z = Ro. Hence must have a zero in DRo by the homotopy invariance of the winding number.

Proposition 1. - There exists a in the u-plane, S = ~ (Mo), such that for each ~ \_ (~1, ~2) E ~2, I ~ I =1, there is a function 03A6 E CZ (Ds) with 03A6 (0) = 0, (0) = 03BE, ~ 03A6 ~C2 (D03B4) C and cp (x)=03A6 (u (x)) satisfies a differential inequality of the form $$|L \varphi| \leq C(|\varphi| + |D\varphi|)$$ in Ixl R, for any solving (19) with u (o) = o, b, -\_K. Proof. - Assume that u E C2 for convenience only and let g =g (u) be a real-valued function. Consider $$\varphi(x) = \Phi(u(x)) = u^1 - g(u^2)$$ and calculate $$L \varphi = \varepsilon^{\alpha\beta} (h_{ij}^1 - g' h_{ij}^2) D_{\alpha} u^i D_{\beta} u^j + g'' | Du^2 |^2$$ $$= b^{\alpha} D_{\alpha} \varphi + c | Du^2 |^2.$$ Here we used the fact that and we let b°' (x), a =1, 2, denote certain bounded expressions in ht, g', D(p, Du2. The term c can easily be computed as $$c(u^1, u^2, g', g'') = g'' + \sum_{i, j=1}^{2} (h_{ij}^1(u) - g'h_{ij}^2(u))(g')^{p_{ij}}$$ where $$p_{ij} = \begin{cases} 2 & (i=j=1) \\ 1 & (i=1, j=2 \text{ or } i=2, j=1). \\ 0 & (i=j=2) \end{cases}$$ Now solve the differential equation $$c(g, t, g', g'') = 0$$ subject to the initial conditions g (o) =g' (o) = 0 and use the above calculation to estimate $$\begin{aligned} |L \varphi| &\leq C \left( |D\varphi| + |\sum_{i,j=1}^{2} ((h_{ij}^{1}(u^{1}, u^{2}) - h_{ij}^{1}(g(u^{2}), u^{2}) - g'(h_{ij}^{2}(u^{1}, u^{2}) - h_{ij}^{2}(g(u^{2}), u^{2})))(g')^{p_{ij}} \right) \\ &\leq C (|D\varphi| + |u^{1} - g(u^{2})|) \\ &\leq C (|\varphi| + |D\varphi|). \end{aligned}$$ The statement then follows from the observation that for any orthogonal matrix T, u(x)=Tu(x) solves a system of the form (19), which satisfies Assumption (A3) with suitable constants Mo, M~. We have now provided all that's necessary to derive the following MAIN THEOREM. - Let u be a homeomorphism from D onto D, D = D 1 (0), of class 2 with u (0) = 0, which solves the Heinz-Lewy system (19). Suppose that the assumptions (A 1), (A3) are satisfied. Then and the Jacobian $$J(u(x)) = D_1 u^1 D_2 u^2 - D_1 u^2 D_2 u^1$$ does not vanish in S~. The following estimates hold in any compact subset 03A9' of 03A9: $$\|u\|_{\mathcal{C}^{1,\mu}(\Omega')} \leq \mathcal{C}, \qquad J(u(x)) \geq c > 0,$$ where the constants C, c depend only on the data ~,, A, L, Mo, M1, M, N and dist (SZ', a~). The non-vanishing of the Jacobian is shown by contradiction as in [15], pp. 88-89. Let us give an outline of the proof for the sake of completeness: Assume that 0~03A9, u(0)=0 and J (o) = o. Then there is a solution ~=(~1~ ~2)~ ~ ~ j =1, to $$D_1 u^1 (0) \xi^1 + D_1 u^2 (0) \xi^2 = 0$$ $$D_2 u^1 (0) \xi^1 + D_2 u^2 (0) \xi^2 = 0.$$ By Proposition 1, there is a mapping 03A6(u) with D03A6(0)=03BE, such that t~ (x) = ~ (u (x)) satisfies a differential inequality of the form (9), and hence, by Corollary 2 to Theorem 2, (p has an asymptotic expansion of the form ( 18) with A ~ 0 and m > 1. This is true because cp ( x) ~ 0 by the univalency of the mapping $$\varphi(x) + i(\xi^1 u^2(x) - \xi^2 u^1(x))$$ in a neighborhood of 0. An argument of Berg's [1], p. 314, yields the asymptotic expansion (18) with m = 0, a contradiction. The estimate for the Jacobian is derived indirectly by a compactness argument as in [11], pp. 142-143, incorporating pp. 138-139, and using Corollary 3 of Theorem 2 of the previous section. Let us finally remark that the Lipschitz condition on the coefficients ht can be weakened to the effect that only certain combinations of the need to be Lipschitz continuous.