Univalent σ-harmonic mappings: applications to composites

Theorem 3.1. Let K > 1 be given, if σ ∈L∞ ♯(R2, Ms K), then the homogenized conductivity σhom satisfies det σhom ≥d2 m (1.16) and, for every λ ∈(−dm, dm) and every A ∈M tr(AσhomAT ) −2λ det A det σhom −λ2 = inf φ∈W 1,2 ♯,A(R2;R2) 1 |Q|

Theorem 3.2. Let K > 1 be given. For σ ∈L∞ ♯(R2, Ms K), set dm and Qdm according to (1.15) and (1.23) respectively. Assume that |Qdm| > 0 and ess inf Q\Qdm √ det σ > dm. (1.25) Then det σhom > d2 m (1.26) and the homogenized conductivity σhom satisfies the following variational principle. For every A ∈m(σhom, dm),

Proposition 2.1. Let σ ∈L∞(Ω, Ms K) and let U ∈W 1,2 loc (Ω, R2) be σ-harmonic. i) If det DU ≥0 almost everywhere in Ω, then, for any λ > 0, φU,λ is Kλ-quasiregular. More precisely zλ(x) ≤DU,λ(x) ≤kλ(x) almost everywhere (2.5) and hence ||DU,λ||L∞(Ω) ≤Kλ. (2.6) In particular φU,1 is K-quasiregular. ii) If, in addition, U is locally univalent, then, for every λ > 0, φU,λ is locally univalent. We will write M+ for the two by two real matrices with positive determinant.

Proposition 2.2. Let σ ∈L∞ ♯(R2, Ms K) and let A ∈M+. If U A ∈W 1,2 ♯,A(R2, R2) (see (1.3)) is a σ-harmonic mapping, then, for every λ > 0, φUA,λ = λU A + J ˜U A is, in addition, an homeomorphism of R2 onto itself and therefore a Kλ-quasiconformal mapping.

Proposition 2.1 is based upon a simple algebraic fact.

Lemma 2.1. Let A ∈M and S ∈Ms. For λ ∈R, set B = λA + Adj(AS). (2.7) Then (det S −λ2)A = −λB + Adj(BS), (2.8) det B = det A(det S + λ2) + λtr(ASAT ), (2.9) (det S −λ2)2 det A = det B(det S + λ2) −λtr(BSBT ). (2.10)

Lemma 3.1. For F, H ∈M, λ ∈R and S ∈Ms and positive definite, we define f(F, S, λ) = tr(FSF T ) + 2λ det F (3.6) and f ∗(H, S, λ) = sup F∈M [2H · F −f(F, S, λ)]. (3.7) Part 1). As a function of the first variable, f is strictly convex if and only if λ2 < det S; it is convex but not strictly convex if and only if λ2 = det S. Part 2). The explicit expression of f ∗is given by f ∗(H, S, λ) =   

Lemma 3.2. Let S ∈Ms and positive definite and set s = √ det S. For λ ≥0, H ∈m(S, λ) (see (1.20)) and f as in (3.6), we define f ∗,+(H, S, λ) = sup F∈M+ [2H · F −f(F, S, λ)]. (3.16) Then f ∗,+(H, S, λ) =    

Lemma 3.3. Under the assumptions of Theorem 3.1, det σhom ≥d2 m. Moreover setting Qdm as in (1.23) we have |Qdm| < 1 ⇒ p det σhom > dm, (3.21) |Qdm| = 1 ⇒ p det σhom = dm.

Lemma 3.4. Let σ ∈L∞ ♯(R2, Ms K), λ ∈[0, dm] and A ∈m(σhom, λ) be given and let φUBλ ,λ be defined by (1.18). Then: i) φUBλ ,λ ∈W(A, σ, λ) and ii) Z Q tr(DφUBλ ,λ(y)σ(y)DφUBλ ,λ(y)T ) −2λ det DφUBλ ,λ(y) det σ −λ2 dy = tr(BλσhomBT λ ) + 2λ det Bλ = tr(AσhomAT ) −2λ det A det σhom −λ2 ·

Corollary 3.1. Under the assumptions of Theorem 3.1, if we assume that A ∈m(σhom, λ), then the minimizers of (1.17) are quasiconformal. Moreover, setting as before φλ = φUBλ ,λ, and recalling (1.9) one has the following inequality zλ(x) ≤Dφλ(x) ≤kλ(x) almost everywhere, (3.32) where zλ and kλ are defined in (2.3) and (2.4).

Corollary 3.2. Under the conditions of Theorem 3.2, the minimizers of (1.27) are quasiconformal. Moreover, setting as before φdm = φUBdm ,dm, one has that their dilatation quotient satisfies the following inequality. zdm(x) ≤Dφdm(x) ≤kdm(x) almost everywhere, where zdm and kdm are defined in (2.3) and (2.4).

Proposition 3.1. Let K > 1 and σ ∈L∞ ♯(R2, Ms K) be given. Assume that the homogenized conductivity σhom is isotropic. Then it satisfies the following family of bounds. For every λ ∈(−dm, dm), for every A ∈M one has h|A|2 −2λ det A h2 −λ2 ≤ Z Q tr[Aσ(y)AT ] −2λ det A det σ(y) −λ2 dy. (3.44) Next we make the following crucial observation.

Proposition 3.2. In the two phase problem for any choice of λ ∈(−dm, dm) and A ∈M, the bound delivered by (3.44) is not “attainable”. The same is true under the much weaker assumption that σ(y) = α(y)I for some scalar function α ∈L∞(R2; [K−1, K]) taking a finite (say N) number of values. Sketch of the proof Step 1. Fix λ ∈(−dm, dm). Optimization over the matrix A shows that the tighter bound is obtained either when A = A+ ̸= 0 and A−= 0 or conversely when A = A−̸= 0 and A+ = 0. Step 2. One checks

Proposition 3.2 gives a strong motivation to pass to the limit in (3.44) sending λ to dm. However, there are two potential obstructions to this. The first one is that, if the set Qdm where σ(y) = dmI has positive measure, then in the right hand side the latter limit is plus infinity unless one requires simultaneously tr[Aσ(y)AT ] −2dm det A = 0, ∀y ∈Qdm. (3.49) Since, in the set Qdm we have σ(y) = dmI, if |Qdm| > 0, equation (3.49) holds if and only if AT A = det A I i.e. if and only if A ∈H+ ! Un

Theorem 3.2 was motivated by these arguments. We will show in Section 4, Example 4.2 that this approach can deliver optimal bounds in cases where the conventional translation principle does not. Here we conclude with an application of Theorem 3.2. Example 3.1. We apply Theorem 3.2 to the two-phase problem. Select again φ = Ax. The condition φ ∈ B(A, σ, dm) implies A ∈H+! Moreover the resulting bound is independent of the specific choice of A ∈H+. It is the same as (3.51) and gives the optimal bou

Theorem 4.1. Let K > 1 be given and let λ ≥0. For σ ∈L∞ ♯(R2, Ms K), the homogenized conductivity σhom satisfies the following inequality. For every matrix A ∈M one has f ∗,+(A, σhom, λ) ≤ inf φ∈B(A,σ,dm) Z Q f ∗,+(AdjDφ(y), σ(y), λ)dy (4.1) where f ∗,+ is defined in (3.17) and B(A, σ, dm) is defined in (1.24).

Theorem 3.2 asserts that for λ = dm and A ∈m(σhom, dm) equality occurs in (4.1) and moreover the left hand side of (4.1) is the same as the left hand side of (1.27). Corollary 3.2 shows, in addition, that the minimizers are quasiconformal mappings. We will see in the next application that also the case λ > dm is very useful. Example 4.1 (Three-phase problem). Given three real numbers 0 < σ1 < σ2 < σ3 and three positive num- bers pi with P i pi = 1, assume that σ(x) = α(x)I with α(x) = 3 X i=1 χi