Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

We study the higher gradient integrability of distributional solutions $u$ to the equation $div(\sigma \nabla u) = 0$ in dimension two, in the case when the essential range of $\sigma$ consists of only two elliptic matrices, i.e., $\sigma\in\{\sigma_1, \sigma_2\}$ a.e. in $\Omega$. In [4], for every pair of elliptic matrices $\sigma_1$ and $\sigma_2$, exponents $p_{\sigma_1,\sigma_2}\in(2,+\infty)$ and $q_{\sigma_1,\sigma_2}\in (1,2)$ have been characterised so that if $u\in W^{1,q_{\sigma_1,\sigma_2}}(\Omega)$ is solution to the elliptic equation then $\nabla u\in L^{p_{\sigma_1,\sigma_2}}_{\rm weak}(\Omega)$ and the optimality of the upper exponent $p_{\sigma_1,\sigma_2}$ has been proved. In this paper we complement the above result by proving the optimality of the lower exponent $q_{\sigma_1,\sigma_2}$. Precisely, we show that for every arbitrarily small $\delta$, one can find a particular microgeometry, i.e., an arrangement of the sets $\sigma^{-1}(\sigma_1)$ and $\sigma^{-1}(\sigma_2)$, for which there exists a solution $u$ to the corresponding elliptic equation such that $\nabla u \in L^{q_{\sigma_1,\sigma_2}-\delta}$, but $\nabla u \notin L^{q_{\sigma_1,\sigma_2}}.$ The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in [2] for the isotropic case.

Thus it suffices to show optimality for this class of coefficients, for which the exponents p σ 1 ,σ 2 and q σ 1 ,σ 2 read as Our main result is the following Theorem 1.2. Let σ 1 , σ 2 be defined by (1.2) for some K > 1 and S 1 , S 2 ∈ [1/K, K].

Connection with the Beltrami equation and explicit formulas for the optimal exponents
For the reader's convenience we recall in this section how to reduce to the case (1.2) starting from any pair σ 1 , σ 2 . We will also give the explicit formulas for p σ 1 ,σ 2 and q σ 1 ,σ 2 .
It is well-known that a solution u ∈ W 1,q loc , q ≥ 1, to the elliptic equation (1.1) can be regarded as the real part of a complex map f : Ω → C which is a W 1,q loc solution to a Beltrami equation. Precisely, if v is such that where the so called complex dilatations µ and ν, both belonging to L ∞ (Ω; C), are given by and satisfy the ellipticity condition The ellipticity (2.4) is often expressed in a different form. Indeed, it implies that there exists 0 ≤ k < 1 such that |µ| + |ν| L ∞ ≤ k < 1 or equivalently that for some K > 1. Let us recall that weak solutions to (2.2), (2.5) are called K-quasiregular mappings. Furthermore, we can express σ as a function of µ, ν inverting the algebraic system (2.3), Conversely, if f solves (2.2) with µ, ν ∈ L ∞ (Ω, C) satisfying (2.4), then its real part is solution to the elliptic equation (1.1) with σ defined by (2.6). Notice that ∇f and ∇u enjoy the same integrability properties. Assume now that σ : Ω → {σ 1 , σ 2 } is a two-phase elliptic coefficient and f is solution to (2.2)-(2.3). Abusing notation, we identify Ω with a subset of R 2 and f = u + iv with the real mapping f = (u, v) : Ω → R 2 . Then, as shown in [4], one can find matrices A, B ∈ SL sym (2) (with SL sym (2) denoting the set of invertible matrices with determinant equal to one) depending only on σ 1 and σ 2 , such that, setting one has that the functionf solves the new Beltrami equatioñ fz =μ f z +νf z a.e. in B(Ω), and the correspondingσ : B(Ω) → {σ 1 ,σ 2 } defined by (2.6) is of the form (1.2): The results in [1] and [5] imply that iff ∈ W 1,q , with q ≥ 2K K+1 , then ∇f ∈ L 2K K−1 weak ; in particular,f ∈ W 1,p for each p < 2K K−1 . Clearly ∇f enjoys the same integrability properties as ∇f and ∇u.
Finally, we recall the formula for K which will yield the optimal exponents. Denote by d 1 and d 2 the determinant of the symmetric part of σ 1 and σ 2 respectively, and by (σ i ) jk the jk-entry of σ i . Set Thus, for any pair of elliptic matrices σ 1 , σ 2 ∈ R 2×2 , the explicit formula for the optimal exponents p σ 1 ,σ 2 and q σ 1 ,σ 2 are obtained by plugging (2.8) into (1.4).

Conformal coordinates. For every real matrix
we write A = (a + , a − ), where a + , a − ∈ C denote its conformal coordinates. By identifying any vector v = (x, y) ∈ R 2 with the complex number v = x + iy, conformal coordinates are defined by the identity Here v denotes the complex conjugation. From (3.1) we have relations and, conversely, Here z and z denote the real and imaginary part of z ∈ C respectively. We recall that and Tr A = 2 a + . Moreover where |A| and A denote the Hilbert-Schmidt and the operator norm, respectively. We also define the second complex dilatation of the map A as (3.6) µ A := a − a + , and the distortion The last two quantities measure how far A is from being conformal. Following the notation introduced in [2], we define for a set ∆ ⊂ C ∪ {∞}; namely, E ∆ is the set of matrices with the second complex dilatation belonging to ∆. In particular E 0 and E ∞ denote the set of conformal and anti-conformal matrices respectively. From (3.4) we have that E ∆ is invariant under precomposition by conformal matrices, that is

Convex integration tools.
We denote by M(R 2×2 ) the set of signed Radon measures on R 2×2 having finite mass. By the Riesz's representation theorem we can identify M(R 2×2 ) with the dual of the space C 0 (R m×n ). Given ν ∈ M(R 2×2 ) we define its barycenter as We say that a map f ∈ C(Ω; R 2 ) is piecewise affine if there exists a countable family of pairwise disjoint open subsets Ω i ⊂ Ω with |∂Ω i | = 0 and such that f is affine on each Ω i . Two matrices A, B ∈ R 2×2 such that rank(B − A) = 1 are said to be rank-one connected and the measure λδ is called a laminate of first order.
and rank(B − C) = 1. Then the probability measure The process of obtaining new measures via (ii) is called splitting. The following proposition provides a fundamental tool to solve differential inclusions by means of convex integration (see e.g. [2, Proposition 2.3] for a proof).
Let Ω ⊂ R 2 be a bounded open set, α ∈ (0, 1) and 0 < δ < min |A i − A j | /2. Then there exists a piecewise affine Lipschitz map f : 3.3. Weak L p spaces. We recall the definition of weak L p spaces. Let f : Ω → R 2 be a Lebesgue measurable function. Define the distribution function of f as Let 1 ≤ p < ∞, then the following formula holds Define the quantity 1/p and the weak L p space as weak is a topological vector space and by Chebyshev's inequality we have [f ] p ≤ f L p . In particular this implies L p ⊂ L p weak .

Proof of Theorem 1.2
For the rest of this paper, σ 1 and σ 2 are as in (1.2)-(1.3). We start by rewriting (1.1) as a differential inclusion. To this end, define the sets where f := (u, v) and v is the stream function of u, which is defined, up to an addictive constant, by (2.1). In order to solve the differential inclusion (4.2), it is convenient to use (3.2) and write our target sets in conformal coordinates: where the operators d j : C → C are defined as Introduce the quantities By (4.5) we have We distinguish three cases. 1. Case s > 0 (corresponding to S > 1). We study this case in Section 5, where we generalise the methods used in [2,Section 3.2]. Observe that this case includes the one studied in [2]. Indeed, for s = k one has that s 1 = s 2 = k and the target sets (4.3) become where E ±k are defined in (3.8). We remark that, in this particular case, the construction provided in Section 5 coincides with the one given in [2, Section 3.2].
2. Case s < 0 (corresponding to S < 1). This case can be reduced to the previous one. Indeed, if we introduceŝ j := −s j ,ŝ := (ŝ 1 +ŝ 2 )/2 > 0 and the operatorsd j (a) := k a + iŝ j a then the target sets (4.3) read as This is the same as the previous case, since the absence of the conjugation does not affect the geometric properties relevant to the constructions of Section 5.
We notice that this case includes s = −k for which the target sets become We remark that in this case, (4.2) coincides with the classical Beltrami equation (see also [2,Remark 3.21]).
. This is a degenerate case, in the sense that the constructions provided in Section 5 for s > 0 are not well defined. Nonetheless, Theorem 1.2 still holds true. In fact, as already pointed out in [4, Section A.3], by an affine change of variables, the existence of a solution can be deduced by [2, Lemma 4.1,Theorem 4.14], where the authors prove the optimality of the lower critical exponent 2K K+1 for the solution of a system in non-divergence form. We remark that in this case Theorem 1.2 actually holds in the stronger sense of exact solutions, namely, there exists u ∈ W 1,1 (Ω; R) solution to (1.5) and such that

The case s > 0
In the present section we prove Theorem 1.2 under the hypothesis that the average s is positive, namely that From (5.1), recalling definitions (4.4), (4.6), (4.7), we have In order to prove Theorem 1.2, we will solve the differential inclusion (4.2) by adapting the convex integration program developed in [2, Section 3.2] to the present context. As already pointed out in the Introduction, the anisotropy of the coefficients σ 1 , σ 2 poses some technical difficulties in the construction of the so-called staircase laminate, needed to obtain the desired approximate solutions. In fact, the anisotropy of σ 1 , σ 2 translates into the lack of conformal invariance (in the sense of (3.9)) of the target sets (4.3), while the constructions provided in [2] heavily rely on the conformal invariance of the target set E {−k,k} . We point out that the lack of conformal invariance was a source of difficulty in [4] as well, for the proof of the optimality of the upper exponent.
This section is divided as follows. In Section 5.1 we establish some geometrical properties of rank-one lines in R 2×2 , that will be used in Section 5.2 for the construction of the staircase laminate. For every sufficiently small δ > 0, such laminate allows us to define (in Proposition 5.9) a piecewise affine map f that solves the differential inclusion (4.2) up to an arbitrarily small L ∞ error. Moreover f will have the desired integrability properties (see (5.59), that is, Finally, in Theorem 5.10, we remove the L ∞ error introduced in Proposition 5.9, by means of a standard argument (see, e.g., [4,Theorem A.2]). Throughout this section c K > 1 will denote various constants depending on K, S 1 and S 2 , whose precise value may change from place to place. The complex conjugation is denoted by J := (0, 1) in conformal coordinates, i.e., Jz = z for z ∈ C. Moreover, R θ := (e iθ , 0) ∈ SO(2) denotes the counter clockwise rotation of angle θ ∈ (−π, π]. Define the the argument function arg z := θ , where z = |z|e iθ , with θ ∈ (−π, π] .

5.1.
Properties of rank-one lines. In this Section we will establish some geometrical properties of rank-one lines in R 2×2 . Lemmas 5.2, 5.3 are generalizations of [2, Lemmas 3.14, 3.15] to our target sets (4.3). In Lemmas 5.4, 5.5 we will study certain rank-one lines connecting T to E ∞ , that will be used in Section 5.2 to construct the staircase laminate.
Proof. The first part of the statement is exactly like in [2, Lemma 3.14]. For the second part, one can easily adapt the proof of [2, Lemma 3.14] to the present context taking into account (5.5) and (5.7). For the reader's convenience we recall the argument. Let A ∈ R 2×2 , Q ∈ T 1 and Q 0 ∈ T 1 such that dist(A, T 1 ) = |A − Q 0 |. By (5.5), we can apply the first part of the lemma to A − Q 0 and Q − Q 0 to get where the last inequality follows from (5.7), since Q − Q 0 ∈ T 1 . Therefore The proof for T 2 is analogous.
lies on a rank-one segment connecting T 1 and E ∞ . Precisely, there exist matrices Q ∈ T 1 {0} and P ∈ E ∞ {0}, with det(P −Q) = 0, such that A ∈ [Q, P ]. We have P = tJR θ A for some t > 0 and θ A as in (5.4). Moreover, there exists a constant c K > 1, depending only on K, S 1 , S 2 , such that Proof. The proof can be deduced straightforwardly from the one of [2, Lemma 3.15]. We decompose any A = (a, b) as with Q ∈ T 1 and P t ∈ E ∞ . The matrices Q and P t are rank-one connected if and only if |a| = |d 1 (a) + t(b − d 1 (a))|. Since det Q > 0 for Q = 0, it is easy to see that there exists only one t 0 > 0 such that the last identity is satisfied. We then set ρ := 1 + 1/t 0 so that The latter is the desired decomposition, since ρ Q ∈ T 1 , ρP t 0 ∈ E ∞ are rank-one connected, ρ > 0 and ρ −1 + (t 0 ρ) −1 = 1. Also notice that ρP t 0 = ρt 0 |b − d 1 (a)|JR θ A as stated. Finally let us prove (5.9). Remark that By the linear independence of T 1 and E ∞ , we get 1 c K |A| ≤ |P − Q| .
We now turn our attention to the study of rank-one connections between the target set T and E ∞ . Then λ j > 0, A j > 0 and det(Q j − JR) = 0. Moreover there exists a constant c K > 1 depending only on K, S 1 , S 2 such that for every a ∈ C {0} and R ∈ SO(2).
This is similar to the previous case. Indeed (5.22) is still true, but for B j we have This implies (5.25) with j = 1. Similarly to the previous case, we can see that (5.25) for j = 2 is equivalent to Notice that f is symmetric, therefore (5.31) is a consequence of (5.27).
We will now turn our attention to the function l. Notice that is the harmonic mean of λ 1 and λ 2 . Therefore H is differentiable and even. By direct computation we have Since λ j > 0, by (5.25) we have (5.33) Moreover H(0) = s 1 + s and H(1) = k 1 + k . Then from (5.32) we deduce l(0) = s, l(1) = k and the rest of the statement for l. The statements for L and p follow directly from the properties of l and from the fact are C 1 and strictly increasing for 0 < t < 1 and t > 1, respectively. Next we prove (5.18). By (5.1) and the properties of λ j , we have in particular where H is defined in (5.32). Since λ j > 0, the inequality M j > 0 is equivalent to H < 1, which holds by (5.34). The inequality M 2 < 2 is instead equivalent to λ 1 (1 − 2λ 2 ) > 0, which is again true by (5.34). The case M 1 < 2 is similar. Finally m > 0 follows from 0 < M 2 < 2 and the continuity of λ j .
Next we split P in order to "climb" one step of the staircase (see Figure 1). Define x := cos θ A , y := sin θ A and a := x k + i y s , as in (5.15). Moreover set d 1 (a)) , Q 2 := λ 2 (−a, d 2 (a)) .
Let us now bound (5.52) from below. We can estimate from below the denominator in the third factor of (5.52) with 2n, since t − n > −ρ by (5.43) and the assumption that ρ < m with m as in (5.17). Therefore The lower bound in (5.37) follows from (5.55) and (5.56). Finally, the last part of the statement follows from a simple geometrical argument, recalling that arg R = θ A = − arg(b − d 1 (a)) and using hypothesis (5.38).
Remark 5.8. In the isotropic case S = K, the laminate ν A provided by Lemma 5.6 coincides with the one in [2,Lemma 3.16]. In particular, the growth condition (5.37) is independent of the initial point A, and it reads as Moreover, by Remark 5.7, for every R θ ∈ SO(2), JR θ is the center of mass of a sequence of laminates of finite order such that (5.57) holds with p(R θ ) ≡ 2K K+1 , which gives the desired growth rate.
In contrast, in the anisotropic case 1 < S < K, the growth rate of the laminates explicitly depends on the argument of the barycenter JR θ . The desired growth rate corresponds to θ = 0, that is, the center of mass has to be J.
In constructing approximate solutions with the desired integrability properties, it is then crucial to be able to select rotations whose angle whose angle lies in an arbitrarily small neighbourhood of θ = 0.
We now proceed to show the existence of a piecewise affine map f that solves the differential inclusion (4.2) up to an arbitrarily small L ∞ error. Such map will have the integrability properties given by (5.59).
Let {ρ n } be a strictly decreasing positive sequence satisfying where m > 0 and c K > 1 are the constants from Lemma 5.6. Define {δ n } as In particular from (5.61),(5.62) it follows that (5.63) δ n < δ 2 , for every n ∈ N .
Assume now that f n and τ n satisfy the inductive hypothesis. We will first define f n+1 by modifying f n on the set Ω n . Since f n is piecewise affine we have a decomposition of Ω n into pairwise disjoint open subsets Ω n,i such that with f n (x) = A i x + b i in Ω n,i , for some A i ∈ R 2×2 and b i ∈ R 2 . Moreover (5.66) dist(A i , S δn n ) < ρ n by (c) and (d). Since (5.66) and (5.61) hold, we can invoke Lemma 5.6 to obtain a laminate ν A i and a rotation R i = R θ A i satisfying, in particular, since δ n+1 = δ n + ρ n by (5.62). By applying Proposition 3.2 to ν A i and by taking into account (5.68), we obtain a piecewise affine Lipschitz mapping g i : Ω n,i → R 2 , such that in Ω n,i . Moreover Since Ω n+1 is well defined, we can also introduce is a direct consequence of (d), (h), and the fact that ρ n is strictly decreasing. Finally let us prove (5.64). First notice that the sets ω n,i are pairwise disjoint. By (5.61), in particular we have ρ n+1 < dist(T, S 1 )/4, so that By (5.67) and (5.63) we have | arg R i | < δ. Then by the properties of β n (see Lemma 5.5), Using (5.71), (5.65), (5.70) in (5.64) yields and (5.64) follows.
We are left to estimate the distribution function of ∇f . By (g) we have that |∇f (x)| > n c K,δ 0 in Ω n and |∇f (x)| < c K,δ 0 n in Ω Ω n .
We remark that the constant c K,δ 0 in (5.59) is monotonically increasing as a function of δ 0 , that is c K,δ 1 ≤ c K,δ 2 if δ 1 ≤ δ 2 .
The boundary condition f n = Jx reads f 1 n = x 1 and f 2 n = −x 2 . We set u n := f 1 n + v n , where v n ∈ H 1 0 (Ω, R) is the unique solution to div(σ n ∇v) = − div(σ n a n − R T π 2 b n ) .
Notice that v n is uniformly bounded in H 1 by (5.79). Since (5.78) holds, it is immediate to check that div(σ n ∇u n ) = div(R T π 2 ∇f 2 n ) = 0, so that u n is a solution of (5.73). Finally, the regularity thesis (5.74), (5.75), follows from the definition of u n and the fact that v n ∈ H 1 0 (Ω; R) and f 1 n satisfies (5.77) with 1 < p n < 2.