THE THREE DIVERGENCE FREE MATRIX FIELDS PROBLEM

MARIAPIA PALOMBARO AND MARCELLO PONSIGLIONE

Abstract. We prove that for any connected open set Ω ⊂ ℝ^{n} and for any set of matrices K = {A_{1}, A_{2}, A_{3}} ⊂ 𝕄^{m×n}, with m ≥ n and rank(A_{i} − A_{j}) = n for i ≠ j, there is no non-constant solution B ∈ L^{∞}(Ω, 𝕄^{m×n}), called exact solution, to the problem

DivB = 0 in D′(Ω, ℝ^{m}) and B(x) ∈ K a.e. in Ω . In contrast, A. Garroni and V. Nesi [10] exhibited an example of set K for which the above problem admits the so-called approximate solutions. We give further examples of this type.

We also prove non-existence of exact solutions when K is an arbitrary set of matrices satisfying a certain algebraic condition which is weaker than simultaneous diagonalizability.

Key words: Differential inclusions, Phase transitions, Homogenization. 2000 Mathematics Subject Classification: 34A60, 82B26, 35B27 .

Contents

1. Introduction 1 2. Preliminaries 5 3. Proofs of the main results 7 4. Approximate solutions 12 References 14

1. Introduction

The problem of characterizing solenoidal matrix fields which take values in a finite set of matrices, has been recently considered by A. Garroni and V. Nesi. This kind of problem is analogous to that on curl free matrix fields in which one asks whether a Lipschitz mapping using a finite number of gradients exists. Here the differential constraint of being the gradient of a mapping, and hence a curl free matrix field, is replaced by that of being a divergence free matrix field (i.e. a matrix valued function whose rows are divergence free in the distributional sense). To describe the problem we begin with some definitions.

Definition 1. Given two integers m, n ≥ 2, a set of real m × n matrices K ⊂ 𝕄^{m×n} and a bounded open set Ω in ℝ^{n}, we say that any B ∈ L^{∞}(Ω, 𝕄^{m×n}) satisfying

DivB = 0  in D′(Ω, ℝ^{m}) ,
B(x) ∈ K  a.e. in Ω ,

B is non-costant , (1.1)

is an exact solution of (1.1). We say that K is rigid for exact solutions if there is no solution to (1.1). Definition 2. We say that Problem (1.1) admits an approximate solution if there exists a uniformly bounded sequence {B_{h}} ⊂ L^{∞}(Ω, 𝕄^{m×n}) such that

DivB_{h} → 0  in W^{−1,∞}(Ω, ℝ^{m}),
dist(B_{h}, K) → 0 in measure,
dist(B_{h}, A) ↛ 0 in measure, for every A ∈ K.

(1.2)

We say that K is rigid for approximate solutions of (1.1) if there is no solution to (1.2). We remark that if K is rigid for approximate solutions and there exists a sequence {B_{h}} satisfying the first two conditions of (1.2), then any accumulation point of the sequence {B_{h}} has to be a constant matrix in K.

Let us briefly describe the situation in the context of the “gradient problem”, that is: find f ∈ W^{1,∞}(Ω, ℝ^{m}) such that Df ∈ K a.e. in Ω and f is not affine. It is well-known in this setting, that a sufficient condition for a set K = {A_{1}, …, A_{N}} ∈ 𝕄^{m×n}, N ≤ 3, to be rigid is that rank(A_{i} − A_{j}) > 1 for i ≠ j. The condition rank(A_{i} − A_{j}) = 1 is called rank-1 connectedness. J. M. Ball and R. D. James studied in detail the case N = 2 and proved, under the latter assumption, a rigidity result both for exact and for approximate solutions (see [4]). For N = 3 rigidity still holds. The following theorem is due to Šverák and will be used later.

Theorem 1.1. (V. Šverák, [14]). Let Ω ⊂ ℝ^{n} be an open connected set and let K = {A_{1}, A_{2}, A_{3}} ⊂ 𝕄^{m×n}, with rank(A_{i} − A_{j}) > 1 for i ≠ j. If f ∈ W^{1,∞}(Ω, ℝ^{m}) satisfies Df ∈ K a.e., then Df is constant. Let p > 2 and let f_{h} ⇀ f in W^{1,p}(Ω, ℝ^{m}). If dist(Df_{h}, K) → 0 in measure, then Df_{h} → A_{i} in measure, for some i ∈ {1, 2, 3}.

The previous result, specialized to the case m = n = 2, will be a crucial tool in the proof of Theorem 1.7. For completeness let us also recall that for N = 4 rigidity still holds for exact solutions (see [8]) but it can fail for approximate ones and a suitable choice of {A_{1}, A_{2}, A_{3}, A_{4}} (see [16], [17]). The case N = 5 is nicely illustrated in [11] by a non-rigid five point configuration without any rank-1 connection.

Remark 1.2. All the previous works provide results also for the “divergence problem” when the working space is 𝕄^{m×2}, since any set of solenoidal matrix fields defines a set of gradients via right-multiplication by J := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

The problems are no longer equivalent if m ≥ n > 2. The right notion of connectedness which comes into play, in this case, is that of the so called rank-(n − 1) connectedness. Given A_{1}, A_{2} ∈ 𝕄^{m×n}, with m ≥ n and rank(A_{1} − A_{2}) ≤ n − 1, one can construct solenoidal matrix fields which take both the values A_{1} and A_{2} on a set of positive measure (indeed one can check that simple laminates work). In contrast, if rank(A_{1} − A_{2}) = n, the following rigidity result holds.

Proposition 1.3. (A. Garroni, V. Nesi, [10]). Let Ω be an open and connected set in ℝ^{n}. Let A_{1}, A_{2} ∈ 𝕄^{m×n}, with m ≥ n and rank(A_{1} − A_{2}) = n. Let B : Ω → {A_{1}, A_{2}} be a measurable function satisfying DivB = 0 in D′(Ω, ℝ^{m}). Then B is constant.

The case when K is made of two matrices has been completely solved and given a negative answer also for what concerns approximate solutions. The following proposition establishes rigidity for approximate solutions under the hypothesis of rank-(n − 1) disconnectedness.

Proposition 1.4. (A. Garroni, V. Nesi, [10]). Let Ω be a bounded open and connected set in ℝ^{n}, and let K = {A_{1}, A_{2}} ⊂ 𝕄^{m×n}, m ≥ n ≥ 1, be such that rank(A_{1} − A_{2}) = n. Let B_{h} be a sequence weakly convergent to B in L^{p}(Ω, M^{m×n}), with p > 1, such that

DivB_{h} → 0 strongly in W^{−1,p}(Ω, ℝ^{m})

and

dist(B_{h}, K) → 0 in measure .

Then

B_{h} → A_{1} or B_{h} → A_{2} in measure .

So far everything seems to be parallel the “gradient problem”, but the case when K consists of three matrices turns out to be different. Indeed one can construct a sequence of matrix fields which are divergence free and whose distance from the set K approaches zero. In other words, approximate solutions exist for a suitable choice of {A_{1}, A_{2}, A_{3}}. The following result clarifies the situation.

4 MARIAPIA PALOMBARO AND MARCELLO PONSIGLIONE

Lemma 1.5. (A. Garroni, V. Nesi, [10]). Given m ≥ n ≥ 3, there exist three pairwise rank-n connected m × n matrices A_{1}, A_{2}, A_{3}, and there exists a sequence B_{h} ∈ L^{∞}(Ω, 𝕄^{m×n}) such that setting K = {A_{1}, A_{2}, A_{3}}, one has

(1.3)  dist(B_{h}, K) → 0  strongly in L^{p}(Ω), ∀p ≥ 1,
(1.4)  DivB_{h} → 0  strongly in W^{−1,p}(Ω; ℝ^{m}), ∀p ≥ 1,

and B_{h} ⇀^{∗} B in L^{∞}, with B ≠ A_{i} for any i = 1, 2, 3.

Remark 1.6. (A. Garroni, V. Nesi, [10]). In Lemma 1.5 one can achieve the stronger requirement DivB_{h} = 0 rather then (1.4), by suitably projecting the fields B_{h} onto Divergence-free matrix fields.

The explicit formula for A_{1}, A_{2}, A_{3}, can be found in the last section of this paper (see Remark 4.3).

Next, we state the main theorem of our paper, namely a rigidity result for three-valued matrix fields under the assumption of rank-(n − 1) disconnectedness.

Theorem 1.7. Let Ω ⊂ ℝ^{n} be a connected open set and let K = {A_{1}, A_{2}, A_{3}} ⊂ 𝕄^{m×n}, with m ≥ n and rank(A_{i} − A_{j}) = n for i ≠ j. If B : Ω → K is a measurable function satisfying DivB = 0 in D′(Ω, ℝ^{m}), then B is constant.

In this paper we will mainly deal with the problem of non-existence of exact solutions. In addition to the previous theorem, we prove a rigidity result for a particular class of matrix fields taking an arbitrary number of values. This is the precise statement.

Theorem 1.8. Let Ω ⊂ ℝ^{n} be a connected open set and let K ⊂ 𝕄^{m×n} be bounded, with m ≥ n and rank(A_{i} − A_{j}) = n for every A_{i}, A_{j} ∈ K, with i ≠ j. Suppose that K satisfies the following condition: there exist n − 1 independent hyperplanes π_{1}, …, π_{n−1} in ℝ^{n}, and n − 1 vector subspaces τ_{1}, …, τ_{n−1} of dimension n − 1 in ℝ^{m}, such that

(1.5) A_{i} : π_{r} → A_{i}(π_{r}) ⊆ τ_{r} , for every A_{i} ∈ K and r = 1, … n − 1 .

Then every measurable matrix field B : Ω → K satisfying DivB = 0 in D′(Ω, ℝ^{m}) is constant.

We remark that Theorem 1.8, specialized to the case m = n = 2, reduces itself to a result which is well-known in the setting of the gradient problem ( see [8], Lemma 5). An easy corollary of this theorem is that a set of simultaneously diagonalizable rank-n connected matrices is rigid for exact solutions.

The study of more general linear differential constraints on the matrix field B is just beginning. The interested reader is referred to [5] for results in this direction.

The plan of the present paper is as follows.

In Section 2, we present an algebraic argument which implies that the right dimension to study the problem is n × n (see Lemma 2.4). Next we remark that the condition of being divergence-free is invariant under any orthogonal change of variables (Remark 2.2). Using this invariance, in order to prove Theorem 1.7 it is enough to consider a very special situation. This kind of argument does not work for an arbitrary number of matrices and one does in fact expect that rigidity fails for a sufficiently large number of them. Yet, under the assumptions of Theorem 1.8, one can prove that rigidity still holds for an arbitrary number of values and actually even for a continuum of them. For the reader’s convenience we give, in Lemma 2.1, the Gauss-Green formula for L^{∞} fields, which will be the main ingredient in the proof of this result.

In Section 3, we give the proofs of Theorem 1.7 and Theorem 1.8.

The final section departs from the main focus of the paper. Indeed, in the spirit of Lemma 1.5, we address the problem of finding approximate solutions to the “three divergence problem”. More precisely, we show that the construction used by Garroni and Nesi actually applies to a larger class of sets K. Theorem 1.9 gives a characterization of all such K’s, which turn out to be non-rigid for approximate solutions.

Theorem 1.9. For every q_{1}, q_{2}, q_{3} ∈ (0, 1), let A ∈ 𝕄^{3×3} be defined as follows

A = \frac{1}{q_{3}} [(1 − ∏_{i=1}^{3}(1 − q_{i})) G^{−1} ( \begin{matrix}λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{3} \end{matrix} ) G − q_{2}(1 − q_{3})I] ,

where λ_{1} = 0 , λ_{2} = 1/(1 − q_{1}) , λ_{3} = q_{2}/(q_{1} + q_{2} − q_{1}q_{2}) , and G is an arbitrary matrix in GL(3). Then, for every M ∈ 𝕄^{3×3} and N ∈ GL(3), the set

K = {M , N + M , NA + M}

is non-rigid for approximate solutions.

2. Preliminaries

In this Section we set some notations and a few preliminary results needed in the proof of the main results of Section 3. Throughout this paper Ω is an open connected subset of ℝ^{n}. We denote by 𝕄^{m×n} the set of the real m × n matrices; 0 and I will indicate the zero matrix and the identity matrix in 𝕄^{n×n} respectively. Left-multiplication of A times a vector v ∈ ℝ^{n} is denoted by A · v. The symbol ⟨v, w⟩ denotes the standard inner product in ℝ^{n}.

6 MARIAPIA PALOMBARO AND MARCELLO PONSIGLIONE

For every measurable subset E of ℝ^{n}, |E| is the n-dimensional Lebesgue measure of E while, for s ∈ ℝ^{+}, H^{s}(E) is its s-dimensional Hausdorff measure. Given a function f ∈ L^{1}(Ω, ℝ), we say that x ∈ Ω is a Lebesgue point for f, and that λ(x) ∈ ℝ is the Lebesgue value of f at x, if

lim_{r→0} − ∫_{B_{r}(x)} |f(y) − λ(x)| dy = 0,

where B_{r}(x) is the open ball of radius r and center x and the symbol − ∫_{B_{r}(x)} stands for \frac{1}{|B_{r}(x)|} ∫_{B_{r}(x)}. This definition extends in the obvious way to vector valued functions. It is well-known that the set of Lebesgue points for f, which from now on we will denote by 𝕃(f), has full measure in Ω. For every k ∈ ℕ and f ∈ L^{1}(Ω, ℝ^{k}), we will denote by f̃ a Lebesgue representative of f (i.e. f̃(x) = λ(x) for every x ∈ 𝕃(f)), so that f̃ coincides with f a.e. in Ω. For more details we refer the reader to [9].

Recall that a vector field f ∈ L^{∞}(Ω, ℝ^{n}) is said to be divergence free if for every φ ∈ C_{0}^{∞}(Ω)

∫_{Ω}⟨f(x), ∇φ(x)⟩ dx = 0.

For the reader’s convenience, we prove a Gauss-Green formula in the particular setting of our problem (much more general results can be found in [1]). In the sequel the symbol ν(x) will denote the outward normal to a given surface at the point x.

Lemma 2.1. Let f ∈ L^{∞}(Ω, ℝ^{n}) be a divergence free vector field, and let U ⊂ \overline{U} ⊂ Ω be open with Lipschitz boundary. Suppose that H^{n−1}(𝕃(f) ∩ ∂U) = H^{n−1}(∂U). Then

∫_{∂U}⟨f̃(s), ν(s)⟩ dH^{n−1}(s) = 0 .

Proof. Consider a sequence {ρ_{n}} of mollifiers and set f_{n} := f ∗ ρ_{n}. We have

divf_{n} = divf ∗ ρ_{n} = 0.

The standard Gauss-Green formula for smooth functions on a Lipschitz domain yields

∫_{∂U}⟨f_{n}(s), ν(s)⟩ dH^{n−1}(s) = 0. (2.1)

It is easy to see that f_{n}(x) → f̃(x) for all x ∈ ∂U ∩ 𝕃(f). Passing to the limit in (2.1) and using the dominated convergence Theorem, we get

∫_{∂U}⟨f̃(s), ν(s)⟩ dH^{n−1}(s) = 0.

Remark 2.2. Let B ∈ L^{∞}(Ω, 𝕄^{m×n}) be a divergence free matrix field and let R be an orthogonal matrix in 𝕄^{n×n}. Using a convolution argument as in Lemma 2.1 and the classical chain rule formula for smooth functions, one can check that for every C ∈ 𝕄^{m×n} and F ∈ 𝕄^{n×m}, the matrix field B̂ : {y ∈ ℝ^{n} | y = R^{T} x, x ∈ Ω} → 𝕄^{n×n} defined by

B̂(y) := R^{T} F (B(Ry) + C)R (2.2)

is divergence free.

The next lemma shows that, given a set K ⊂ 𝕄^{m×n}, the property of rank-(n − 1) disconnectedness is preserved under left multiplication by suitable matrices in 𝕄^{n×m}.

Lemma 2.3. Let K ⊂ 𝕄^{m×n} be at most countable, with m > n and rank(A_{i}−A_{j}) = n for every A_{i}, A_{j} ∈ K, with i ≠ j. Then there exists F ∈ 𝕄^{n×m} such that rank(F A_{i} −F A_{j}) = n for every A_{i}, A_{j} ∈ K with i ≠ j.

Proof. Since the image of A_{i} − A_{j} is a n-dimensional subspace of ℝ^{m}, we can always find a linear operator F : ℝ^{m} → ℝ^{n} such that Ker(F (A_{i} − A_{j})) = Ker(F) ∩ Im(A_{i} − A_{j}) = {0} for every i ≠ j. Then rank(F A_{i} − F A_{j}) = n for every A_{i}, A_{j} ∈ K with i ≠ j.

The previous results show that, as long as we consider discrete valued matrix fields, we can always set the problem in the space of square matrices. This is the claim of the next Lemma.

Lemma 2.4. Let K ⊂ 𝕄^{m×n} be at most countable, with m > n and rank(A_{i} − A_{j}) = n for every A_{i}, A_{j} ∈ K, with i ≠ j. Suppose we are given a divergence free matrix field B : Ω → K. Then there exist a divergence free matrix field B′ : Ω → K′, where K′ ⊂ 𝕄^{n×n}, card(K′) = card(K), and rank(A′_{i} − A′_{j}) = n for every A′_{i}, A′_{j} ∈ K′, with i ≠ j.

Proof. Apply Lemma 2.3 and use Remark 2.2 with R = I and C = 0.

3. Proofs of the main results

In this section we give the proofs of Theorem 1.7 and Theorem 1.8.

Proof of Theorem 1.7. By Lemma (2.4) it is enough to consider the case m = n > 2. Moreover, due to the local character of our problem, we can make any convenient change of variables. Hence, as it is customary in this kind of problems, we begin with some reductions to special cases. We use Remark 2.2 choosing F = (A_{2} − A_{1})^{−1} and C = −A_{1}. In this way we can assume that A_{1} = 0 and A_{2} = I. Moreover, since for any linear operator in ℝ^{n} there exists a two-dimensional invariant subspace, we can choose the orthogonal matrix R in (2.2) so that A_{3} is of the form

A_{3} := A = ( \begin{matrix}a_{11} & a_{12} & 0 & ⋯ & 0 \\ a_{21} & a_{22} & 0 & ⋯ & 0 \\ a_{31} & a_{32} & a_{33} & ⋯ & a_{3n} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{n1} & a_{n2} & a_{n3} & ⋯ & a_{nn} \end{matrix} ) . (3.1)

and rank(A) = rank(A − I) = n. By assumptions we can write

B = χ_{E_{1}}0 + χ_{E_{2}}I + χ_{E_{3}}A ,

where E_{i} are disjoint measurable sets and E_{1} ∪ E_{2} ∪ E_{3} = Ω. Next remark that, due to (3.1), the first two equations of DivB = 0 only involve derivatives with respect to directions e_{1} and e_{2}. Roughly speaking, the idea is to use these informations to conclude that B does not depend on the variables x_{1} and x_{2}. This allows us to make a “projection” of the original problem into a lower dimensional space (actually a section of Ω) and to proceed by induction on the dimension n, using, at the final step, that by Theorem 1.1 and Remark 1.2, rigidity for exact solutions holds for n = 2.

Let us proceed with the formal proof. Set n > 3 and suppose that rigidity holds in 𝕄^{(n−1)×(n−1)}. We want to prove that B is constant. For every x ∈ Ω, let Q be a open coordinate cube centered in x and such that \overline{Q} ⊂ Ω. Without loss of generality we assume that x = 0, so that Q = (−l, l)^{n} for some positive l.

Let {ρ_{i}} be a sequence of mollifiers and set B_{i} := ρ_{i} ∗ B. For i large enough we have that B_{i} is well defined in Q, B_{i} ∈ C^{∞}(Q, 𝕄^{n×n}) and DivB_{i} = 0 in Q in the classical sense. Then fix \overline{x} := (\overline{x}_{3}, ⋯, \overline{x}_{n}) ∈ (−l, l)^{n−2} such that for H^{2}-a.e. (x_{1}, x_{2}) ∈ (−l, l)^{2}, (x_{1}, x_{2}, \overline{x}) ∈ 𝕃(B). By Fubini’s Theorem, this is possible for H^{n−2}-a.e. \overline{x} ∈ (−l, l)^{n−2}, since, as already remarked, 𝕃(B) has full measure in Ω. Now consider the field \underline{B} : (−l, l)^{2} → 𝕄^{2×2} defined by

\underline{B}(x_{1}, x_{2}) := ( \begin{matrix}b̃_{11} & b̃_{12} \\ b̃_{21} & b̃_{22} \end{matrix} ) (x_{1}, x_{2}, \overline{x}), (3.2)

where b̃_{kl} is a Lebesgue representative of the kl-entry of the matrix B. Then let b_{kl}^{i} be the kl-entry of B_{i} and set

\underline{B_{i}}(x_{1}, x_{2}) := ( \begin{matrix}b_{11}^{i} & b_{12}^{i} \\ b_{21}^{i} & b_{22}^{i} \end{matrix} ) (x_{1}, x_{2}, \overline{x}). (3.3)

Since \underline{B}_{i} ∈ C^{∞}((−l, l)^{2}, 𝕄^{2×2}) and Div \underline{B}_{i} = 0 in (−l, l)^{2}, we have

∫_{(−l,l)^{2}} \underline{B}_{i} · ∇φ dx_{1}dx_{2} = 0 ∀φ ∈ C_{0}^{∞}(−l, l)^{2}. (3.4)

THE THREE DIVERGENCE FREE MATRIX FIELDS PROBLEM 9

Moreover \underline{B}_{i}(x_{1}, x_{2}) converges to \underline{B}(x_{1}, x_{2}) at every (x_{1}, x_{2}) ∈ (−l, l)^{2} such that (x_{1}, x_{2}, \overline{x}) ∈ 𝕃(B). Then passing to the limit for i → ∞ in (3.4) and using the dominated convergence Theorem we get

∫_{(−l,l)^{2}} \underline{B}(x_{1}, x_{2}) · ∇φ(x_{1}, x_{2}) dx_{1}dx_{2} = 0 ∀φ ∈ C_{0}^{∞}(−l, l)^{2}. (3.5)

By Theorem 1.1 we have that \underline{B} is constant and hence B̃ is constant on the section (x_{1}, x_{2}, \overline{x}). Since this is true for H^{n−2}-a.e. \overline{x} ∈ (−l, l)^{n−2}, we deduce that B̃ does not depend on (x_{1}, x_{2}) in Q. In particular we have

\frac{∂χ_{E_{i}}}{∂x_{1}} = 0 in D′(Q). (3.6)

From (3.6), it is easy to see that there exist three measurable sets E′_{1}, E′_{2}, E′_{3} in (−l, l)^{n−1} such that

E_{i} = (−l, l) × E′_{i} a.e. (3.7)

Now call 0′, I′, A′ the n × (n − 1)-minors of the matrices 0, I, A respectively, obtained by eliminating the first column of each matrix. Notice that rank(A′) = rank(A′ −I′) = n−1. Then set

B′ := 0′χ_{E′_{1}} + I′χ_{E′_{2}} + A′χ_{E′_{3}} .

Let us emphasize that B′ leaves in a space of dimension n − 1. Combining (3.6) with the equation DivB = 0 in D′(Q, ℝ^{n}), one concludes that B′ satisfies

DivB′ = 0 in D′((−l, l)^{n−1}, ℝ^{n}) . (3.8)

By Lemma 2.3, there exists F ∈ M^{(n−1)×n} such that

(3.9) rank(F I′) = rank(F A′ − F I′) = n − 1 .

Then set

A′_{2} := F I′ ,
A′_{3} := F A′ ,
B_{(n−1)} := 0χ_{E′_{1}} + A′_{2}χ_{E′_{2}} + A′_{3}χ_{E′_{3}} .

By (3.8) and (3.9), it follows that B_{(n−1)} is an exact solution of the problem

DivB_{(n−1)} = 0 in D′((−l, l)^{n−1}, ℝ^{n−1}) , B_{(n−1)} ∈ {0, A′_{2}, A′_{3}} ⊂ 𝕄^{(n−1)×(n−1)} . (3.10)

By the inductive assumption, problem (3.10) is rigid. Then |(−l, l)^{n−1}| = |E′_{i}| for some i and hence, by (3.7), |Q| = |E_{i}|. By the arbitrariness of Q we conclude that B is constant.

10 MARIAPIA PALOMBARO AND MARCELLO PONSIGLIONE

Remark 3.1. Before giving the proof of Theorem 1.8, we make some considerations about its assumption (1.5). We want to show that, even in this case, we can reduce to a very special situation. Indeed, condition (1.5) implies that Im(A_{i}) = Im(A_{j}) for every A_{i}, A_{j} ∈ K. We can thus apply the same argument as in Lemma 2.4, although no requirement is made on the cardinality of the set K. More precisely, we fix any two of the matrices in K, say A_{1} and A_{2}, and choose a matrix F ∈ 𝕄^{n×m} such that F (A_{1} − A_{2}) = I. For such an F, we have that F A_{i} ∈ 𝕄^{n×n} and rank(F A_{i}−F A_{j}) = n for every A_{i}, A_{j} ∈ K. Moreover, the hyperplanes π_{1}, …, π_{n−1} are preserved under the action of every F A_{i}.

We are now ready to prove Theorem 1.8.

Proof of Theorem 1.8. By Remark 2.2 and Remark 3.1, we can assume that K ⊂ 𝕄^{n×n} and A_{i}(π_{r}) ⊆ π_{r} for every A_{i} ∈ K and r = 1, …, n − 1. For every r, let v_{r} be the unit vector orthogonal to π_{r}. We want to prove that B does not depend on any of the directions v_{r}, which are independent by assumptions. Then B would only depend on one direction, but the condition of being divergence free will imply that B is constant. We will only check the statement for one vector v_{r}. Choose any of the vectors v_{r} and a real number q, and by contradiction assume that there exist two points P, Q ∈ 𝕃(B), with P − Q = qv_{r} and such that B̃(P) ≠ B̃(Q); for instance suppose that B̃(P) = A_{1} and B̃(Q) = A_{2}.

Let us briefly digress to explain the idea of the proof in an informal way. We want to apply the Gauss-Green formula in a small cylinder with axis parallel to v_{r} and bases centered in P and Q respectively. Computing the flux of B through the boundary of the cylinder we will check that its contribution through the bases can never compensate the contribution through the lateral boundary, the former being a vector parallel to (A_{1} − A_{2})v_{r}, the latter belonging to the hyperplane π_{r}. This idea, however, will require some technical efforts, as B may be non-constant on the bases. What we do, in fact, is to consider a sequence of cylinders with vanishing radii. The contradiction will arise for a sufficiently small radius.

Now we continue the formal proof. To simplify the notations we will assume that P = (0, …, 0) and v_{r} = e_{n}, so that Q = (0, …, 0, q). Let ρ(x) := (x_{1}^{2} + ⋯ + x_{n−1}^{2})^{1/2} and, for every r ∈ ℝ^{+}, set

C(r) := {x ∈ ℝ^{n} : 0 ≤ ρ(x) ≤ r, 0 ≤ x_{n} ≤ q}.

Since B̃(P) = A_{1} and B̃(Q) = A_{2}, we can find a cylinder on whose bases the mean value of B is A′_{1} and A′_{2} respectively, with |A_{1} − A′_{1}| and |A_{2} − A′_{2}| arbitrarily small (this can be checked by using Fubini’s theorem). More precisely, for every given δ > 0, we can find a radius r_{δ} ∈ ℝ^{+} and a vector w_{δ} ∈ ℝ^{n} such that, setting

C_{δ} := C(r_{δ}) + w_{δ}

Q + w_{δ}^{O + w_{δ}}^{r_{δ}} D_{δ}^{2} D_{δ}^{1} L_{δ}  ^{e_{n} = v_{r}}  ^{ν(s)}

Figure 1. The cylinder C_{δ}

and denoting by D_{δ}^{1} and D_{δ}^{2} the bases of C_{δ} and by L_{δ} its lateral boundary (see Figure 1), the following hold:

C_{δ} ⊂ Ω ,

H^{n−1}(∂C_{δ} ∩ 𝕃(B)) = H^{n−1}(∂C_{δ}) ,

− ∫_{D_{δ}^{1}} |B̃(s) − A_{1}|dH^{n−1}(s) + − ∫_{D_{δ}^{2}} |B̃(s) − A_{2}|dH^{n−1}(s) < δ . (3.11)

By Lemma 2.1 we get

∫_{∂C_{δ}} B̃(s) · ν(s) dH^{n−1}(s) = 0 . (3.12) We write (3.12) as the sum of three contributions as follows

∫_{∂C_{δ}} B̃(s) · ν(s) dH^{n−1}(s) = ∫_{D_{δ}^{1}} B̃(s) · (−e_{n}) dH^{n−1}(s) +

∫_{D_{δ}^{2}} B̃(s) · e_{n} dH^{n−1}(s) + ∫_{L_{δ}} B̃(s) · ν(s) dH^{n−1}(s) , (3.13) where ν(s) = \frac{1}{ρ(s)}(s_{1}, …, s_{n−1}, 0) on L_{δ}. On the other hand we have

∫_{D_{δ}^{1}} B̃(s) · (−e_{n}) dH^{n−1}(s) = H^{n−1}(D_{δ}^{1}) A_{1}^{δ} · (−e_{n}) ,
(3.14)  ∫_{D_{δ}^{2}} B̃(s) · e_{n} dH^{n−1}(s) = H^{n−1}(D_{δ}^{1}) A_{2}^{δ} · e_{n} ,

12 MARIAPIA PALOMBARO AND MARCELLO PONSIGLIONE

where A_{1}^{δ}, A_{2}^{δ} ∈ 𝕄^{n×n} and, by (3.11), are such that |A_{1}^{δ} − A_{1}| + |A_{2}^{δ} − A_{2}| < δ. Then we set

u_{δ} := ∫_{L_{δ}} B̃(s) · ν(s) dH^{n−1}(s) .

Dividing the right hand side in (3.13) by H^{n−1}(D_{δ}^{1}) and using (3.12) and (3.14), we obtain

(A_{2}^{δ} − A_{1}^{δ})e_{n} + \frac{u_{δ}}{H^{n−1}(D_{δ}^{1})} = 0 . (3.15)

Now recall that e_{n} is orthogonal to π_{r}, rank(A_{2} − A_{1}) = n and (A_{2} − A_{1})(π_{r}) = π_{r}. It follows that (A_{2} − A_{1}) · e_{n} ∉ π_{r}, and hence (A_{2}^{δ} − A_{1}^{δ}) · e_{n} ∉ π_{r} for δ small enough. On the other hand we have that u_{δ} ∈ π_{r}, the hyperplane π_{r} being preserved under the action of every A_{i} ∈ K. Then, for sufficiently small δ, (3.15) gives a contradiction.

Remark 3.2. The assumption of boundedness required for the set K, in Theorem 1.8, can be actually removed. The previous proof, indeed, can be adapted to L_{loc}^{1} matrix fields by suitable modifications.

Remark 3.3. Theorem 1.8, specialized to the case m = n = 2, gives a rigidity result which was already known in the setting of the “gradient problem” (see [8], Lemma 5).

4. Approximate solutions

In this section we give a refinement of Lemma 1.5 in [10] in which the authors give an explicit example of a set which is non-rigid for approximate solutions. Their construction is set in 𝕄^{3×3}, but it can be extended to the case m, n ≥ 3 by slight modifications. It actually provides an algorithm (similar to that of Tartar’s for the gradients, [17]), which allows us to find approximate solutions for a large class of sets K. Similar constructions can be found in the works of several authors, see [2], [7], [12], [13], [15]. The particular case here resembles the construction in [12]. The key point is the following lemma.

Lemma 4.1. Let K = {A_{1}, A_{2}, A_{3}} ⊂ 𝕄^{3×3} be a set of pairwise rank-3 connected matrices. If there exist three matrices S_{1}, S_{2}, S_{3} ∈ 𝕄^{3×3} which satisfy the conditions

det(A_{i} − S_{i}) = 0, for i = 1, 2, 3 , (4.1) S_{i} = q_{i−1}A_{i−1} + (1 − q_{i−1})S_{i−1} mod 3 , for i = 1, 2, 3 , (4.2)

for some q_{i} ∈ (0, 1), then the set K is non-rigid for approximate solutions.

The proof of this lemma relies on works on multiple scales. We refer to [3] and [6] for a general treatment and to [10] and [17] for the case of interest here. We simply remark that condition (4.1) is used to laminate A_{i} and S_{i} in some direction belonging to Ker(A_{i} − S_{i}), while (4.2) is used to construct a sequence which “approaches” the set K in the sense of

THE THREE DIVERGENCE FREE MATRIX FIELDS PROBLEM 13

Lemma 1.5. This is the strategy used in [10] where the authors make an explicit choice of the matrices A_{i} and S_{i}. Theorem 1.9, given in the introduction, characterizes all possible triples {A_{1}, A_{2}, A_{3}} which one can obtain in this way, and it is a corollary of the following proposition.

Proposition 4.2. Let q_{1}, q_{2}, q_{3} ∈ (0, 1) be given. Let A_{1} = 0 and A_{2} = I in 𝕄^{3×3}. Then there exist S_{1}, S_{2}, S_{3}, A_{3} ∈ 𝕄^{3×3} satisfying conditions (4.1) and (4.2) of Lemma 4.1, if and only if A_{3} is of the form

A_{3} = \frac{1}{q_{3}} [(1 − ∏_{i=1}^{3}(1 − q_{i})) G^{−1} ( \begin{matrix}λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{3} \end{matrix} ) G − q_{2}(1 − q_{3})I] , (4.3)

where G is an arbitrary matrix in GL(3) and the λ_{i}’s are defined as follows

λ_{1} = 0 , λ_{2} = \frac{1}{1 − q_{1}} , λ_{3} = \frac{q_{2}}{q_{1} + q_{2} − q_{1}q_{2}} .

Proof. We rewrite (4.2) more explicitly:

S_{2} = q_{1}A_{1} + (1 − q_{1})S_{1} = (1 − q_{1})S_{1},

S_{3} = q_{2}A_{2} + (1 − q_{2})S_{2} = q_{2}I + (1 − q_{2})(1 − q_{1})S_{1}, (4.4)

S_{1} = q_{3}A_{3} + (1 − q_{3})S_{3} = q_{3}A_{3} + q_{2}(1 − q_{3})I + (1 − q_{1})(1 − q_{2})(1 − q_{3})S_{1} .

Now let λ_{i} be the eigenvalues of S_{1}, then by (4.1) and (4.4) we get

det(A_{1} − S_{1}) = 0 ⟺ det(−S_{1}) = 0 ⟺ λ_{1} = 0 ,

det(A_{2} − S_{2}) = 0 ⟺ det(I − (1 − q_{1})S_{1}) = 0 ⟺ λ_{2} = \frac{1}{1 − q_{1}} .

Moreover one can check that

det(A_{3} − S_{3}) = 0 ⟺ det[(q_{1} + q_{2} − q_{1}q_{2})S_{1} − q_{2}I] = 0 ⟺ λ_{3} = \frac{q_{2}}{q_{1} + q_{2} − q_{1}q_{2}} .

Note that the λ_{i}’s are all distinct, since q_{1}, q_{2}, q_{3} ∈ (0, 1). Therefore the matrix S_{1} is diagonalizable. Hence for any S_{1} ∈ {G^{−1}diag(λ_{1}, λ_{2}, λ_{3})G, G ∈ GL(3)}, the matrices A_{3}, S_{2}, S_{3} are uniquely determined by (4.4). In particular A_{3} is of the form (4.3). Conversely, for any A_{3} of the form (4.3), the matrices S_{1}, S_{2}, S_{3} are uniquely determined and conditions (4.1) and (4.2) are satisfied.

Proof of Theorem 1.9. Let A, M, N ∈ 𝕄^{3×3} satisfy the assumptions of Theorem 1.9. Without loss of generality we can assume that M = 0 and N = I. Notice that A is of the form (4.3). Then, by Proposition 4.2 and Lemma 4.1, the set K = {0, I, A} is non-rigid for approximate solutions.

Remark 4.3. If we choose G = I in (4.3) the set K reduces itself to that given in [10], that is

A_{3} = diag(\frac{−q_{2}(1 − q_{3})}{q_{3}} , \frac{q_{1} + q_{3} − q_{1}q_{3}}{q_{3}(1 − q_{1})} , \frac{q_{2}}{q_{1} + q_{2} − q_{1}q_{2}} ) ,

S_{1} = diag(0, \frac{1}{1 − q_{1}} , \frac{q_{2}}{q_{1} + q_{2} − q_{1}q_{2}} ) ,

S_{2} = diag(0, 1, \frac{q_{2}(1 − q_{2})}{q_{1} + q_{2} − q_{1}q_{2}} ) ,

S_{3} = diag(q_{2}, 1, \frac{q_{2}}{q_{1} + q_{2} − q_{1}q_{2}} ) .

Acknowledgements

The authors wish to thank Enzo Nesi for having proposed the subject of this paper, and for many interesting discussions and suggestions.

References

[1] G. Anzellotti, On the existence of the rates of stress and displacement for Prandtl-Reuss plasticity, Quart. Appl. Math. 41 (1983/84), no.2, 181-208.

[2] R. Aumann, S. Hart, Bi-convexity and bi-martingales, Israel J. Math. 54 (1986), 159-180.

[3] G. Allaire, M. Briane, Multiscale convergence and reiterated homogenization, Proc. Roy. Edinburgh A 126 (1996), 297-342.

[4] J. Ball, R. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., 100 (1987), 13-52.

[5] M. Barchiesi, Inclusioni differenziali per EDP: rigidità e mancanza di rigidità, Tesi di Laurea, Dipartimento di Matematica, Università di Roma“La Sapienza”, 2003.

[6] M. Briane, Corrector for the homogenization of a laminate, Adv. Math. Sci. Appl., 45 (1994), 357-379.

[7] E. Casadio Tarabusi, An algebraic characterization of quasi-convex functions, Ricerche Mat. 42 no. 1 (1993), 11-24.

[8] M. Chlebik, B. Kirchheim, Rigidity for the four gradient problem, J. Reine und Angew. Math., 551 (2002), 1-9.

[9] L. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press (1992).

[10] A. Garroni, V. Nesi, Rigidity and lack of rigidity for solenoidal matrix fields, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. (to appear).

[11] B. Kirchheim, D. Preiss, Construction of Lipschitz mappings with finitely many non rank-one connected gradients, in preparation.

[12] V. Nesi, G. W. Milton, Polycrystalline configurations that maximize electrical resistivity, J. Mech. Phys. Solids, 39 no.4 (1991), 525-542.

[13] V. Scheffer, Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities. Dissertation, Princeton University, 1974.

[14] V. Šverák, New examples of quasiconvex functions Arch. Rat. Mech. Anal., 119 (1992), 293-300.

[15] D. R. S. Talbot, J. R. Willis, Variational principles for inhomogeneous nonlinear media, MA J. Appl. Math. 35 no. 1 (1985), 39-54.

[16] L. Tartar, A note on separately convex functions (II), Note 18, Carnegie-Mellon University, 1987.

[17] L. Tartar, Some remarks on separately convex functions, Microstructure and phase transition, 191-204, IMA Vol. Math. Appl., 54, Springer, New York, 1993.

(Mariapia Palombaro) Dipartimento di Matematica, Università “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy

E-mail address: palombar@mat.uniroma1.it.

(Marcello Ponsiglione) S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy

E-mail address: ponsigli@sissa.it