The three divergence free matrix fields problem

We prove that for any connected open set $\Omega\subset \R^n$ and for any set of matrices $K=\{A_1,A_2,A_3\}\subset M^{m\times n}$, with $m\ge n$ and rank$(A_i-A_j)=n$ for $i\neq j$, there is no non-constant solution $B\in L^{\infty}(\Omega,M^{m\times n})$, called exact solution, to the problem Div B=0 \quad \text{in} D'(\Omega,\R^m) \quad \text{and} \quad B(x)\in K \text{a.e. in} \Omega. In contrast, A. Garroni and V. Nesi \cite{GN} 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.Comment: 15 pages, 1 figur

On a differential inclusion related to the Born-Infeld equations

We study a partial differential relation that arises in the context of the Born-Infeld equations (an extension of the Maxwell's equations) by using Gromov's method of convex integration in the setting of divergence free fields

Rank-(n – 1) convexity and quasiconvexity for divergence free fields

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Gradient integrability and rigidity results for two-phase conductivities in two dimensions

This paper deals with higher gradient integrability for σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of div(σ∇u)=0 in dimension two. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and σ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries. We focus also on two-phase conductivities, i.e., conductivities assuming only two matrix values, σ1 and σ2, and study the higher integrability of the corresponding gradient field |∇u| for this special but very significant class. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement described by the sets Ei=σ−1(σi). We find the optimal integrability exponent of the gradient field corresponding to any pair {σ1,σ2} of elliptic matrices, i.e., the worst among all possible microgeometries. We also treat the unconstrained case when an arbitrary but finite number of phases are present

Dislocations in nanowire heterostructures: from discrete to continuum

We discuss an atomistic model for heterogeneous nanowires, allowing for dislocations at the interface. We study the limit as the atomic distance converges to zero, considering simultaneously a dimension reduction and the passage from discrete to continuum. Employing the notion of Gamma-convergence, we establish the minimal energies associated to defect-free configurations and configurations with dislocations at the interface, respectively. It turns out that dislocations are favoured if the thickness of the wire is sufficiently large