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

Phase transitions and minimal hypersurfaces in hyperbolic space]

The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behaviour at infinity. Then, we study the limit as the length scale tends to zero through a $\Gamma$-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in M. Anderson and U. Lang using geometric measure theory

Stability of some unilateral free-discontinuity problems in two-dimensional domains

The purpose of this paper is to study the stability of some unilateral free-discontinuity problems, under perturbations of the discontinuity sets in the Hausdorff metric.Comment: 16 page