Some microstructures in three dimensions (Q2776391)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^3\) and \(W^{1,\infty}(\Omega)\) mean the Lipschitz continuous functions from \(\Omega\) into \(\mathbb{R}^3\). The authors consider minimization problems of the type: \(\inf_{u\in W^{1,\infty}_0(\Omega)} \int_\Omega \varphi(\nabla v) dx\), or \(\inf_{v\in W^{1,\infty}(\Omega)}\int_\Omega(\varphi(\nabla v)+|v|^2) dx\), where \(\varphi\) is a function which admits four wells at \(3\times 3\) matrices \(W_i\) which are chosen to be diagonal but incompatible, that is satisfying \(\text{rank}(W_i- W_j)> 1\). Using the quasi-convex envelope \(Q\varphi\) of \(\varphi\), the authors prove the non-existence of a solution of the above minimization problems. Moreover, any bounded minimizing sequence generates a Young measure which can be explicitely computed in terms of Dirac measures at these wells.NEWLINENEWLINENEWLINEThe rest of the paper is devoted to the numerical approximation of these minimization problems replacing \(W^{1,\infty}(\Omega)\) (or \(W^{1,\infty}_0(\Omega)\)) by the finite element space \(V^h\) (or \(V^h_0\)) of continuous and piecewise affine functions on a triangulation of \(\Omega\). The authors prove the existence of minimizers and build minimizing sequences using the classical ``saw-tooth'' functions. They prove energy estimates with respect to the mesh size for these minimizing sequences.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00022].