Relaxation of some multi-well problems

Relaxation of Some Multi { Well Problemskaushik Bhattacharya

Mathematical models of phase transitions in solids lead to the variational problem, minimize R W (Du)dx where W has a multi-well structure: W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e., for K = denotes the (3 2)-matrix formed with the rst two columns of U i. We characterize generalized convex hulls, including the...

Multiscale Modelling and Computation of Microstructures in Multi-well Problems

A multiscale model and numerical method for computing microstructures with large and inhomogeneous deformation is established, in which the microscopic and macroscopic information is recovered by coupling the finite order rank-one convex envelope and the finite element method. The method is capable of computing microstructures which are locally finite order laminates. Numerical experiments on a...

Boundary Conditions for Multi-dimensional Hyperbolic Relaxation Problems

Coupling Requirements for Well Posed and Stable Multi-physics Problems

Abstract. We discuss well-posedness and stability of multi-physics problems by studying a model problem. By applying the energy method, boundary and interface conditions are derived such that the continuous and semi-discrete problem are well-posed and stable. The numerical scheme is implemented using high order finite difference operators on summation-by-parts (SBP) form and weakly imposed boun...

Bmo and Uniform Estimates for Multi–well Problems

We establish optimal local regularity results for vector-valued extremals and minimizers of variational integrals whose integrand is the squared distance function to a compact setK in matrix space MN×n. The optimality is illustrated by explicit examples showing that, in the nonconvex case, minimizers need not be locally Lipschitz. This is in contrast to the case when the set K is suitably conve...