Local minimizers and planar interfaces in a phase-transition model with interfacial energy

Abstract

Interfacial energy is often incorporated into variational solid-solid phase transition models via a perturbation of the elastic energy functional involving second gradients of the deformation. We study consequences of such higher-gradient terms for local minimizers and for interfaces. First it is shown that at slightly sub-critical temperatures, a phase which globally minimizes the elastic energy density at super-critical temperatures is an L ¹-local minimizer of the functional including interfacial energy, whereas it is typically only a W ^1,∞-local minimizer of the purely elastic functional. The second part deals with the existence and uniqueness of smooth interfaces between different wells of the multi-well elastic energy density. Attention is focussed on so-called planar interfaces, for which the deformation depends on a single direction x · N and the deformation gradient then satisfies a rank-one ansatz of the form ${Dy(x) = A + u(x \cdot N) \otimes N}$ , where A and ${B=A+a \otimes N}$ are the gradients connected by the interface.