Asymptotic analysis of a second-order singular perturbation model for phase transitions

TL;DR

This paper studies the asymptotic behavior of a family of energies related to phase transitions in nonlinear second-order materials, establishing convergence to a sharp interface model and bounds on oscillations of minimizers.

Contribution

It introduces a new nonlinear interpolation inequality and provides improved bounds on parameters preventing oscillations in minimizers for a specific energy model.

Findings

Existence of a positive constant k_0 for Gamma-convergence to a sharp interface functional.

Upper bounds on k for which minimizers do not oscillate on fine scales.

Improved estimates over previous results by Mizel, Peletier, and Troy.

Abstract

We consider the problem of the asymptotic description of a family of energies introduced by Coleman and Mizel in the theory of nonlinear second-order materials depending on an extra parameter k. By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k_0 such that, for k<k_0, these energies Gamma-converge to a sharp interface functional. Moreover, for a special choice of the potential term in the energy, we provide an upper bound on the values of k such that minimizers cannot develop oscillations on some fine scale, thus improving previous estimates by Mizel, Peletier and Troy.