Global Continuation via Higher-Gradient Regularization and Singular Limits in Forced One-Dimensional Phase Transitions

Abstract

We consider a standard "higher-gradient" model for forced phase transitions in one-dimensional, shape-memory solids. We prescribe a parameter-dependent body forcing. The component of the potential energy corresponding to conventional elasticity is characterized by a nonconvex stored energy function of the strain. Our main goal is to show that global solution branches of the regularized problem converge to a global branch of weak solutions in the limit of vanishing "capillarity" (the coefficient of the higher-gradient term). The existence of global branches for the regularized, semilinear problem is routine, based upon the Leray--Schauder degree. In the physically meaningful case when the body force is everywhere nonnegative, we obtain uniform a priori bounds via a subtle maximum principle. This together with topological connectivity arguments yields the existence of global branches of weak solutions to the zero-capillarity problem. Moreover, by examining the singular limits of various supplementary conservation laws (satisfied by all solutions of the regularized problem), we show that the above-mentioned weak solutions also minimize the potential energy of the zero-capillarity problem.