Global continuation via higher-gradient regularization and singular limitsin forced one-dimensional phase transitions

We consider a standard higher-gradient model for forced phase transitions i n one-dimensional, shape-memory solids. We prescribe a parameter-dependent body forcing. The component of the potential energy corresponding to conven tional 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 li mit of vanishing capillarity (the coefficient of the higher-gradient term). The existence of global branches for the regularized, semilinear problem i s routine, based upon the Leray-Schauder degree. In the physically meaningf ul case when the body force is everywhere nonnegative, we obtain uniform a priori bounds via a subtle maximum principle. This together with topologica l connectivity arguments yields the existence of global branches of weak so lutions to the zero-capillarity problem. Moreover, by examining the singula r limits of various supplementary conservation laws (satisfied by all solut ions of the regularized problem), we show that the above-mentioned weak sol utions also minimize the potential energy of the zero-capillarity problem.