Classical Solutions for Nonelliptic Euler–Lagrange Equations via Continuation

Abstract

We consider a higher-gradient model in one-dimensional nonlinear elasticity. Therefore we consider a physically reasonable stored-energy density W such that $W(\nu)$ goes to infinity for $\nu\searrow 0$ and $\nu\nearrow\infty$. We prescribe a parameter-dependent body-forcing. Our main goal is to show under a certain sign condition for the body-force that global solution branches of the singular perturbed problem converge to a global branch of weak solutions of vanishing capillarity. Moreover, the solution satisfies the first and the second Weierstrass–Erdmann corner conditions, and the strain field is uniformly pointwise positive.