Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation

TL;DR

This paper establishes the existence of weak solutions for obstacle problems involving quasilinear wave equations in one dimension, extending linear case results and addressing both one and two obstacle scenarios.

Contribution

It demonstrates the existence of continuous, variational weak solutions satisfying entropy conditions for quasilinear wave obstacle problems, extending prior linear results.

Findings

Existence of weak solutions for one obstacle quasilinear wave problems.

Solutions are continuous and satisfy a weak entropy condition.

Solutions are variational in a hyperbolic sense without viscosity.

Abstract

We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two obstacles problem. In contrast with the linear setting, for both strictly quasilinear cases we obtain continuous solutions in a weak complementary sense, which moreover satisfy a weak entropy condition in the free region where the string is not in contact with the obstacles. We further show that, in both the one and two obstacle cases, these solutions are variational solutions in a hyperbolic sense without the viscosity term.