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

João Paulo Dias, Wladimir Neves, José Francisco Rodrigues

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.

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

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.

Paper Structure

This paper contains 7 sections, 8 theorems, 189 equations.

Table of Contents

  1. Introduction and the Physical Problem
  2. Structural and physical hypothesis
  3. Statement of the Main Results
  4. Entropy inequality and Continuity of Solutions
  5. Regularisation and Penalisation Approximation
  6. One obstacle string problem
  7. Two obstacles string problem

Key Result

Theorem 2.1

Under the structural hypothesis H1-H3, and initial data for the initial conditions with $u_0\geq 0$ in $\mathbb R$, there exists a complementary weak solution $u\in C(0,T;C^{0,\alpha}_{loc}(\mathbb R))$, with $0<\alpha<1/2$, in the sense of Defintion DefCompOne. Moreover, this solution is also a var

Theorems & Definitions (22)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1: One Obstacle String Problem
  • Theorem 2.2: Two Obstacles String Problem
  • Definition 2.5
  • Theorem 2.3
  • proof
  • ...and 12 more