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Initial-boundary value problem for second order hyperbolic operator with mixed boundary conditions

Djamel Ait-Akli

Abstract

We deal with a linear hyperbolic differential operator of the second order on a bounded planar domain with a smooth boundary. We establish a well-posedness result in case where a mixed, Dirichlet-Neumann, condition is prescribed on the boundary. We focus on the case of a non-homogeneous Dirichlet data and a homogeneous Neumann one. The presented proof is based on a functional theoretical approach and on an approximation argument. Moreover, this work discuss an improvement of a result concerning the range of some operators related to the considered hyperbolic PDE yielding characterizations for the range space of these operators.

Initial-boundary value problem for second order hyperbolic operator with mixed boundary conditions

Abstract

We deal with a linear hyperbolic differential operator of the second order on a bounded planar domain with a smooth boundary. We establish a well-posedness result in case where a mixed, Dirichlet-Neumann, condition is prescribed on the boundary. We focus on the case of a non-homogeneous Dirichlet data and a homogeneous Neumann one. The presented proof is based on a functional theoretical approach and on an approximation argument. Moreover, this work discuss an improvement of a result concerning the range of some operators related to the considered hyperbolic PDE yielding characterizations for the range space of these operators.
Paper Structure (13 sections, 3 theorems, 84 equations)

This paper contains 13 sections, 3 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminary facts
  3. Auxiliary results
  4. Observation 1.
  5. Observation 2.
  6. Observation 3.
  7. Observation 4.
  8. Step 1: completeness of $\mathcal{D}$.
  9. Step 2: proof of Estimate (\ref{['propestim']}).
  10. Proof of the main result
  11. Step 1: construction of the approximating solution.
  12. Step 2: passing to the limit using Lemma \ref{['prop']}.
  13. Conclusion.

Key Result

Theorem 1.1

Let $f\in C^1([0,T]; L^2(\Omega))$. Assume that $(\Psi^0, \Psi^1)$ satisfy the weak boundary condition (wb). Let $G\in H^1((0,T)\times\Gamma_d)$ such that there exists $\tilde{G}\in H^1((0,T)\times\partial\Omega)$ satisfying: Let $\lbrace s_i\rbrace_i$ be defined as in (dn). We assume further that $G$ satisfy point-wisely a domination condition near $s_i$, that is: Then, there is a unique soluti

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • proof
  • proof
  • Remark 4.1
  • Remark 4.2