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Singular Klein-Gordon equation on a bounded domain

Michael Ruzhansky, Alibek Yeskermessuly

TL;DR

The paper studies the wave equation with a spatial potential $V(x)$, initial data, and nonhomogeneous Dirichlet boundary data on a bounded domain, allowing singular coefficients and data. It develops a rigorous very weak solution framework using trace and extension domain theory to handle singularities and boundary conditions, and proves existence, uniqueness, and consistency results. The approach combines Galerkin approximations and energy estimates for weak solutions under homogeneous boundaries, extends to nonhomogeneous boundaries via trace extensions, and introduces $L^{\infty}$- and $H^1$-moderate regularizations to define very weak solutions, showing they converge to classical solutions when regular data are available. These results broaden well-posedness theory for hyperbolic equations with singular coefficients and irregular boundaries, with potential applications in PDEs where distributions and point sources arise.

Abstract

In this paper, we consider the wave equation for the Laplace operator with potential, initial data, and nonhomogeneous Dirichlet boundary condition. We establish a weak solution by using traces and extension domains. We also establish the existence, uniqueness and consistency of the very weak solution for the wave equation with singularities in the potential, initial data, source term, boundary and boundary condition.

Singular Klein-Gordon equation on a bounded domain

TL;DR

The paper studies the wave equation with a spatial potential , initial data, and nonhomogeneous Dirichlet boundary data on a bounded domain, allowing singular coefficients and data. It develops a rigorous very weak solution framework using trace and extension domain theory to handle singularities and boundary conditions, and proves existence, uniqueness, and consistency results. The approach combines Galerkin approximations and energy estimates for weak solutions under homogeneous boundaries, extends to nonhomogeneous boundaries via trace extensions, and introduces - and -moderate regularizations to define very weak solutions, showing they converge to classical solutions when regular data are available. These results broaden well-posedness theory for hyperbolic equations with singular coefficients and irregular boundaries, with potential applications in PDEs where distributions and point sources arise.

Abstract

In this paper, we consider the wave equation for the Laplace operator with potential, initial data, and nonhomogeneous Dirichlet boundary condition. We establish a weak solution by using traces and extension domains. We also establish the existence, uniqueness and consistency of the very weak solution for the wave equation with singularities in the potential, initial data, source term, boundary and boundary condition.
Paper Structure (7 sections, 12 theorems, 127 equations)

This paper contains 7 sections, 12 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Traces and extension domains
  3. Homogeneous Dirichlet boundary condition
  4. Galerkin approximations
  5. Energy estimates
  6. Non-homogeneous Dirichlet condition
  7. Very weak solutions

Key Result

Theorem 2.1

Let $1<p<\infty$, $k\in \mathbb{N}^*$ be fixed. Let $\Omega$ be an admissible domain in $\mathbb{R}^n$. Then, for $\beta=k-(n-d)/p>0$, the following trace operators are linear continuous and surjective with linear bounded right inverse, i.e. with bounded extension operators $E: B^{p,p}_\beta (\partial\Omega)\to W^k_p(\mathbb{R}^n)$, $E_\Omega:W^k_p(\Omega)\to W^k_p(\mathbb{R}^n)$ and $E_{\partial

Theorems & Definitions (40)

  • Definition 1: $W^k_p$-extension domains JW-84, Af-Roz
  • Definition 2: $d$-measure JW-84
  • Definition 3: Ahlfors $d$-regular set or $d$-set JW-84, Af-Roz
  • Definition 4: Markov's local inequality JW-84, Af-Roz
  • Definition 5: JW-84, Af-Roz, DRP-22
  • Definition 6
  • Definition 7: Admissible domain DRP-22, Af-Roz
  • Theorem 2.1: JW-84
  • Definition 8: JW-84
  • Definition 9: JW-84
  • ...and 30 more