Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A nonvariational form of the acoustic single layer potential

M. Lanza de Cristoforis

TL;DR

The paper develops a nonvariational framework for the acoustic single layer potential to address the Helmholtz Neumann problem with boundary data in the distributional space $V^{-1,\alpha}(\partial\Omega)$ and solutions in $C^{0,\alpha}(\overline{\Omega})_\Delta$, where the Laplacian lies in $C^{-1,\alpha}(\overline{\Omega})$. It establishes the mapping properties of the acoustic volume and single layer potentials for such densities, proves distributional jump formulas for the single layer potential, and derives a distributional quasi-symmetrization principle that connects layer potentials and their transposes. A key result is the compactness of the boundary operator $W_\Omega^t[S_{n,\lambda},\cdot]$ on $V^{-1,\alpha}(\partial\Omega)$, enabling a Fredholm-type analysis of boundary integral formulations in this nonvariational setting. Collectively, these results lay the groundwork for existence and uniqueness results for Helmholtz Neumann problems with rough boundary data and pave the way for exterior-domain treatments in follow-up work. The framework extends classical potential theory to negative-exponent Schauder spaces, accommodating Hölder-continuous solutions with potentially infinite Dirichlet integrals near the boundary.

Abstract

We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$ and the space $V^{-1,α}(\partialΩ)$ of (distributional) normal derivatives on the boundary of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in the Schauder space with negative exponent $C^{-1,α}(\overlineΩ)$. Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in $Ω$ with boundary data in $V^{-1,α}(\partialΩ)$ and solutions in the space of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in $C^{-1,α}(\overlineΩ)$, \textit{i.e.}, in a space of functions that may have infinite Dirichlet integral. Namely, a Neumann problem that does not belong to the classical variational setting.

A nonvariational form of the acoustic single layer potential

TL;DR

The paper develops a nonvariational framework for the acoustic single layer potential to address the Helmholtz Neumann problem with boundary data in the distributional space and solutions in , where the Laplacian lies in . It establishes the mapping properties of the acoustic volume and single layer potentials for such densities, proves distributional jump formulas for the single layer potential, and derives a distributional quasi-symmetrization principle that connects layer potentials and their transposes. A key result is the compactness of the boundary operator on , enabling a Fredholm-type analysis of boundary integral formulations in this nonvariational setting. Collectively, these results lay the groundwork for existence and uniqueness results for Helmholtz Neumann problems with rough boundary data and pave the way for exterior-domain treatments in follow-up work. The framework extends classical potential theory to negative-exponent Schauder spaces, accommodating Hölder-continuous solutions with potentially infinite Dirichlet integrals near the boundary.

Abstract

We consider a bounded open subset of of class for some and the space of (distributional) normal derivatives on the boundary of -Hölder continuous functions in that have Laplace operator in the Schauder space with negative exponent . Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in with boundary data in and solutions in the space of -Hölder continuous functions in that have Laplace operator in , \textit{i.e.}, in a space of functions that may have infinite Dirichlet integral. Namely, a Neumann problem that does not belong to the classical variational setting.

Paper Structure

This paper contains 9 sections, 30 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. A distributional outward unit normal derivative for the functions of $C^{0,\alpha}_{ {\mathrm{loc}} }(\overline{\Omega^-})_\Delta$
  4. Preliminaries to the acoustic volume potential
  5. Acoustic layer potentials with densities in the space $V^{-1,\alpha}(\partial\Omega)$
  6. Jump formulas for the acoustic single layer potential
  7. A distributional form of a quasi-sym-metri-zation principle
  8. A compactness result for $W_\Omega^t[S_{n,\lambda} ,\cdot]$ in $V^{-1,\alpha}(\partial\Omega)$
  9. Appendix

Key Result

Proposition 2.3

Let $\alpha\in]0,1[$. Let $\Omega$ be a bounded open Lipschitz subset of ${\mathbb{R}}^{n}$. There exists one and only one linear and continuous extension operator $E^\sharp_\Omega$ from $C^{-1,\alpha}(\overline{\Omega})$ to $\left(C^{1,\alpha}(\overline{\Omega})\right)'$ such that for all $f= f_{0}+\sum_{j=1}^{n}\frac{\partial}{\partial x_{j}}f_{j}\in C^{-1,\alpha}(\overline{\Omega})$. Moreover

Theorems & Definitions (39)

  • Definition 2.1
  • Proposition 2.3
  • Lemma 2.7
  • Definition 2.8
  • Lemma 2.10
  • Lemma 2.13
  • Lemma 2.14
  • Lemma 2.17
  • Definition 2.18
  • Theorem 2.21
  • ...and 29 more