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The volume potential for elliptic differential operators in Schauder spaces

M. Lanza de Cristoforis

TL;DR

This work develops a comprehensive regularity theory for volume potentials associated with the fundamental solution of a second-order elliptic operator with constant coefficients in Schauder spaces, including negative-order cases. It provides a detailed structural expansion of the fundamental solution, introduces an extension framework $E^\sharp$ to treat densities in $C^{-1,\alpha}(\overline{\Omega})$, and proves continuity results upgrading regularity by two derivatives (to $C^{m+2,\alpha}$ or the generalized $C^{m+2,\omega_1}$ when $\alpha=1$). A key technical tool is a new integral-operator lemma for kernels in $\mathcal{K}^{m,\alpha}_h$, establishing mapping properties into Hölder and $\omega_1$-Hölder spaces and enabling the extension of Miranda65-type results to nonhomogeneous operators. The results cover both interior and exterior volume potentials and include limiting cases, with potential applications to Neumann and Poisson-type problems in elliptic PDE potential theory. Overall, the paper broadens classical Schauder-space continuity results to negative-exponent densities and nonhomogeneous operators, enriching the potential-theoretic toolkit for elliptic PDEs.

Abstract

The aim of this paper is to prove continuity results for the volume potential corresponding to the fundamental solution of a second order differential operator with constant coefficients in Schauder spaces of negative exponent and to generalize some classical results in Schauder spaces of positive exponents.

The volume potential for elliptic differential operators in Schauder spaces

TL;DR

This work develops a comprehensive regularity theory for volume potentials associated with the fundamental solution of a second-order elliptic operator with constant coefficients in Schauder spaces, including negative-order cases. It provides a detailed structural expansion of the fundamental solution, introduces an extension framework to treat densities in , and proves continuity results upgrading regularity by two derivatives (to or the generalized when ). A key technical tool is a new integral-operator lemma for kernels in , establishing mapping properties into Hölder and -Hölder spaces and enabling the extension of Miranda65-type results to nonhomogeneous operators. The results cover both interior and exterior volume potentials and include limiting cases, with potential applications to Neumann and Poisson-type problems in elliptic PDE potential theory. Overall, the paper broadens classical Schauder-space continuity results to negative-exponent densities and nonhomogeneous operators, enriching the potential-theoretic toolkit for elliptic PDEs.

Abstract

The aim of this paper is to prove continuity results for the volume potential corresponding to the fundamental solution of a second order differential operator with constant coefficients in Schauder spaces of negative exponent and to generalize some classical results in Schauder spaces of positive exponents.
Paper Structure (9 sections, 27 theorems, 164 equations)

This paper contains 9 sections, 27 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Preliminaries on the fundamental solution
  4. The distributional volume potential
  5. A technical lemma on an integral operator
  6. The volume potential with density in a Schauder space of positive exponent
  7. The volume potential with density in a Schauder space of negative exponent
  8. Appendix: a formula of integration by parts
  9. Appendix: a limiting case of two theorems of C. Miranda

Key Result

Lemma 2.4

Let $\alpha\in ]0,1]$. Let $\Omega$ be a bounded open Lipschitz subset of ${\mathbb{R}}^{n}$. Then the canonical inclusion ${\mathcal{J}}$ from the Lebesgue space $L^1(\Omega)$ of integrable functions in $\Omega$ to $(C^{1,\alpha}(\overline{\Omega}))'$ that takes $f$ to the functional ${\mathcal{J}} is linear continuous and injective.

Theorems & Definitions (32)

  • Remark 2.2
  • Lemma 2.4
  • Definition 2.6
  • Proposition 2.9
  • Lemma 2.11
  • Lemma 2.12
  • Proposition 2.16
  • Proposition 2.18
  • Proposition 2.22
  • Proposition 2.23
  • ...and 22 more