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On the tangential gradient of the kernel of the double layer potential

M. Lanza de Cristoforis

Abstract

In this paper we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we deduce the validity of a continuity property of the double layer potential in Hölder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator.

On the tangential gradient of the kernel of the double layer potential

Abstract

In this paper we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we deduce the validity of a continuity property of the double layer potential in Hölder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator.
Paper Structure (5 sections, 11 theorems, 131 equations)

This paper contains 5 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Technical preliminaries on the differential operator
  3. The maximal function on the boundary of the kernel associated to a positively homogeneous function of degree $-(n-1)$
  4. Boundedness of the maximal function of the tangential gradient of the kernel of the double layer potential
  5. A classical lemma for the computation of the principal value

Key Result

Theorem 1.7

Let $\beta\in]0,1]$. Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,1}$. Let ${\mathbf{a}}$ be as in (introd0), (ellip), (symr). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Then the following statements hold.

Theorems & Definitions (11)

  • Theorem 1.7
  • Proposition 2.3
  • Lemma 2.6
  • Proposition 2.8
  • Lemma 2.12
  • Theorem 3.3
  • Theorem 3.12
  • Theorem 3.14
  • Proposition 3.17
  • Theorem 3.29
  • ...and 1 more