A survey on the boundary behavior of the double layer potential in Schauder spaces in the frame of an abstract approach

TL;DR

This paper reviews the boundary behavior of the double layer potential operator for elliptic PDEs in Schauder spaces, consolidating known results and extending them through an abstract metric space framework.

Contribution

It compiles existing continuity results for the double layer potential in Schauder spaces and demonstrates their derivation via an abstract metric space approach.

Findings

Summarizes known boundary continuity properties of the double layer potential.

Shows that many properties can be deduced from abstract metric space results.

Includes non-doubling measure cases in the analysis.

Abstract

We provide a summary of the continuity properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in H\"{o}lder and Schauder spaces on the boundary of a bounded open subset of ${\mathbb{R}}^n$. The purpose is two-fold. On one hand we try present in a single paper all the known continuity results on the topic with the best known exponents in a H\"{o}lder and Schauder space setting and on the other hand we show that many of the properties we present can be deduced by applying results that hold in an abstract setting of metric spaces with a measure that satisfies certain growth conditions that include non-doubling measures as in a series of papers by Garc\'{\i}a-Cuerva and Gatto in the frame of H\"{o}lder spaces and later by the author.