A limiting case of a theorem of C. Miranda for layer potentials in Schauder spaces

TL;DR

This paper extends C. Miranda's theorem for layer potentials to a limiting case involving Schauder spaces, addressing the scenario where the open set and densities have specific regularity classes.

Contribution

It proves a new version of Miranda's theorem for layer potentials in Schauder spaces under limiting regularity conditions, using generalized Schauder spaces.

Findings

Established a theorem for layer potentials in limiting Schauder space cases.

Handled the case where the open set is of class C^{m,1} and densities are in C^{m-1,1} or C^{m,1}.

Introduced generalized Schauder spaces for the analysis.

Abstract

The aim of this paper is to prove a theorem of C.~Miranda for the single and double layer potential corresponding to the fundamental solution of a second order differential operator with constant coefficients in Schauder spaces in the limiting case in which the open set is of class $C^{m,1}$ and the densities are of class $C^{m-1,1}$ for the single layer potential and of class $C^{m,1}$ for the double layer potential for some nonzero natural number $m$. The treatment of the limiting case requires generalized Schauder spaces.