A real analytic family of fundamental solutions of elliptic partial differential operators with real constant coefficients

TL;DR

This paper constructs a real analytic family of fundamental solutions for elliptic PDEs with constant coefficients, highlighting their regularity properties and expressing them via real analytic functions of coefficients and spatial variables.

Contribution

It introduces a novel family of fundamental solutions expressed through real analytic functions, with detailed regularity analysis in Schauder spaces.

Findings

Family of solutions expressed by real analytic functions

Regularity properties established in Schauder spaces

Potential applications in solving elliptic PDEs

Abstract

We construct of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. We show regularity properties in the frame of Schauder spaces for the corresponding single layer potentials.