Exterior extension problems for strongly elliptic operators: solvability and approximation using fundamental solutions

TL;DR

This paper investigates the solvability and approximation of three exterior extension problems for strongly elliptic PDEs, demonstrating the effectiveness of boundary integral methods for numerical solutions.

Contribution

It introduces new results on the existence, dense solvability, and approximation of solutions using fundamental solutions for a broad class of elliptic systems.

Findings

Solutions can be approximated by single layer potentials.

The problems are densely solvable and conditionally well-posed.

Results support the use of boundary integral methods for numerical solutions.

Abstract

In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of the Sobolev spaces over a domain with relatively smooth boundaries. We consider the existence of solutions to these problems, the dense solvability and conditional well-posedness of these problems for a wide class of strongly elliptic systems. We also consider the approximation of solutions to these problems by a single layer potential and by a linear combination of "discrete" fundamental solutions in relation to a narrower class of strongly elliptic operators of the second order. The obtained results justify the applicability of the indirect method of boundary integral equations and for numerical solving the exterior extension problems.