Exterior boundary-value Poincare problem for elliptic systems of the second order with two independent variables

TL;DR

This paper investigates the conditions under which elliptic systems of second-order partial differential equations with two variables are normally solvable in exterior boundary-value problems, highlighting exceptions and specific cases where Noether's theorems apply.

Contribution

It demonstrates that uniform ellipticity does not always guarantee normal solvability and identifies conditions under which systems are normally solvable, especially for decomposed elliptic systems.

Findings

Uniform ellipticity does not always imply normal solvability.

Certain classes of elliptic systems are normally solvable under additional conditions.

Noether theorems hold for decomposed elliptic systems in exterior regions.

Abstract

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid.