Second-Order Elliptic Equations on a Stratified Set

Abstract

A series of problems in mathematical physics connected with complex systems consisting of elements having different dimensionalities or different physical characteristics are conveniently modeled by boundary-value problems on stratified sets Ω. The approaches to equations on graphs described in other articles in this issue also allow development in this case. The geometry of a stratified set is notably more complex than the geometry of a graph (Ω is now a connected union of a finite number of manifolds of different dimensions), but, on the whole, the methodological principles developed for equations on graphs can be realized. Among these principles, the interpretation of all differential relations associated with the constituent elements of the set as a single equation on Ω of the divergent type is fundamental. Divergence, as in the classical case, is the density of the current of a vector field on Ω with respect to a special “stratified” measure. But to obtain substantial results, it is necessary to isolate a special class of so-called solid stratified sets. The nature of the results obtained is determined by the type of solidity of Ω. Two types of solidity of Ω are described in this paper. In connection with this, it is divided into two parts.