Elliptic Problems with Power Singularities on the Right-Hand Sides. Degenerate Elliptic Problems

Abstract

In this chapter, we study the elliptic problem (7.1.1)-(7.1.2) (or, in abstract notation, (7.1.3)) in the case where the function $$ F\left( x \right) = \left( {f\left( x \right),\phi _1 \left( x \right), \ldots ,\phi _m \left( x \right)} \right) $$ may have arbitrary power singularities in the vicinities of manifolds of various dimensionalities. In our investigation, we use the following procedure: The function F(x) is replaced by its regularization F, i.e., by a certain element of the space \(\tilde K_{s,p} \) with some s <0. As a result, we arrive at a situation already encountered in our study of Green’s functions. The theorem on complete collection of isomorphisms enables us to determine and study solutions of the considered problem near the manifold of singularities of the right-hand sides. Note that the regularization F of the function F(x) is determined ambiguously. Thus, even in the case where the defect of problem (7.1.3) is equal to zero \(\left( {\eta = 0\,{\text{and}}\,\eta {\text{* = 0}}} \right)\), the problem with power singularities on the right-hand sides admits numerous solutions. To choose a unique solution, it is necessary to impose additional restrictions.