Differential equations in spaces with asymptotics on manifolds with cusp singularities.

*(English. Russian original)*Zbl 1068.58014
Differ. Equ. 38, No. 12, 1764-1773 (2002); translation from Differ. Uravn. 38, No. 12, 1664-1672 (2002).

Summary: Traditionally, elliptic operators on smooth manifolds are considered in classical Sobolev spaces. The weighted version of these spaces is used in elliptic theory on manifolds with singularities. At the same time, in the latter case, one can consider equations in quite different spaces, e.g., in spaces of formal series given by expansions in special function systems associated with the singularity type.

A deformation of these two opposite situations (Sobolev spaces and resurgent functions) naturally leads to spaces of intermediate type, which are spaces where a function is defined by a finite segment of its asymptotic expansion in a neighborhood of a singularity and by the fact that it belongs to some Sobolev space. The advantage of this type of spaces is that, as a result, we not only arrive at the conclusion that the solution belongs to a specific type of space, but also obtain its asymptotics in a neighborhood of the singularity. The paper reports on an investigation of elliptic equations in such spaces (which are referred to as spaces with asymptotics) on manifolds with cusp singularities.

A deformation of these two opposite situations (Sobolev spaces and resurgent functions) naturally leads to spaces of intermediate type, which are spaces where a function is defined by a finite segment of its asymptotic expansion in a neighborhood of a singularity and by the fact that it belongs to some Sobolev space. The advantage of this type of spaces is that, as a result, we not only arrive at the conclusion that the solution belongs to a specific type of space, but also obtain its asymptotics in a neighborhood of the singularity. The paper reports on an investigation of elliptic equations in such spaces (which are referred to as spaces with asymptotics) on manifolds with cusp singularities.