Spectral boundary value problems and elliptic equations on manifolds with singularities.

*(English. Russian original)*Zbl 0962.58008
Dokl. Math. 60, No. 1, 97-99 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 367, No. 5, 597-599 (1999).

From the text: We describe a class of general boundary value problems, which we call spectral boundary value problems. Boundary conditions in these problems are set in the space of all boundary values by projectors onto a “positive subspace” defined by the conormal symbol (see below) of the elliptic operator in question, i.e., onto the subspace corresponding to the points of the spectrum of the conormal symbol with a positive imaginary part.

Spectral boundary value problems are closely related to equations on manifolds with singular points. Indeed, attaching a cone to the boundary of the manifold and extending the elliptic operator under consideration in a natural way, we obtain an elliptic operator on a manifold with conic singularities. It is found that the spectral boundary value problem for the original operator is equivalent to the corresponding equation for the constructed operator on the manifold with singularities in the weighted Sobolev spaces in the sense that natural isomorphisms exist between the kernels and cokernels of two described problems.

Spectral boundary value problems are closely related to equations on manifolds with singular points. Indeed, attaching a cone to the boundary of the manifold and extending the elliptic operator under consideration in a natural way, we obtain an elliptic operator on a manifold with conic singularities. It is found that the spectral boundary value problem for the original operator is equivalent to the corresponding equation for the constructed operator on the manifold with singularities in the weighted Sobolev spaces in the sense that natural isomorphisms exist between the kernels and cokernels of two described problems.

##### MSC:

58J32 | Boundary value problems on manifolds |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |