The polynomial property of elliptic problems and an algebraic description of their attributes
The polynomial property of a self-adjoint system of differential operators means that the corresponding sesquilinear form degenerates on a finite-dimensional subspace of polynomials only. Almost all stationary problems in mathematical physics provide an example of such a system. In the lecture it will be shown that this simple property implies that the boundary value problem is elliptic while the kernel and co-kernel of the corresponding operator in Sobolev and Hlder spaces can be described in terms of the polynomial subspace for different kind of domains such as bounded domains, with smooth or piecewise smooth boundaries, and unbounded ones, with conical, cylindrical and quasi-cylindrical outlets to infinity.
Serguei Nazarov, St. Petersburg University, Russia
May 19, 2000