Weighted Sobolev spaces for elliptic equations on non-smooth domains
Henguang Li

Abstract: It is well known that elliptic equations have singular solutions in Sobolev spaces due to non-smooth points on the boundary, changes of boundary conditions, and discontinuities of the coefficients. The presence of the singularity causes difficulties both in the theoretical analysis and in the practical computation. In this talk, we discuss our analysis on the well-posedness, regularity, and the Fredholm property for elliptic equations of this type in a class of weighted Sobolev spaces. With a careful selection of the index for the Fredholm property, we show the uniqueness of the solution, and no loss of regularity for the singular solution in these weighted Sobolev spaces. Applications of this technique on a Schroedinger-type operator and on numerical methods will be mentioned.