Pseudodifferentia Equations on Manifolds with Boundary: Fredholm Property and Asymptotic

Abstract

The main purpose of the present paper is the investigation of systems of pseudo-differential equations (PsDEs) with symbols from extended Hormander classes on a manifold with smooth boundary. Equations are treated in anisotropic Bessel potential spaces with weight (BPSwW). Theorem about factorization of symbols, proved earlier by E. Shamir, R. Duduchava and E. Shargorodsky is revised and general criteria is obtained for PsDEs in BPSwW on manifolds with smooth boundary to possess the Fredholm property. It is proved that the criteria is invariant with respect to the weight exponents and the conormal smoothness parameter, which participate in the definition of the spaces. In the second part of the paper results of G. Eskin and J. Bennish on asymptotic of solutions to systems of PsDEs (L2–theory) are extended and complete asymptotic expansion of a solution to near the boundary is obtained (Lp–theory). More precise description of exponents and of logarithmic terms of the expansion is presented. Investigations are carried out in connection with problems arising in elasticity (crack problems) and some other fields of mathematical physics when the potential method is applied. In forthcoming papers asymptotic of a function represented by a potential will be presented when asymptotic of a density on the boundary of the domain is known.