Partial differential equations in Sobolev spaces with or without weights.

摘要

First, we study traces of functions in Sobolev-Slobodeckij spaces with or without weights. We show that the well-known trace theorem for weighted Sobolev spaces holds true under minimal regularity assumptions on the domain. Using this result, we prove the existence of a bounded linear right inverse of the trace operator for Sobolev-Slobodeckij spaces Wsp (O) when s - 1/p is an integer.; Second, we introduce another weighted Sobolev spaces H&d15;gp,q (O) and study some of their properties. Then we prove a trace theorem and interpolation theorem for the spaces H&d15;gp,q (O). Using the trace theorem, we solve non-zero boundary value problems for elliptic equations in weighted Sobolev spaces H&d15;gp,q (O).; Third, we prove the unique solvability of second order elliptic equations in the usual Sobolev spaces when the coefficients of second order terms are allowed to have discontinuity at finitely many parallel hyper-planes in Rd . We find solutions the derivatives of which satisfy a given jump condition at the hyper-planes.; Finally, we discuss parabolic equations with leading coefficients being discontinuous at a hyper-plane in Rx Rd . Then by using the unique solvability of such parabolic equations, we establish the weak uniqueness of diffusion processes with time-dependent discontinuous coefficients.