On proper formulation of boundary condition for degenerated PDEs when trace embedding theorems are missing and application to nonhomogeneous BVPs

摘要

We propose certain interpretation of boundary conditions f = g on partial derivative Omega, the boundary of Omega, when f belongs to the weighted Sobolev space W-rho(1,p) (Omega) subordinated to the integrable weight rho(x) and g is defined on partial derivative Omega, where Omega subset of R-n is a given domain. It may happen that the trace embedding theorems are missing. Our proposed interpretation requires to have defined an extension operator Ext : X --> W-rho(1,p) (Omega), where X is the relevant function space of functions defined on partial derivative Omega. We show that the proposed interpretation does not depend on the choice of the extension operator Ext. Moreover, we apply our result to obtain existence and uniqueness of solutions to an example of the boundary value problem involving degenerated p-Laplacian with nonhomogeneous boundary condition. Existence of the extension operator within the class of weights rho(x) = tau( dist (x, partial derivative Omega)) when X = W-omega(1-1/p,p) (partial derivative Omega) is certain weighted Slobodetskii space and Omega is a Lipschitz boundary domain is confirmed by our previous results.