Uniqueness of weighted Sobolev spaces with weakly differentiable weights

We prove that weakly differentiable weights w which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order p-Sobolev space, that isH1,p(Rd,wdx)=V1,p(Rd,wdx)=W1,p(Rd,wdx), where d∈N and p∈[1,∞). If w admits a (weak) logarithmic gradient ∇w/w which is in Llocq(wdx;Rd), q=p/(p−1), we propose an alternative definition of the weighted p-Sobolev space based on an integration by parts formula involving ∇w/w. We prove that weights of the form exp(−β|⋅|q−W−V) are p-admissible, in particular, satisfy a Poincaré inequality, where β∈(0,∞), W, V are convex and bounded below such that |∇W| satisfies a growth condition (depending on β and q) and V is bounded. We apply the uniqueness result to weights of this type. The associated nonlinear degenerate evolution equation is also discussed.