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 is H^1,p(R^d, w dx) = V^1,p(R^d, w dx) = W^1,p(R^d, w dx), where d∈N and p∈[1, ∞). If w admits a (weak) logarithmic gradient ∇w/w which is in L^q_loc(w dx; R^d), 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.