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 H1,p Rd,w dx = V 1,p Rd,w dx =W1,p Rd,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;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 padmissible, 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. © 2012 Elsevier Inc. All rights reserved.