We give necessary and sufficient conditions for the equality in weighted Sobolev spaces. We also establish a Rellich-Kondrachov compactness theorem as well as a Lusin type approximation by Lipschitz functions in weighted Sobolev spaces.
In this paper I discuss two questions on -Laplacian type operators: I characterize sets that are removable for Hölder continuous solutions and then discuss the problem of existence and uniqueness of solutions to with zero boundary values; here is a Radon measure. The joining link between the problems is the use of equations involving measures.
We discuss the uniqueness of solutions to problems like ⎧ λ |u|s-1u - div (|∇u|p-2∇u) = μ on Ω, ⎨ ⎩ u = 0 in ∂Ω, where λ ≥ 0 and μ is a signed Radon measure.
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