Uniqueness of weighted Sobolev spaces with weakly differentiable weights

Jonas M. Tölle*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

13 Citations (Scopus)

Abstract

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) = V1,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 Lqloc(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(-βq-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.

Original languageEnglish
Pages (from-to)3195-3223
Number of pages29
JournalJournal of Functional Analysis
Volume263
Issue number10
DOIs
Publication statusPublished - 15 Nov 2012
MoE publication typeA1 Journal article-refereed

Keywords

  • Density of smooth functions
  • H=W
  • Nonlinear degenerate parabolic equation
  • Nonlinear Kolmogorov operator
  • p-Laplace operator
  • Poincaré inequality
  • Smooth approximation
  • Weighted p-Laplacian evolution
  • Weighted Sobolev spaces

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