Boundary value problems for the Laplacian in convex and semiconvex domains

Dorina Mitrea*, Marius Mitrea, Lixin Yan

*Corresponding author for this work

    Research output: Contribution to journalArticle

    32 Citations (Scopus)

    Abstract

    We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in Rn, when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel-Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates. As a corollary, sharp estimates for the Green potential are obtained in a variety of contexts, including local Hardy spaces. A substantial part of this analysis applies to bounded semiconvex domains (i.e., Lipschitz domains satisfying a uniform exterior ball condition).

    Original languageEnglish
    Pages (from-to)2507-2585
    Number of pages79
    JournalJournal of Functional Analysis
    Volume258
    Issue number8
    DOIs
    Publication statusPublished - 15 Apr 2010

    Keywords

    • Besov and Triebel-Lizorkin spaces
    • Convex domain
    • Green operator
    • Laplacian
    • Lipschitz domain satisfying a uniform exterior ball condition
    • Nontangential maximal function
    • Poisson problem
    • Semiconvex domain

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