A+ CATEGORY SCIENTIFIC UNIT

Weighted inequalities for gradients on non-smooth domains

Volume 471 / 2010

Caroline Sweezy, J. Michael Wilson Dissertationes Mathematicae 471 (2010), 1-53 MSC: 35J25, 42B25, 42B37. DOI: 10.4064/dm471-0-1

Abstract

We prove sufficiency of conditions on pairs of measures $\mu $ and $\nu $, defined respectively on the interior and the boundary of a bounded Lipschitz domain $\Omega $ in $d$-dimensional Euclidean space, which ensure that, if $u$ is the solution of the Dirichlet problem. $$\eqalign{ \Delta u &=0\quad\ \hbox{in }\Omega, \cr u\vert _{\partial \Omega } &=f, \cr}$$% with $f$ belonging to a reasonable test class, then $$ \bigg( \int_{\Omega }|\nabla u|^{q}\,d\mu \bigg) ^{1/q}\leq \bigg( \int_{\partial \Omega }|f|^{p}\,d\nu \bigg) ^{1/p}, $$ where $1

Authors

  • Caroline SweezyDepartment of Mathematical Sciences
    New Mexico State University
    Box 30001, 3MB
    Las Cruces, New Mexico 88003-8801, USA
    e-mail
  • J. Michael WilsonDepartment of Mathematics and Statistics
    University of Vermont
    16 Colchester Ave.
    Burlington, VT 05405, USA
    e-mail

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