Weighted inequalities for gradients on non-smooth domains
Volume 471 / 2010
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
