Hardy’s inequality for functions vanishing on a part of the boundary

Mardi 19 janvier 2016 14:00-15:00 - Moritz Egert - Université Paris-Sud

Résumé : Consider the usual Sobolev spaces $W^1,p$$(\Omega)$, $1<p<$$\infty$, on a bounded domain $\Omega$ in $\mathbbR^d$. To each closed subset $D$ of the boundary $\partial$$\Omega$ there corresponds a subspace $W_D^1,p$$(\Omega)$ defined as the closure of all smooth functions whose support stays away from $D$. These spaces naturally occur, for example, in the study of second-order divergence-form equations with a Dirichlet condition on $D$ and a Neumann condition on the rest of the boundary.
In my talk I will discuss the validity of Hardy’s inequality
$
\int_\Omega \frac|u(x)|^p\mathrmdist(x, D)^p ; d x \leq c \int_\Omega |\nabla u(x)|^p ; d x
$
for functions $u \in W_D^1,p$$(\Omega)$. A main result is that if $D$ is sufficiently thick in terms of Hausdorff content and the rest of the boundary of $\Omega$ can be described by Lipschitz coordinates, then this inequality holds true. In fact, $W_D^1,p$$(\Omega)$ then is the largest subspace of $W^1,p$$(\Omega)$ on which the left-hand side of Hardy’s inequality is finite. These results yield an intrinsic characterization of the Sobolev spaces with partially vanishing traces that can be used, for example, to set up a coherent interpolation theory on non-smooth domains $\Omega$ where most of the usual approaches fail.

Lieu : Salle 113-115 (Bâtiment 425)

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