• Haïm BREZIS
• Moshe MARCUS

Code(s) de Classification MSC:

• 26D10 Inequalities involving derivatives and differential and integral operators

Résumé: On montre que pour tout domaine $\Omega \subset \R^n$, borné
régulier, il existe une constante réelle
$\lambda^\star \in \R$ telle que $\int_\Omega |\nabla u|^2 \geq {1
\over 4} \int_\Omega (u^2/\delta^2) + \lambda^\star \int_\Omega
u^2, \quad \forall u \in H^1_0(\Omega),$ où $\delta(x) =$
dist$(x,\partial\Omega)$. Si $\Omega$ est convexe on montre que $\lambda^\star > 0$.


Abstract: We prove that for any smooth bounded domain $\Omega$ in
$\R^n$ there is a constant $\lambda^\star \in \R$ such that
$\int_\Omega |\nabla u|^2 \geq {1 \over 4} \int_\Omega
(u^2/\delta^2) + \lambda^\star \int_\Omega u^2, \quad \forall u
\in H^1_0(\Omega),$ where $\delta(x) =$
dist$(x,\partial\Omega)$. If $\Omega$ is convex we prove that $\lambda^\star > 0$.

Mots Clés: inégalité de Hardy

Date: 1998-01-01