Two sharp Poincaré inequalities in $W^{1,\infty}(\Omega)$ for convex domains $\Omega \subset \mathbb{R}^d$.

Abstract: Two sharp Poincaré inequalities $$ ||u-Lu||_{L^\infty(\Omega)} \leq C(\Omega)||\nabla u||_{L^{\infty}(\Omega)}$$ are proved for Lipschitz functions $u$ on convex domains $\Omega \subset \mathbb{R}^d$. Here, $Lu$ is either the integral mean $\langle u \rangle$ of $u$ over $\Omega$ or its median value $\mathrm{med} \, u$. The sharp constants in both cases can be interpreted geometrically, and the case that $Lu = \langle u \rangle$ we prove that (i) among convex domains of equal measure balls have the best, i.e. the smallest, Poincar\'{e} constants, and (ii) since $C(\Omega)$ is the maximum of a certain convex function $\zeta$, calculating $C(\Omega)$ for polygonal convex domains is a matter of ordering the valus $\zeta(V_i)$ where $V_1,\ldots,V_n$ are the vertices of $\Omega$. In this context, we introduce the idea of `magic points' and explore evidence for their existence in concrete cases. This talk is based on joint works with J. Deane and S. Zelik.

MathematicsPhysics

Audience: researchers in the topic

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor)