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SUMMARY:Jonathan Bevan (University of Surrey)
DTSTART:20260420T134000Z
DTEND:20260420T151000Z
DTSTAMP:20260427T121927Z
UID:NSCM/205
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NSCM/205/">T
 wo sharp Poincaré inequalities in $W^{1\,\\infty}(\\Omega)$ for convex do
 mains $\\Omega \\subset \\mathbb{R}^d$.</a>\nby Jonathan Bevan (University
  of Surrey) as part of Nečas Seminar on Continuum Mechanics\n\nLecture he
 ld in Room K3\,  Faculty of Mathematics and Physics\, Charles University\,
  Sokolovská 83  Prague 8..\n\nAbstract\nTwo sharp Poincaré inequalities\
 n$$ ||u-Lu||_{L^\\infty(\\Omega)} \\leq C(\\Omega)||\\nabla u||_{L^{\\inft
 y}(\\Omega)}$$\nare proved for Lipschitz functions $u$ on convex domains $
 \\Omega \\subset \\mathbb{R}^d$.\nHere\, $Lu$ is either the integral mean 
 $\\langle u \\rangle$ of $u$ over $\\Omega$ or its median value $\\mathrm{
 med} \\\, u$. \nThe sharp constants in both cases can be interpreted geome
 trically\, and the case that $Lu = \\langle u \\rangle$ we prove that\n(i)
  among convex domains of equal measure balls have the best\, i.e. the smal
 lest\, Poincar\\'{e} constants\, and\n(ii) since $C(\\Omega)$ is the maxim
 um of a certain convex function $\\zeta$\, calculating $C(\\Omega)$ for po
 lygonal convex domains is a matter of ordering the valus $\\zeta(V_i)$ whe
 re $V_1\,\\ldots\,V_n$ are the vertices of $\\Omega$.  In this context\, w
 e introduce the idea of `magic points' and explore evidence for their exis
 tence in concrete cases.  This talk is based on joint works with J. Deane 
 and S. Zelik.\n
LOCATION:https://researchseminars.org/talk/NSCM/205/
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