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UID:7465@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20171026T153000
DTEND;TZID=Europe/Paris:20171026T163000
DTSTAMP:20241120T204337Z
URL:https://www.i2m.univ-amu.fr/evenements/variational-approach-to-the-reg
 ularity-of-the-free-boundaries/
SUMMARY: (...): Variational approach to the regularity of the free boundari
 es
DESCRIPTION:: In this talk we present some recent results on the structure 
 of the free boundary of the local minimizers of the following problems: -t
 he Bernoulli problem in dimension two\\min\\{\\int_{B_1} \\big(|\nabla u|^
 2+\\ind_{\\{u>0\\}}\\big)\\\,:\\\, u\\in H^1(B_1)\\ +Dirichlet\\ Boundary\
 \  Conditions\\}\;-the obstacle problem in any dimension\\min\\{\\int_{B_1
 } \\big(|\nabla u|^2+u\\big)\\\,:\\\, u\\ge 0\,\\  u\\in H^1(B_1)\\ +\\ D.
 \\\, B.\\\,C.\\}-the thin-obstacle problem in any dimension\\min\\{\\int_{
 B_1} |\nabla u|^2\\\,:\\\, u(0\,0\,\\dots\,0\,x_d)\\ge 0\,\\  u\\in H^1(B_
 1)\\ +\\ D.\\\, B.\\\,C.\\}Our approach is based on variational inequaliti
 es for the boundary adjusted energies of G.S. Weiss. In particular\, -at t
 he {flat regular points} of the free boundary $\\partial\\{u>0\\}$ we intr
 oduce a direct method\, that allows to compare the energy of the minimizer
  to the energy of its homogeneous extension\, obtaining the so called {epi
 perimetric inequality}-at the singular points\, where the classical epiper
 imetric inequality fails\, we introduce a new tool which we call {logarith
 mic epiperimetric inequality} that allows to prove the $C^{1\,\\log}$ rect
 ifiability of the singular set for the obstacle problem and $2m$-singular 
 set for the thin obstacle problem.http://www.velichkov.it
CATEGORIES:Groupe de travail,Calcul des Variations & EDP
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