Variational approach to the regularity of the free boundaries

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Date(s) - 26/10/2017
15 h 30 min - 16 h 30 min


In this talk we present some recent results on the structure of the free boundary of the local minimizers of the following problems:

-the 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 inequalities for the boundary adjusted energies of G.S. Weiss. In particular,

-at the {flat regular points} of the free boundary $\partial\{u>0\}$ we introduce a direct method, that allows to compare the energy of the minimizer to the energy of its homogeneous extension, obtaining the so called {epiperimetric inequality}

-at the singular points, where the classical epiperimetric inequality fails, we introduce a new tool which we call {logarithmic epiperimetric inequality} that allows to prove the $C^{1,\log}$ rectifiability of the singular set for the obstacle problem and $2m$-singular set for the thin obstacle problem.

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