Free Boundary Regularity for Almost-Minimizers
Year: 2017
Type: preprint
Abstract: In this paper we study the free boundary regularity for almost-minimizers of the functional \begin{equation*} J(u)=\int_{\mathcal O} |\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x)\ dx \end{equation*} where $q_\pm \in L^\infty(\mathcal O)$. Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [AC], [ACF], [CJK], [W]). Nevertheless we succeed in proving that, under a non-degeneracy assumption on $q_\pm$, the free boundary is uniformly rectifiable. Furthermore, when $q_-\equiv 0$, and $q_+$ is Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a $C^{1,α}$ function (thus extending the results of [AC] to almost-minimizers).
Authors: David, Guy, Engelstein, Max, Toro, Tatiana
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Geometry of Measures, G202495085, Free boundaries and extremal inequalities, G3469923543, G3502766958, Geometric measure theory and applications, Graduate Research Fellowship Program (GRFP), RTG: Analysis and Differential Equations, G5752146209, G5969482202, G6702880738, G7908957721, Mathematical Sciences Research Institute (MSRI), PostDoctoral Research Fellowship, G8906270148
OpenAlex ID: W2952218037