Calculus of Variations and Geometric Measure Theory

L. Spolaor - B. Velichkov

An epiperimetric inequality for the regularity of some free boundary problems: the $2$-dimensional case

created by spolaor on 08 Feb 2017
modified by velichkov on 08 Nov 2017


Accepted Paper

Inserted: 8 feb 2017
Last Updated: 8 nov 2017

Journal: Comm. Pure Appl. Math.
Pages: 32
Year: 2017


Using a direct approach, we prove a $2$-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension $2$, the $C^{1,\alpha}$ regularity of the free-boundary in the scalar one-phase and double-phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore we show that in the vectorial case the free boundary can end in a cusp.

Keywords: Regularity of free boundary