Calculus of Variations and Geometric Measure Theory

M. Carducci

Optimal regularity of the thin obstacle problem by an epiperimetric inequality

created by carducci on 18 Sep 2023
modified on 07 Dec 2023


Published Paper

Inserted: 18 sep 2023
Last Updated: 7 dec 2023

Journal: Annali di Matematica Pura ed Applicata (1923 -)
Year: 2023
Doi: 10.1007/s10231-023-01403-1

ArXiv: 2307.12658 PDF


The key point to prove the optimal $C^{1,\frac12}$ regularity of the thin obstacle problem is that the frequency at a point of the free boundary $x_0\in\Gamma(u)$, say $N^{x_0}(0^+,u)$, satisfies the lower bound $N^{x_0}(0^+,u)\ge\frac32$. In this paper we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies $W_\frac32$. It allows to say that there are not $\lambda-$homogeneous global solutions with $\lambda\in (1,\frac32)$, and by this frequancy gap, we obtain the desired lower bound, thus a new self contained proof of the optimal regularity.