The two-phase fractional obstacle problem

TL;DR

This paper investigates the regularity and free boundary separation in a two-phase fractional obstacle problem related to heat flow with boundary control, extending understanding of non-local obstacle problems.

Contribution

It establishes the optimal regularity of minimizers and proves the separation of free boundaries for the case when the parameter a is non-negative.

Findings

Minimizers exhibit optimal regularity.

Free boundaries are separated when a ≥ 0.

Connection to fractional obstacle and heat flow boundary control.

Abstract

We study minimizers of the functional $$ \int_{B_1^+}|\nabla u|^2 x_n^a\,d x +2\int_{B_1'} (\lambda_+ u^++\lambda_- u^-)\,d x', $$ for $a\in(-1,1)$. The problem arises in connection with heat flow with control on the boundary. It can also be seen as a non-local analogue of the, by now well studied, two-phase obstacle problem. Moreover, when $u$ does not change signs this is equivalent to the fractional obstacle problem. Our main results are the optimal regularity of the minimizer and the separation of the two free boundaries $\Gamma^+=\partial'\{u(\cdot,0)>0\}$ and $\Gamma^-=\partial'\{u(\cdot,0)<0\}$ when $a\geq 0$.