Two-phase almost minimizers for a fractional free boundary problem

Abstract

In this paper, we study almost minimizers to a fractional Alt–Caffarelli–Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries $F^+(u)=\partial \{u(\cdot ,0)>0\}$ and $F^-(u)=\partial \{u(\cdot ,0)<0\}$ cannot touch, that is, $F^+(u)\cap F^-(u)=\emptyset$. Lastly, we prove a flatness implies $C^{1,\gamma }$ result for the free boundary.

Minimizers are viscosity solutions

We employ standard techniques to show that minimizers are viscosity solutions. This will show a necessary step in the proof of Lemma 5.5. As this result is not explicitly shown in the literature, we provide the result here.

Proposition A.1

Let V and ${\overline{V}}$ be given as in the proof of Lemma 5.5. Then ${\overline{V}}$ is the minimizer of the functional

$$\begin{aligned} {\tilde{J}}(w,B_1)=J(w,B_1)+ \int _{B_1} 2\mu ^2 w |y|, \end{aligned}$$

among all $w \in H^1(a,B_1)$ satisfying $\min \{V, {\overline{V}}\}\le w \le {\overline{V}}$.

Proof

Let ${\tilde{w}}$ be the minimizer, and let $V_t=V_{{\overline{M}}, \xi ', {\overline{\zeta }}}(x+te_n,y)$ with $t \in [0,\mu /(8n)]$. Notice that $V_0 \le \min \{V, {\overline{V}}\}$ and $V_{\mu /(8n)}={\overline{V}}$. Furthermore, $V_t <{\overline{V}}$ on $\partial B_1 \cap \{{\overline{V}}>0\}$. We will show that if $V_t$ touches ${\tilde{w}}$ by below on $\{{\tilde{w}}>0\}$ or $F({\tilde{w}})$, then necessarily ${\tilde{w}}=V_t$, so that ${\overline{V}}={\tilde{w}}$.

Suppose $V_t(X)={\tilde{w}}(X)>0$. Now $|y|^a \mathcal {L}_a {\tilde{w}}=\mu ^2 |y|$. Since $\mathcal {L}_a V_t \ge \mathcal {L}_a {\tilde{w}}$ in $\{{\tilde{w}}>0\}$, then from the strong maximum principle, we conclude that $V_t \equiv {\tilde{w}}$.

Now suppose that $x \in F(V_t) \cap F({\tilde{w}})$. From the standard proofs of [1] adapted to this nonhomogeneous situation, we have that ${\tilde{w}}$ has both the $C^{0,s}$ regularity as well as the nondegeneracy. Since $V_t$ touches ${\tilde{w}}$ by below at x, then every blow-up of ${\tilde{w}}$ at x must be the same rotation of the prototypical 2D solution. Then

$$\begin{aligned} \frac{\partial {\tilde{w}} - V_t}{\partial \tau ^s}(x,0)=0. \end{aligned}$$

Consequently,

$$\begin{aligned} \lim _{y \rightarrow 0}y^a \partial _y ({\tilde{w}} - V_t)(x,y)=0. \end{aligned}$$

Since ${\tilde{w}}-V_t \ge 0$, ${\tilde{w}}(x,0)-V_t(x,0)=0$, then this will violate the Hopf-type lemma (4.2). $\square$