On a two-phase free boundary condition for p-harmonic measures

Abstract

Let ${\Omega^i\subset {\bf R}^n, i\in\{1,2\}}$ , be two (δ, r ₀)-Reifenberg flat domains, for some ${0 < \delta < \hat \delta}$ and r ₀ > 0, assume ${\Omega^1\cap\Omega^2=\emptyset}$ and that, for some ${w\in {\bf R}^n}$ and some 0 < r, ${w\in\partial\Omega^1\cap\partial\Omega^2, \partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}$ . Let p, 1 < p < ∞, be given and let u ⁱ, ${i\in\{1,2\}}$ , denote a non-negative p-harmonic function in Ω ⁱ, assume that u ⁱ, ${i\in\{1,2\}}$, is continuous in ${\bar\Omega^i\cap B(w,2r) }$ and that u ⁱ = 0 on ${\partial\Omega^i\cap B(w,2r)}$ . Extend u ⁱ to B(w, 2r) by defining ${u^i\equiv 0}$ on ${B(w,2r) {\setminus} \Omega^i}$. Then there exists a unique finite positive Borel measure μ ⁱ, ${i\in\{1,2\}}$ , on R ⁿ, with support in ${\partial\Omega^i\cap B(w,2r)}$ , such that if ${\phi \in C_0^\infty (B(w,2r))}$ , then

$$\int\limits_{\mathbf R^n} \,|\nabla u^i|^{ p-2} \,\langle \nabla u^i, \,\nabla \phi \rangle \,dx =- \int\limits_{\mathbf R^n} \,\phi \,d \mu^i.$$

Let ${\Delta(w,2r)=\partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}$ . The main result proved in this paper is the following. Assume that μ ² is absolutely continuous with respect to μ ¹ on Δ(w, 2r), d μ ² = kd μ ¹ for μ ¹-almost every point in Δ(w, 2r) and that ${\log k\in VMO(\Delta(w,r),\mu^1)}$ . Then there exists ${\tilde \delta = \tilde \delta(p,n) > 0}$ , ${\tilde \delta < \hat \delta}$ , such that if ${\delta\leq\tilde\delta}$ , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).