Variational p-harmonious functions: existence and convergence to p-harmonic functions

Abstract

In a recent paper, the last three authors showed that a game-theoretic p-harmonic function v is characterized by an asymptotic mean value property with respect to a kind of mean value $\nu _p^r[v](x)$ defined variationally on balls $B_r(x)$. In this paper, in a domain $\Omega \subset \mathbb {R}^N$, $N\ge 2$, we consider the operator $\mu _p^\varepsilon$, acting on continuous functions on $\overline{\Omega }$, defined by the formula $\mu _p^\varepsilon [v](x)=\nu ^{r_\varepsilon (x)}_p[v](x)$, where $r_\varepsilon (x)=\min [\varepsilon ,\mathop {\mathrm {dist}}(x,\Gamma )]$ and $\Gamma$ denotes the boundary of $\Omega$. We first derive various properties of $\mu ^\varepsilon _p$ such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function $u^\varepsilon \in C(\overline{\Omega })$ satisfying the Dirichlet-type problem:

$$\begin{aligned} u(x)=\mu _p^\varepsilon [u](x) \ \text{ for } \text{ every } \ x\in \Omega ,\quad u=g \ \hbox { on } \ \Gamma , \end{aligned}$$

for any given function $g\in C(\Gamma )$. This result holds, if we assume the existence of a suitable notion of barrier for all points in $\Gamma$. That $u^\varepsilon$ is what we call the variational p-harmonious function with Dirichlet boundary data g, and is obtained by means of a Perron-type method based on a comparison principle. We then show that the family $\{ u^\varepsilon \}_{\varepsilon >0}$ gives an approximation for the viscosity solution $u\in C(\overline{\Omega })$ of

$$\begin{aligned} \Delta _p^G u=0 \ \text{ in } \Omega , \quad u=g \ \hbox { on } \ \Gamma , \end{aligned}$$

where $\Delta _p^G$ is the so-called game-theoretic (or homogeneous) p-Laplace operator. In fact, we prove that $u^\varepsilon$ converges to u, uniformly on $\overline{\Omega }$ as $\varepsilon \rightarrow 0$.