On the definition and properties of $p$-harmonious functions

Manfredi, Juan J.; Parviainen, Mikko; Rossi, Julio D.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 215-241.

We consider functions that satisfy the identity

$$u_{ϵ}\left(x\right)=\frac{\alpha }{2}\left\{\underset{{\overline{B}}_{ϵ}\left(x\right)}{sup}{u}_{ϵ}+\underset{{\overline{B}}_{ϵ}\left(x\right)}{inf}{u}_{ϵ}\right\}+\beta -{\int }_{B_{ϵ}\left(x\right)}{u}_{ϵ}dy$$

for a bounded domain in ${ℝ}^n$. Here $ϵ>0$ and $\alpha$, and $\beta$ are suitable nonnegative coefficients such that $\alpha +\beta =1$. In particular, we show that these functions are uniquely determined by their boundary values, approximate $p$-harmonic functions for $2\le p<\infty$ (for a choice of $p$ that depends on $\alpha$ and $\beta$), and satisfy the strong comparison principle. We also analyze their relation to the theory of tug-of-war games with noise.

Classification : 91A15, 35B50, 35J25, 35J70, 49N70, 91A24

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