On the definition and properties of $p$-harmonious functions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, p. 215-241

We consider functions that satisfy the identity

${u}_{ϵ}\left(x\right)=\frac{\alpha }{2}\left\{\underset{{\overline{B}}_{ϵ}\left(x\right)}{sup}{u}_{ϵ}+\underset{{\overline{B}}_{ϵ}\left(x\right)}{inf}{u}_{ϵ}\right\}+\beta -\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\int }_{{B}_{ϵ}\left(x\right)}{u}_{ϵ}dy$

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.

Published online : 2018-06-21
Classification:  91A15,  35B50,  35J25,  35J70,  49N70,  91A24
@article{ASNSP_2012_5_11_2_215_0,
author = {Manfredi, Juan J. and Parviainen, Mikko and Rossi, Julio D.},
title = {On the definition and properties of $p$-harmonious functions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {2},
year = {2012},
pages = {215-241},
zbl = {1252.91014},
mrnumber = {3011990},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_215_0}
}

Manfredi, Juan J.; Parviainen, Mikko; Rossi, Julio D. On the definition and properties of $p$-harmonious functions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, pp. 215-241. http://www.numdam.org/item/ASNSP_2012_5_11_2_215_0/

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