Another way to say harmonic

Authors:
Michael G. Crandall and Jianying Zhang

Journal:
Trans. Amer. Math. Soc. **355** (2003), 241-263

MSC (2000):
Primary 35J70, 35J05, 35B50

DOI:
https://doi.org/10.1090/S0002-9947-02-03055-6

Published electronically:
August 28, 2002

MathSciNet review:
1928087

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Abstract | References | Similar Articles | Additional Information

Abstract: It is known that solutions of $-\Delta _\infty u=-\sum _{i,j=1}^nu_{x_i} u_{x_j}u_{x_ix_j}=0$, that is, the $\infty$-harmonic functions, are exactly those functions having a comparison property with respect to the family of translates of the radial solutions $G(x)=a|x|$. We establish a more difficult linear result: a function in ${\mathbb R^n}$ is harmonic if it has the comparison property with respect to sums of $n$ translates of the radial harmonic functions $G(x)=a|x|^{2-n}$ for $n\not =2$ and $G(x)=b\ln (|x|)$ for $n=2$. An attempt to generalize these results for $-\Delta _\infty u=0$ ($p=\infty$) and $-\Delta u=0$ ($p=2$) to the general $p$-Laplacian leads to the fascinating discovery that certain sums of translates of radial $p$-superharmonic functions are again $p$-superharmonic. Mystery remains: the class of $p$-superharmonic functions so constructed for $p\not \in \{2,\infty \}$ *does not* suffice to characterize $p$-subharmonic functions.

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Additional Information

**Michael G. Crandall**

Affiliation:
Department of Mathematics, University of California, Santa Barbara, Santa Barbara, California 93106

Email:
crandall@math.ucsb.edu

**Jianying Zhang**

Affiliation:
Department of Mathematics, University of California, Santa Barbara, Santa Barbara, California 93106

Email:
zjyjenny@math.ucsb.edu

Received by editor(s):
August 17, 2001

Received by editor(s) in revised form:
February 20, 2002

Published electronically:
August 28, 2002

Article copyright:
© Copyright 2002
American Mathematical Society