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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: August 28, 2002
MathSciNet review: 1928087
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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 $G(x)=a\vert x\vert$. 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 $G(x)=a\vert x\vert^{2-n}$ for $n\not=2$ and $G(x)=b\ln(\vert x\vert)$ 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

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03055-6
PII: S 0002-9947(02)03055-6
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