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

Full-text PDF

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Abstract: It is known that solutions of , that is, the -harmonic functions, are exactly those functions having a comparison property with respect to the family of translates of the radial solutions . We establish a more difficult linear result: a function in is harmonic if it has the comparison property with respect to sums of translates of the radial harmonic functions for and for . An attempt to generalize these results for () and () to the general -Laplacian leads to the fascinating discovery that certain sums of translates of radial -superharmonic functions are again -superharmonic. Mystery remains: the class of -superharmonic functions so constructed for *does not* suffice to characterize -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:
https://doi.org/10.1090/S0002-9947-02-03055-6

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