Another way to say harmonic
Authors:
Michael G. Crandall and Jianying Zhang
Journal:
Trans. Amer. Math. Soc. 355 (2003), 241263
MSC (2000):
Primary 35J70, 35J05, 35B50
Published electronically:
August 28, 2002
MathSciNet review:
1928087
Fulltext PDF Free Access
Abstract 
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Additional Information
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.
 1.
E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of functionals, Arch. Rat. Mech. Anal., 157 (2001), 255283.
 2.
T. Battharchaya, E. Di Benedetto, and J. Manfredi, Limits as of and related extremal problems, Rend. Sem. Mat. Unvers. Politecn. Torino., Fasciocolo Speciale (1989), Nonlinear PDE's, 1568.
 3.
M. G. Crandall, An efficient derivation of the Aronsson equation, preprint.
 4.
M. G. Crandall, L. C. Evans and R. Gariepy, Optimal Lipschitz Extensions and the Infinity Laplacian, Calculus Var. Partial Differential Equations 13 (2001), 123139, DOI 10.1007/s005260000065.
 5.
M. G. Crandall, H. Ishii, and P.L. Lions, User's guide to viscosity solutions of secondorder partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 167. MR 92j:35050
 6.
J. T. Heinonen, T. Kilpeläinen and O. Marten, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993. MR 94e:31003
 7.
R. Jensen, Uniqueness of Lipschitz extensions minimizing the supnorm of the gradient, Arch. Rat. Mech. Anal. 123 (1993), 5174. MR 94g:35063
 8.
P. Juutinen, P. Lindqvist, and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM J. Math. Anal. 33 (2001), 699717.
 9.
P. Lindqvist, On the definition and properties of psuperharmonic functions, J. für die Reine und Angewandte Mathematik 365 (1986), 6779. MR 87e:31011
 10.
P. Lindqvist, On the Growth of the solutions of the equation in dimensional space, J. Diff. Equations 58 (1985), 307317. MR 86k:35043
 1.
 E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of functionals, Arch. Rat. Mech. Anal., 157 (2001), 255283.
 2.
 T. Battharchaya, E. Di Benedetto, and J. Manfredi, Limits as of and related extremal problems, Rend. Sem. Mat. Unvers. Politecn. Torino., Fasciocolo Speciale (1989), Nonlinear PDE's, 1568.
 3.
 M. G. Crandall, An efficient derivation of the Aronsson equation, preprint.
 4.
 M. G. Crandall, L. C. Evans and R. Gariepy, Optimal Lipschitz Extensions and the Infinity Laplacian, Calculus Var. Partial Differential Equations 13 (2001), 123139, DOI 10.1007/s005260000065.
 5.
 M. G. Crandall, H. Ishii, and P.L. Lions, User's guide to viscosity solutions of secondorder partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 167. MR 92j:35050
 6.
 J. T. Heinonen, T. Kilpeläinen and O. Marten, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993. MR 94e:31003
 7.
 R. Jensen, Uniqueness of Lipschitz extensions minimizing the supnorm of the gradient, Arch. Rat. Mech. Anal. 123 (1993), 5174. MR 94g:35063
 8.
 P. Juutinen, P. Lindqvist, and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM J. Math. Anal. 33 (2001), 699717.
 9.
 P. Lindqvist, On the definition and properties of psuperharmonic functions, J. für die Reine und Angewandte Mathematik 365 (1986), 6779. MR 87e:31011
 10.
 P. Lindqvist, On the Growth of the solutions of the equation in dimensional space, J. Diff. Equations 58 (1985), 307317. MR 86k:35043
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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/S0002994702030556
PII:
S 00029947(02)030556
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
