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

 

Fundamental solutions and two properties of elliptic maximal and minimal operators


Authors: Patricio L. Felmer and Alexander Quaas
Journal: Trans. Amer. Math. Soc. 361 (2009), 5721-5736
MSC (2000): Primary 35J60; Secondary 35B05, 35B60
Published electronically: June 16, 2009
MathSciNet review: 2529911
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Abstract: For a large class of nonlinear second order elliptic differential operators, we define a concept of dimension, upon which we construct a fundamental solution. This allows us to prove two properties associated to these operators, which are generalizations of properties for the Laplacian and Pucci's operators. If $ {\mathcal M}$ denotes such an operator, the first property deals with the possibility of removing singularities of solutions to the equation

$\displaystyle {\mathcal M}(D^2 u)-u^p=0,$   in$\displaystyle \quad B\setminus \{0\}, $

where $ B$ is a ball in $ \mathbb{R}^N$. The second property has to do with existence or nonexistence of solutions in $ R^N$ to the inequality

$\displaystyle {\mathcal M}(D^2 u)+u^p\le 0,$   in$\displaystyle \quad \mathbb{R}^N. $

In both cases a common critical exponent defined upon the dimension number is obtained, which plays the role of $ N/(N-2)$ for the Laplacian.


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Additional Information

Patricio L. Felmer
Affiliation: Departamento de Ingeniería Matemática, and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Alexander Quaas
Affiliation: Departamento de Matemática, Universidad Santa María, Casilla: V-110, Avda. España 1680, Valparaíso, Chile

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04566-8
PII: S 0002-9947(09)04566-8
Keywords: Extremal operators, viscosity solutions, fundamental solutions, removability of singularities, Liouville-type theorems
Received by editor(s): January 4, 2006
Received by editor(s) in revised form: May 10, 2007
Published electronically: June 16, 2009
Article copyright: © Copyright 2009 American Mathematical Society