Abstract

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized p-Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. Currently, asymptotic nonlinear mean value formulas are rare in the literature and our goal is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Ampère equation.

Author information

Contact details are reproduced from the original publication and may be historical.

Pablo Blanc

Dept. of Mathematics and Statistics, University of Jyväskylä, Finland

pblanc@dm.uba.ar

Fernando Charro

Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.

fcharro@wayne.edu

Juan J. Manfredi

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

manfredi@pitt.edu

Julio D. Rossi

Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Argentina

jrossi@dm.uba.ar

Keywords

  • Monge-Ampère
  • Mean Value Formulas
  • viscosity solutions

Mathematics Subject Classification

35J60, 35J96, 35B05

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Suggested citation

P. Blanc, F. Charro, J. J. Manfredi, J. D. Rossi. “A Nonlinear Mean Value Property for the Monge-Ampère Operator.” Journal of Convex Analysis 28 (2021), No. 2, 353–386. https://doi.org/10.68381/jca28022

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