Invited talk

Thursday, July 22, 14:45 ~ 15:45 UTC-3

Asymptotic Mean Value Properties for Non Linear Partial Differential Equations

Julio Daniel Rossi

Buenos Aires Univ., Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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. Our goal in this talk is to show that an asymptotic nonlinear mean value formula characterizes weak solutions (in the viscosity sense) for the wide class of partial differential equations, including eigenvalues of the Hessian and the classical Monge-Amp\`ere equation.

References

P. Blanc -- J. D. Rossi. Game Theory and Partial Differential Equations. De Gruyter Series in Nonlinear Analysis and Applications Vol. 31. 2019.

P. Blanc -- F. Charro -- J. J. Manfredi -- J. D. Rossi. A nonlinear mean value property for Monge-Ampere. To appear in Journal of Convex Analysis.

P. Blanc -- C. Esteve -- J. D. Rossi. The evolution problem associated with eigenvalues of the Hessian. Journal of the London Mathematical Society. Vol. 102(3), 1293--1317, (2020).

P. Blanc -- J. D. Rossi. Games for eigenvalues of the Hessian and concave/convex envelopes. Journal de Math\'ematiques Pures et Appliquees. Vol. 127, 192--215, (2019).

Joint work with P. Blanc (Jyvaskyla, Finland), F. Charro (Detroit, USA), C. Esteve (Bilbao, Spain) and J.J. Manfredi (Pittsburgh, USA)..

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