Problems related to the infinity Laplacian operator, the weak KAM theory and singularities of solutions of Monge-Ampere equations

University Of California-Irvine, Irvine CA

Investigators

Abstract

Yu The PI proposes to continue his study of problems related to the infinity Laplacian operator, the weak KAM theory and singularities of solutions of the Monge-Ampere equation. (1) The infinity Laplacian operator arises from minimizing the L-infinity norm of the gradient and a two person differential game called ?tug-of-war?. The PI intends to solve several problems from the?tug-of-war? game. One of the important questions is to see how we can use the game theory interpretation to understand more about the infinity Laplacian equation, a highly degenerate nonlinear elliptic equation. The PI also intends to characterize asymptotic behaviors of principle eigenfunctions of p-Laplacian operators as p goes to infinity. Other problems concern properties of classical solutions of the infinity Lapalcian equation and uniqueness of absolute minimizers from minimizing more general norms of the gradient. (2) The aim of the weak KAM theory is to use pde approaches to study the Aubry-Mather theory. Our major goal here is to find a variational method to identify the Aubry set. (3) It was known that generalized solutions of Monge-Ampere equations from the optimal mass transfer problems might have singularities. The PI plans to use some tools developed with P. Cannarsa to explore the regularity of the set of singularities. (1) Equations involving the infinity Laplacian operator are very different from elliptic PDEs that people knew before. On one hand, they are second order. On the other hand, the infinity Laplacian operator is so degenerate that those equations sometimes behave as first order PDEs, for example, their solutions even possess some sort of characteristics. Proposed problems in this topic require new methods and ideas which will enhance people's knowledge of elliptic PDEs. Beside its extreme mathematical interest, the infinity Laplacian operator also has important applications in practical issues, for example, to restore images with poor dynamical range, to determine the optimal strategy in the tug-of-war game which is applicable to economic and political modeling, etc. (2) Very little has been known about the structure of the Aubry-Mather set when the dimension is bigger than two. The research proposed in the weak KAM theory part may provide a numerical method to approximate the Aubry set. (3) Monge-Ampere equations from optimal transfer problems have interesting applications in meteorology. The semigeostrophic equations from meteorology can be formulated as a coupled Monge-Ampere/transport problem. The results about the set of singularities of generalized solutions of the Monge-Ampere equation should help people understand how fronts arise in large scale weather pattern.