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.
The major goal of this research grant is to study some significant and interesting analytic problems related to the infinity Laplacian operator and the weak KAM theory. Summary of outcomes: Over the past four years, the PI is very productive. A total of 14 publications have been generated within this period. The PI has solved quite a few proposed problems including several outstanding open problems. The infinity Laplacian equation is a highly nonlinear partial differential equation. It originally arose from finding an optimal way to interpolate data within a domain by minimizing some kind of energy (so called "Lipschitz constant"). Besides its significance in the analytical theory of partial differential equations (PDE), the infinity Laplacian operator also has important applications in practical issues; for example, to restore partially damaged images. Major result 1: In a paper by four probablists (Peres, Schramm, Sheffield, and Wilson), the authors discovered that the infinity Laplacain equation could be used to identify the optimal strategy in an interesting differential game (called "tug-of-war" game), which might be used to model competitions in economy and politics. The PI gave an affirmative answer to an important open problem proposed by those probablists, which asks whether the running cost in this "tug-of-war" game is determined by the final payoff. Major result 2: In another paper, the PI further studied the connections between the original variational formulation of infinity Laplacian type equations and the "tug-of-war" game when the running cost is a constant. He showed that, the smallest solution of the PDE is equal to the value function of the game and the largest solution is given by the variational formulation. Major result 3: The first eigenvalue of the Laplacian operator is related to the basic frequency of idealized drums and is known to be simple (i.e., the corresponding normalized eigenfunction is unique). The problem that whether the first eigenvalue of the infinity Laplacian operator is simple has been open for more than one decade. This is analytically very challenging. The PI together with Ryan Hund and Charles Smart provided a counterexample which shows that the simplicity of the first eigenvalue is in general false for the infinity Laplacian operator. Major result 4: It was proved by Ovidiu Savin that the interpolation of data given by the infinity Laplacian equation is smooth in the interior of any domain on the plane. In a joint work with Changyou Wang, we extended Savin’s result and showed that such kind of interpolation is also smooth near and along the boundary of the domain. This is called "boundary regularity" in the theory of PDE. The weak KAM theory is to use PDE approaches to study a kind of integrable structure in the Hamiltonian system, which plays an essential role in classical mechanics. For particles travel within those structures, we are able to provide more information about their positions and velocities over a long time. Major result 5: In a joint work with Jack Xin, we used techniques in weak KAM theory to solve an open Problem Posed by Embid, Majda and Souganidis regarding modeling of turbulent flame speed in turbulent combustion. Roughly speaking, the turbulent flame speed is the burning velocity under the influence of strong ambient fluid (think of the spread of wildfire fanned by strong wind). To determine the turbulent flame speed is one of most important unsolved problems in turbulent combustion. A significant project is to study the dependence of turbulent flame speed on flow intensity A (e.g. the wind velocity). Many simplified models have been introduced to study this. In a paper by Embid, Majda and Souganidis, the authors compared turbulent flame speeds predicted by a model proposed by Majda and Souganidis and by the well-known G-equation model in turbulent combustion. They did explicit computations for the shear flow and raised the question to compare these two models in flows with periodically arrayed vortices, which is analytically much more difficult. The PI and Jack Xin are able to answer this question by identifying the sharp growth laws of turbulent flame speeds given by these two models. Precisely speaking, our results says that under the G-equation model, the turbulent flame speed grows as (A/log A) and under the model proposed by Majda-Souganidis, it grows as A/(logA-loglogA). Major result 6: Besides analytic studies, it is at least equivalently important to find efficient numerical schemes to compute the turbulent flame speed, which, under the G-equation model, is corresponding to the effective Hamiltonian in the weak KAM. In a joint work with Songting Luo and Hongkai Zhao, we introduced a numerical scheme, which is a based on a different theoretical framework from other existing schemes. The main advantage of our method is that we only need to solve a single equation in order to obtain values of the effective Hamiltonian along all directions. .
