In many problems in physics and engineering the relevant energies are proportional to the square of the velocity. The resulting equations are linear and model very well small variations from equilibrium. But more substantial variations are better modeled by considering non-quadratic energies. In this proposal we study some aspects of the mathematical theory of the equations that result from the minimization of energies (or other quantities in physics and engineering) that are given by power laws of the type "Energy is approximatley the sum of the absolute values of velocity raised to the pth power", for p different from 2. The minimizers of these non-quadratic energies are p-harmonic functions. When the energy is quadratic (p=2) the resulting equations are linear, but for non-quadratic energies we are necessarily led to non-linear equations. Recently Peres and Sheffield found an unexpected connection between random Tug-of-War games and p-harmonic functions. Motivated by these considerations we introduce the class of p-harmonious functions and explore its relationship to the classical p-harmonic functions. In particular, p-harmonious functions serve as very good discrete approximations to p-harmonic functions and suggest novel approaches to some long-standing open problems.
The possible impact on other sciences comes from the connection between p-harmonic and p-harmonious functions and stochastic games. It may shine new light into some optimization problems that can be formulated in spaces quite more general than Euclidean space (graphs, trees, length spaces). The more immediate impact will be on human resources. The PI will use the formulation of Tug-of-War games in simple graphs to mentor several freshman students who used computer simulation to run these games, and are exposed to mathematical thinking early in their undergraduate career. The PI current graduate student will graduate in December 2010. The PI will mentor a postdoc in Analysis funded by the University of Pittsburgh for the duration of this project.
The PI has studied the relation between partial differential equations and games. It turns out that you can solve complicated non-linear equations by running a certain kind of stochastic games. Imaging replacing the regular game of chess by random chess, where a coin is tossed, and the outcome determine who is the player that move next. In many cases these random games are easier to analyze mathematically, and their value functions are solutions of complex nonlinear partial differential equations. One such equation is obtained by minimiing non-quadratic energies (proportial to a power of the velocity that is not quadratic). These equations appear in plasticity, glaceology, and fluid mechanics. The games involved are Tug-of-War games and regular random walks. Suppose that we have a token in a board and that we toss a coin. When the outcomes is head, player I moves the token a fixed distance d in any directions he wishes. If the outcome is tails, player II moves. When they hit the boundary of the board, player II plays player I a predetermine amount (the pay-off or bounary data). It you playthis game many times and average out the resutls the expected payment that player I would received from player II starting at x solves the infinity-Laplacian u_i u_j u_ij =0 If instead of decided where to move, the players move at random the expected payment solves the familiar Laplace equaton u_ii =0. Combining both games we would move randomly with probability a and using Tug-of-War with probabilty b (Here a and be are positive and a+b=1), we would get solutions to the equation (1) a u_ii + b u_i u_j u_ij = 0 The PI and his collaborators have studied how to solve euation (1) in grapsh, Euclidean space, and the Heisenberg group.
