Analysis of the p-Laplacian

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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.