Collaborative Research: L-Infinity Varieational Problems and the Aronsson Equation

PROJECT SUMMARY The main themes of this project study the Loo-variational problems and associated Euler- Lagrange equations, called the Aronsson equation. There are many problems in which the appropriate cost functional must be the supermum (or infimum) of integrand functions. To attack these problems, we plan to systematically study the variation problems in Loo and its analysis by PDE methods in the following aspects: (1) complete characterization of absolute minimizers of Loo-functionals by the Aronsson equations, i.e. finding generic sufficient conditions on the Hamiltonian functions H such that absolute minimizers of H and viscosity solutions to the Aronsson equation are equivalent, (2) Uniqueness problems for viscosity solutions to the Aronsson equation and absolute minimizers, (3) Regularity issues of viscosity solutions to the Aronsson equation in dimensions two or higher, (4) applications of LCXJ-variational problems to the study of weak KAM theory in the Hamiltonian dynamics, and (5) LCXJ-variational problems with constraints such as the principal eigenvalue problem for the infinity Laplace operator. The proposed problems belong to the field of nonlinear PDEs which provide basic laws governing most natural phenomena and play crucial roles in studying problems arising from calculus of variations, analysis, probality, and applied sciences including optimal control and imagine processing. Variational problems in L= are very important from point of views of both practical issues and mathematical theories. For example, find a chemotherapy regimen for treatment of cancer seeking to minimize the maximum tumor load, recover partially damaged or lost imagines by optimal extensions in appropriate sense of observed boundary imagines, and seek strategies to maximize wining rate in a two person "tug-of-war" games. etc. On the other hand, LOG-variational problems usually yield extremely degenerate elliptic PDEs, such as oo-Laplace equation or the Aronsson equation, in which the existing theory of partial differential equations usually reaches its limit and new ideas and techniques must be developed in order to better understand the problems. Moreover, better understanding of LCXJ-variational problems will be important to understand the Hamiltonian dynamics in which the Hamiltonian functions are far-from-integrable. Over the past few years, the PIs have been actively involved and made several interesting contributions to these problems, and we would like to continrme our studies in this direction within the proposed project. 1 TPI6519672