This project identifies for intensive study several classes of interesting problems concerning the properties of solutions of certain nonlinear partial differential equations (PDE). The specific topics are analyzing solutions of nonconvex Hamilton-Jacobi equations, improving PDE methods for weak KAM theory, studying regularity of solutions of the infinity Laplacian equation, developing PDE techniques for "inconsistent" stochastic optimal control problems, formulating variational principles for optimal symplectic maps, and studying certain strongly nonlinear parabolic systems. The various topics cited above have widely differing structures but are unified by reason of being accessible to either variational, maximum principle and/or energy methods, although usually in certain singular limits.
Vast research experience has shown that simple-looking nonlinear partial differential equations, with mathematically natural structures, appear and reappear in the pure and applied sciences. In particular, most of the fundamental equations of the physical and engineering sciences are partial differential equations, of which the most difficult, and arguably most important, are nonlinear. So-called perturbation techniques can handle various "close-to-linear" equations, but there remains the real necessity of discovering general principles and methods for various important nonlinear equations in the large. Understanding the existence, uniqueness, and regularity of solutions, and ascertaining as well their behavior, are fundamental mathematical tasks that should have practical implications in view of the various interpretations of these equations. The particular equations singled out for study in the project are all natural in their structures, and consequently can be expected to arise in various applications. For instance, the nonconvex Hamilton--Jacobi equation is the basic equation for two-person, zero-sum differential games, so understanding the singular structure of its solutions helps directly in the design of optimal strategies. The proposed extension of partial differential equations methods to the nonstandard stochastic optimal control problems should likewise be fundamental in framing optimality conditions.
Project Outcomes for General Public NSF Grant DMS-1001724 Lawrence C. Evans This grant supported me and several of my PhD students a wide ranging research program on properties of solutions of various nonlinear partial differential equations(PDEs). RESEARCH. A. Envelopes and nonconvex Hamilton-Jacobi PDE. I discovered a new general ``envelope'' representation formula for solutions of the Hamilton-Jacobi PDEwith nonconvex Hamiltonians H.My new formula generalizes both the Hopf-Lax representation (for convex H) and Hopf's formula (for convex initial data). B. Sup-norm variational problems. My then student Charlie Smart and I applied a nonlinear adjoint method to the so-called ``infinity Laplacian'' equation, a highly degenerate PDE that arises in variational problemsin the supremum-norm. We resolved a long standing problem by showing that appropriately rescaled blow-up limits are unique and thus solutions are everywhere differentiable. C. Differential games. Optimal strategies for two--person, zero--sum differential games can be computed in principle by solving certain highly nonlinear PDE, but there are many technical problems. In a recent paper I extended a famous condition of R Isaacs for the solution alonga surface of discontinuity and discovered that certain ``entropy type'' inequalities must hold has well. These should make it easier to construct explicit solutions. f D. Liquid crystal models. Some recently proposed models for liquid crystals give highly singular variational problems that do not fall within standard theory. My former student H Tran, my former postdoc O. Kneuss and I have shown partial regularity for minimizers for several of these physically realistic models. EDUCATION: I have recently revised some online notes into the book ``An Introduction to Stochastic DifferentialEquations'', published by the American Math Society in 2014. I have also written a fairly long article on ``Partial differential equations" for the forthcoming ``The Princeton Companion to Applied Mathematics''. (I earlier wrote the entry on ``Variational methods'' for ``The Princeton Companion to Mathematics''.) Recent PhD students: A. Chen (University of California, Berkeley, 2012, coadvisor: X. Guo ), ``Impulse Control and Optimal Stopping''. H. Tran (University of California, Berkeley, 2012), ``Some New Methods for Hamilton--Jacobi Type Nonlinear Partial Differential Equations''. J. DeIonno (University of California, Berkeley, 2013), ``Quasi-Variational Inequalities for Source-Expanding Hele-Shaw Problems'' Current PhD students: Khalilah Beal, Woo-Hyun Cook, Te Zhang, Peyam Tabrizian, Peter Vinella, Christoph Kroener Recent postdocs: Hiroyoshi Mitake (Hiroshima University, 2010-2011) Olivier Kneuss (Ecole Polytechnique Federale de Lausanne, 2012--2013)
