This project is an in depth analysis of quasiconvex functions (a.k.a. level set convex functions), including nonsmooth quasiconvex functions. It focuses on their connections with second order, fully nonlinear partial differential equations, and the application of these connections to diverse problems such as worst case design, stochastic targeting, motion by curvature, and optimal transport. The primary idea is to use the theory of viscosity solutions of partial differential equations to characterize when a function has convex level sets. From this point several branches of inquiry will be followed. For example, the characterization of quasiconvex functions through a nonlinear partial differential equation suggests a PDE construction of the quasiconvex envelope of a given function ? analogous to the well known PDE construction of the convex envelope of a given function. Another possible construction is the quasiconvex-quasiconcave envelope of a function of several variables (which would have significant application in continuous game theory). Interestingly, there is a very strong connection between the characterization of quasiconvex functions using partial differential equations and the characterization of surfaces evolving by the motion of the boundary by mean curvature. This project could yield a more stable type of approximate motion by curvature. This project will also study of the connection between the quasiconvex characterizing differential equation, stochastic target problems, and deterministic differential tug-of-war games. Finally, quasiconvex functions arise naturally in the calculus of variations with a worst case cost. This project will consider several of problems of this type including an extension to optimal transport theory with worst case cost of transfer, and variational problems with various constraints.
Quasiconvexity (a.k.a. level set convexity) is an important generalization of convexity. Quasiconvex functions have been widely used for decades in economics and management science, where they arise naturally as preference/objective functions. A very important application and extension was found in optimal control and differential games with a worst case cost. Significant applications to structural engineering, materials science, image processing, semiconductor design, aircraft landing strategies, and chemotherapy dosing for carcinoma management all fall in the category of worst case design and involve functions of the type considered in this project. Expansion of a region in space governed by the physics at the boundary such as the edge of conflagration plasma (e.g., the front of a forest fire) is determined by equations which are derived and studied in this project. The connection with stochastic target problems is an important approach to control problems in which one seeks to reach an objective with much more certainty than simply the average objective value. Such problems have applications ranging from evolutionary biology to image reconstruction. In classical optimal transport one tries to minimize the average cost of transporting one distribution of particles to another while maintaining the total mass. This project will study minimizing the maximum cost of transportation of the particles. Such problems have applications in the physics of the rearrangement of molecules, the construction of public transportation networks, and in the biochemistry of how malignant tumors metastasize.
The project involved basic research in mathematical analysis. The investigators collaboratively solved problems and formulated new theories; pursued applications of the theoretical results; helped to organize and participated in seminars and conferences; and disseminated the results of the research through coauthored publications and public lectures. The main focus of the proposal was nonlinear partial differential equations and generalized convexity concepts, with applications to optimal transport, to theoretical economics, and to control engineering. The key problems formulated in the proposal, of characterizing quasiconvex functions through nonlinear partial differential equations and of establishing uniqueness of solutions results for such equations, have been solved. The results have been published in several articles. Overall, the project led to nine articles in peer-refereed journals and further conference proceedings publications. The research has led to new open problems and new applications, and further publications are expected. The investigators disseminated the new results through several seminar and conference presentations and through extended research visits, for example to the Mathematical Sciences Research Institute. The grant support led to participation and lectures by the investigators in several conferences with international impact, like the conference organized by the Society for Industrial and Applied Mathematics on the Analysis of Partial Differential Equations, the innaugural Mathematical Congress of the Americas, and more. The investigators contributed to organization of such conferences as well.
