This project is concerned with the study of nonlinear elliptic and parabolic equations in connection with more complex problems of differential geometry and with physical applications. Such problems include the evolution of a hypersurface in Euclidean (N+1)-space by functions of its principal curvatures, the Ricci flow on both Riemannian and Lorentzian manifolds, the Yamabe flow on surfaces, and the Weyl problem with nonnegative Gaussian curvature. Most of the proposed problems involve equations that are either singular or degenerate, hence problems for which the classical results fail. New analytical techniques will be developed in the project.
The project links a wide range of active fields of mathematics, in particular, nonlinear partial differential equations, geometry, and classical analysis. The proposed research activity on the geometry and regularity of degenerate nonlinear parabolic equations and free-boundary problems may lead to significant geometric and even topological applications. The principal investigator intends to study the applications of the mathematical problems to other disciplines such as quantum field theory, relativity theory, plasma physics, image analysis, and thin liquid film dynamics. Results will be disseminated to the research community at various meetings and by publication of research articles. New courses linking partial differential equations and geometric analysis for graduate students will be designed and implemented.
The outcomes of this award linked a wide range of active fields of mathematics, in particular nonlinear partial differential equations, differential geometry and classical analysis. Special emphasis was given the study of solutions to nonlinear geometric flows, including the Ricci flow, the Yamabe flow as well as the evolution of hypersurfaces by functions of their principal curvatures. The latter plays a role in a number of applied fileds, in particular on image processing. A major part of the performed research concerned with ancient solutions to geometric equations, including the curve shortening flow, the Ricci flow on surcaces and the Yamabe flow. These are rather special solutions yet important that often appear as limits of singularities. Their classification is found to play a role in understanding the singulariries of the flow. In addtion, the classification of solutions to the 2-dim Ricci flow appears to be relevant in quantum filed theory. The PI and colaborators established the classification of ancient solutions in varioaus geometric evolution equations and they also constructed new models of ancient solutions that reveal new phenomena. In a different line of reseach, t suported by this award, the PI and collaborators establised the optimal regularity in elliptic and parabolic obstacle problems, involving degenerate operators that are modeled on the Heston operator, the well known generator of the two-dimensional Heston stochastic volatility process with killing. Such problems originate from the American option problem in mathematical finance. Students and postdocs were trained. Special emphasis was l be given to the encouragement of talented female undergraduate students, graduate students and postdocs to pursue a successful career in mathematics. Results were disseminated to the research community at various meetings and by publication of research articles. New courses linking partial differential equations and geometric analysis for graduate students were designed and implemented.
