Monotonicity formulas, nonlinear PDE's and sub-Riemannian Geometry

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This proposal is concerned with a number of questions at the interface of nonlinear partial differential equations and geometry, with particular emphasis on sub-Riemannian manifolds. The unifying theme is the systematic search of some basic monotonicity properties of the solutions of the problem at hand. Such properties play a special role in analysis and geometry and often lead to a remarkable insight in the nature of the relevant equations. One of the main directions in this proposal is a new notion of curvature in sub-Riemannian geometry. It represents a generalization of the Ricci curvature tensor from Riemannian geometry. Combining new Bochner identities with the monotonicity of some entropy-like functionals, for manifolds for which such generalized Ricci tensor is nonnegative one is led to a priori gradient bounds of Li-Yau type, Harnack inequalities, Gaussian upper bounds, isoperimetric inequalities, and a sub-Riemannian Bonnet-Myers compactness theorem in the strictly positive case. In another direction the proposal aims at furthering the present knowledge of minimal surfaces in sub-Riemannian geometry with particular emphasis on the sub-Riemannian Bernstein problem. The PI and his co-authors have recently solved this problem in the first (three-dimensional) Heisenberg group. The proposed research revolves around the analysis of the higher dimensional problem as well as the study of new monotonicity properties of the relevant area functionals. In yet another direction the proposal is concerned with the study of some new monotonicity properties of solutions of variational inequalities of elliptic and parabolic type with an obstacle confined to lie in alower dimensional manifold. Such monotonicity formulas are then applied to the study of the regularity of the relevant free boundary problems.This proposal can be placed at the confluence of two major areas of research in mathematics known as partial differential equations and Riemannian geometry. Partial differential equations are relations between an unknown function and a certain number of its derivatives. They govern the observable phenomena of the physical world. Riemannian geometry provides with a framework which is necessary to understand what happens when we are confronted with phenomena which fall outside the classical mechanics of Newton and Galilei. For instance, in Einstein?s theory of relativity the description of the curved space-time requires the use of Riemannian manifolds, with their intrinsic geometry. The past decade has witnessed an explosion of interest in a far reaching generalization of Riemannian geometry, as well as in the relevant partial differential equations which are needed to describe the new phenomena which arise in this area. Since this proposal is at the forefront of some of these developments it has the potential to impact those areas of mathematics and of the applied sciences (robotics, mechanical engineering, neuroscience) which are at the origin of these advances. In view of the extensive involvement and training of doctoral students and post-doctoral advisee, and the systematic dissemination of the relevant research through seminars, lectures, conferences, publications and websites, this proposal presents a strong component of human resources development.