Nonlinear Partial Differential Equations in Sub-Riemannian Geometry

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Nonlinear Partial Differential Equations in sub-Riemannian Geometry Abstract of Proposed Research Nicola Garofalo Sub-Riemannian spaces model media with a constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem at hand. Typical examples are crystalline structures, or the movements of the arm of a robot. Models of sub-Riemannian spaces appear in diverse areas of both pure and applied sciences. These include harmonic analysis, several complex variables, group representation, calculus of variations and control theory, geometry (collapsing of Riemannian spaces, CR geometry), geometric measure theory, (Alexandrov spaces, Lie group theory, complex manifolds), quantum mechanics, robotics, mathematical finance, material sciences (crystalline structures), medicine (neurophysiology of the cerebral cortex). The development of analysis and geometry during the past century has been greatly influenced by questions arising from the analysis of systems of partial differential equations; often nonlinear systems. While most of these problems have by now been settled in the classical Euclidean or Riemannian settings, their sub-Riemannian counterparts presently form a body of fundamental and challenging new directions in mathematics. The appropriate mathematical formulation of the problems at hand involves the framework of sub-Riemannian spaces, whose basic prototype is the Heisenberg group (also known to physicists as Weyl group). The class of Carnot groups, is the geometric framework for most problems to be studied under this award. Specific topics include (i) To continue the study of minimal surfaces with particular emphasis on the Bernstein problem and on the question of their regularity including Poincare-Sobolev inequalities and Liouville type theorems on minimal surfaces; (ii) The development of a regularity theory for new variational inequalities with non-holonomic constraints arising in various branches of the applied sciences and the investigation of monotonicity formulas for the relevant constrained energies associated with these problems; (iii) Isoperimetric inequalities for the Gaussian measures associated with the heat semigroup; (iv) To continue the development of the theory of convexity in connection with a maximum principle of Alexandrov-Bakelman-Pucci type; (v) To investigate a CR positive mass theorem using the theory of minimal surfaces; (vi) To study the sharp interior regularity of solutions of nonlinear subelliptic equations. The basic laws that describe most natural phenomena are usually stated as partial differential equations or systems of equations. An understanding of the physical world also requires use of the underlying geometric structure. The present proposal belongs to the mainstream of research which sits at the confluence of the theory of partial differential equations with an emerging type of geometry, called sub-Riemannian geometry. Both theories have witnessed an explosion of interest in the last decade and now attract the interest of various schools of mathematicians both in the US and abroad. Under this award we will investigate problems from mathematical physics and geometry where symmetry plays an important role. Symmetry is present everywhere in nature, including in the fundamental laws of gravitation and electrostatic attraction. The study of conditions under which a natural phenomenon develops symmetries has both practical consequences and intrinsic interest.