Non-linear equations in analysis and geometry

Purdue Research Foundation, West Lafayette IN

Investigators

Abstract

This proposal is concerned with two main projects. The former focuses on various questions in sub-Riemannian geometry and in the closely connected analysis of sub-elliptic pde's and systems. The PI proposes to investigate the classification of non-negative entire solutions to non-linear equations in groups of Heisenberg type, and compute the best constants in the Folland-Stein Sobolev embedding. This program is instrumental to a possible attack of the compact CR Yamabe problem in the open case of CR manifolds of co-dimension higher than one. The geometric case of such embedding will also be investigated along with the relative isoperimetric inequalities. The PI also proposes to study the regularity of minimal surfaces, the question of traces on lower dimensinal sub-manifolds of functions having integrable horizontal derivatives. The basic boundary value problems, such as the Dirichlet and the Neumann problem will also be investigated, and a theory of variational inequalities and regularity of "free boundaries" will be developed. The second project is concerned with various problems in which symmetry plays an important role. One of them is concerned with the determination of the extremal functions in the Tomas-Stein restriction theorem for the Fourier transform. Other problems are connected with symmetry in the exterior obstacle problem, a conjecture of De Giorgi connected to minimal surfaces, and symmetry in the evolution of surfaces driven by mean curvature. Partial differential equations and systems formed by the latter are the basic laws which describe most natural phenomena. An understanding of the physical world also requires grasping the underlying geometric structure of the latter in its various forms. The present proposal belongs to that mainstream of research which sits at the confluence of the theory of partial differential equations and systems, both linear and non-linear, and their connections with an emerging type of geometry, called sub-Riemannian geometry. Both theories have witnessed an explosion of interest in the last decade and they continue to attract the interest of various schools of mathematicians both nationwide and abroad. Another main part of this proposal is devoted to the study of physical and mathematical problems in which symmetry plays an important role. Symmetry is present everywhere in nature, a remarkable instance being the fundamental laws of gravitation and electrostatic attraction. The study of conditions under which a given natural phenomenon develops symmetries is both important for its practical consequences (the presence of symmetries drastically reduces the human effort) and for its implications in the furthering of our knowledge.