Non-linear equations in analysis and geometry

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This proposal is concerned with two main projects. The former focuses onvarious questions in sub-Riemannian geometry and in the closely connectedanalysis of sub-elliptic pde's and systems. The PI proposes to investigatethe classification of non-negative entire solutions to non-linearequations in groups of Heisenberg type, and compute the best constants inthe Folland-Stein Sobolev embedding. This program is instrumental to apossible attack of the compact CR Yamabe problem in the open case of CRmanifolds of co-dimension higher than one. The geometric case of suchembedding will also be investigated along with the relative isoperimetricinequalities. The PI also proposes to study the regularity of minimalsurfaces, the question of traces on lower dimensinal sub-manifolds offunctions having integrable horizontal derivatives. The basic boundaryvalue problems, such as the Dirichlet and the Neumann problem will also beinvestigated, and a theory of variational inequalities and regularity of"free boundaries" will be developed. The second project is concerned withvarious problems in which symmetry plays an important role. One of them isconcerned with the determination of the extremal functions in theTomas-Stein restriction theorem for the Fourier transform. Other problemsare connected with symmetry in the exterior obstacle problem, a conjectureof De Giorgi connected to minimal surfaces, and symmetry in the evolutionof surfaces driven by mean curvature.Partial differential equations and systems formed by the latter are thebasic laws which describe most natural phenomena. An understanding of thephysical world also requires grasping the underlying geometric structureof the latter in its various forms. The present proposal belongs to thatmainstream of research which sits at the confluence of the theory ofpartial differential equations and systems, both linear and non-linear,and their connections with an emerging type of geometry, calledsub-Riemannian geometry. Both theories have witnessed an explosion ofinterest in the last decade and they continue to attract the interest ofvarious schools of mathematicians both nationwide and abroad. Another mainpart of this proposal is devoted to the study of physical and mathematicalproblems in which symmetry plays an important role. Symmetry is presenteverywhere in nature, a remarkable instance being the fundamental lawsof gravitation and electrostatic attraction. The study of conditions underwhich a given natural phenomenon develops symmetries is both important forits practical consequences (the presence of symmetriesdrastically reduces the human effort) and for its implicationsin the furthering of our knowledge.