Some nonlinear problems in analysis and geometry

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PI: Nicola Garofalo, Purdue UniversityDMS-0300477Abstract:The development of analysis and geometry during the past century has been greatly influenced by the desire of solving various basic problems involving some special partial differential equations, mostly of nonlinear type. 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 open questions. One of the broader objectives of this proposal is to study some of them. This PI is concerned with developing a new theory of minimal surfaces, or more in general surfaces with bounded mean curvature, in sub-Riemannian spaces, study their regularity and classify the isoperimetric sets in some model spaces with symmetries. He proposes a calculus on hypersurfaces which hinges on the idea of horizontal Gauss map, and leads to a new notion of mean curvature The analysis of the ensuing nonlinear equations and systems constitutes a challenging new avenue of study. Within such calculus, minimal surfaces are thus hypersurfaces of zero mean curvature, and a problem of fundamental interest is a sub-Riemannian version of the famous conjecture of Bernstein. The latter displays a marked discrepancy with its classical ancestor and there is a host of new geometric phenomena connected with the singularities of the Gauss map which generically occur at those points where the subbundle which generates the sub-Riemannian structure becomes part of the tangent space to the hypersurface. Given the role of the classical Bernstein problem in the development of last century's mathematics, it is foreseeable that the theory of sub-Riemannian minimal surfaces and the corresponding Bernstein problem will sparkle a broad development. The PI also proposes to find the minimizers in the Folland-Stein embedding for groups of Heisenberg type and Siegel domain of type 2, and thereby compute the best constants. This program is instrumental to attacking the compact CR Yamabe problem for CR manifolds of higher codimension. In connection with the CR Yamabe problem the PI proposes to investigate a CR version of the positive mass theorem from relativity due to Schoen and Yau. It is expected that the theory of minimal surfaces previously mentioned will play an important role. Another emerging theory in sub-Riemannian geometry is that of equations of Monge-Amp\`ere type, which occupy a central position in geometry as well as in the calculus of variations in view of their tight connection with the problem of mass transport. The PI proposes to investigate a new estimate connected with a sub-Riemannian version of the geometric maximum principle of Alexandrov, Bakelman, and Pucci. In joint work he has recently obtained results for the appropriate class of ``convex" functions, and, inspired by N.Krylov's approach, established monotonicity type results for a functional involving the symmetrized horizontal Hessian along with some appropriate commutators. Another problem included in this proposal is the optimal regularity for nonlinear equations arising in the study of quasiregular mappings between Carnot groups. This is presently a fundamental open question and, without its solution, it will be impossible to make substantial advances in nonlinear potential theory for sub-Riemannian spaces. In this connection the PI also plans to analyze the delicate question of the uniqueness of the fundamental solution and Green function, and study the geometric properties of their level sets. Other directions of investigation are the analysis of boundary value problems (Dirichlet, Neumann) for subelliptic equations and their associated heat flows, the study of free boundary problems, and the analysis of global properties of solutions to some pde's arising in geometry and mathematical physics. 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 the mainstream of research, which sits at the confluence of the theory of partial differential equations and systems, mostly of nonlinear type, 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. This proposal is also concerned with problems from mathematical physics and geometry 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 natural phenomenon develops symmetries is important both for practical consequences and for its implications in the furthering of our knowledge.