Degenerate Elliptic Equations: Regularity of weak solutions with applications

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My research programme concerns itself with the study of second order quasilinear and nonlinear partial differential equations using methods that lay at a cross section between classical, functional and harmonic analysis, three major branches of mathematics with applications in the physical sciences and engineering. Partial differential equations (PDEs) are closely connected to all of the sciences (with emphasis on physics and mathematics) because of their connection to the behavior of physical systems via Newton's laws. I study classes of non-elliptic or degenerate elliptic PDEs connected to Geometry and Physics in areas studying surfaces with prescribed curvature and fluid dynamics. Some familiar equations that fall in the scope of my work are the famous Laplace and p-Laplace equations that are used, for example, to model fluid flow associated to non-Newtonian fluids in Physics. I study these equations to find conditions on their structure under which a solution of the equation exists and/or is regular (bounded, continuous, differentiable). I do this taking inspiration from the deep techniques of mathematicians like DeGiorgi, Nash, Moser, Serrin, and Trudinger. My research programme is also concerned with applications of new regularity results, particularly in mathematics. Two aspects of particular interest: I am keenly interested in the exploration of deep connections between the existence of regular solutions of boundary value problems for degenerate elliptic PDEs and the validity of important norm-inequalities like Poincare and Sobolev estimates. Secondly, I am interested in further extending the classical Myers-Serrin H=W result to Sobolev spaces defined with respect to classes of vector fields that may not be Lipschitz continuous. This research progamme provides opportunities for both undergraduate and graduate students while also promoting both national and international collaborations with mathematicians working in universities in the United States and Europe.