Mass transport, Geometric inequalities and partial differential systems

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Partial Differential Equations and Systems provide major bridges between mathematics and many other disciplines in basic and applied sciences and engineering. This central field of Mathematics is used to model physical phenomena, biological systems, quantum materials and solid state devices, to name only a few areas of applications. My research program is focused on developing novel, preferably general and encompassing methods to deal with the analysis of classes of such systems, including evolution equations. The goal is to either apply these methods to problems that are not amenable to standard techniques, or to exhibit universal features, which could aid in the understanding and the deployment of the often-required hard analysis. The methods normally address questions of existence, multiplicity, regularity and other qualitative properties of solutions of stationary or dynamic differential equations and systems. They include: • Self-dual variational calculus on linear as well as curved spaces, to deal with equations, which are not of Euler-Lagrange type, such as those involving non self-adjoint terms or/and non-linear operators. • Mass transport theory to handle the still elusive variational problems arising in non-linear elasticity theory.  • Monge-Kantorovich theory to decouple certain partial differential systems, such as those of De Giorgi-type arising as a limiting elliptic system describing phase separation of multiple state Bose-Einstein condensates. • Its equivariant form for applications to matching problems in economics, and to Density Functional Theory, a well-established method for tackling the quantum mechanics of many-body systems. • Infinite dimensional critical point theory for borderline problems that lack compactness. These often require a fine analysis of the linear operators involved, such as the second and fourth-order Hardy-Schrodinger operators, the fractional Laplacians, and appropriate perturbations of them. The singularities could lie either in the interior, or on the boundary of the domain under consideration. The latter can be Euclidean or curved. • Novel approaches for the identification of best constants and/or the establishment of the existence of steady states and other extremals for various functional and geometric inequalities, such as those of Caffarelli-Kohn-Nirenberg, Moser-Aubin-Onofri, and Gagliardo-Nirenberg inequalities. • New insight for addressing issues of regularity, stability, and critical dimensions in various non-linear eigenvalue problems (self-adjoint or not), especially those dealing with the biharmonic operator, and the fractional Laplacians.