Research description

My research lies in calculus of variations and partial differential equations, with a particular emphasis on regularity, stability, and geometric structure. Many of the problems I study ask a common question: under weak or natural assumptions, can one prove that solutions, minimizers, interfaces, or transport maps are more regular, more stable, or more rigid than expected?

This point of view connects several areas of analysis, including optimal transport, Monge–Ampère equations, free boundary problems, geometric and functional inequalities, elliptic and parabolic PDE, geometric measure theory, transport equations, and evolution equations.

Optimal transport and Monge–Ampère equations

Optimal transport studies the most efficient way of moving one distribution of mass to another. Besides its original formulation, it has become a powerful tool in analysis, geometry, probability, and mathematical physics.

A central theme of my work in this area is the regularity of optimal maps and its connection with the Monge–Ampère equation. These questions involve understanding when transport maps are smooth, when singularities may occur, and how the geometry of the source and target measures affects the structure of the map.

The Monge–Ampère equation also appears in several geometric and variational problems. It provides a bridge between convex analysis, nonlinear PDE, and optimal transport. Questions of existence, uniqueness, stability, and partial regularity are central to this area and continue to play an important role in my work.

Free boundary problems

A major direction of my recent research concerns free boundary problems. These are problems in which the unknown is not only a function, but also an interface determined by the solution itself. Such interfaces arise naturally in obstacle problems, phase transitions, Bernoulli-type problems, Stefan-type problems, and nonlocal models.

The main questions concern the regularity of the free boundary, the structure of singularities, and the classification of possible blow-up limits or global solutions. A typical difficulty is that singularities may form even when the equation is very regular. Understanding which singularities can occur, how large the singular set can be, and whether singularities disappear under generic assumptions are central problems in this field.

Many of the methods used in this area combine variational arguments, monotonicity formulas, blow-up analysis, dimension-reduction techniques, and fine PDE estimates. These tools reveal the local geometry of the free boundary and help distinguish regular points from singular ones.

Geometric and functional inequalities

Another important part of my research concerns sharp geometric and functional inequalities. These include isoperimetric-type inequalities, Wulff inequalities, Brunn–Minkowski-type inequalities, Prékopa–Leindler and Borell–Brascamp–Lieb inequalities, Sobolev inequalities, and log-Sobolev inequalities.

A recurring question is quantitative stability. Once the optimal constant and equality cases are known, one can ask whether a nearly optimal set or function must be close to an exact optimizer. This leads to refined estimates that measure the distance from equality in terms of a natural deficit.

This area brings together ideas from optimal transport, convex geometry, calculus of variations, and PDE. It is also closely connected to questions of rigidity: when equality or near equality occurs, the underlying object is forced to have a special structure.

Elliptic PDE and stable solutions

I am also interested in regularity, rigidity, and classification problems for nonlinear elliptic PDE. A central class of problems concerns stable solutions of semilinear elliptic equations. Stability is a variational condition that often forces strong regularity properties, but in high dimensions singularities may still appear.

The study of stable solutions raises several natural questions: in which dimensions are solutions smooth? If singularities occur, how large can the singular set be? Can one describe the behavior of solutions near singular points? These questions connect elliptic estimates, variational methods, geometric analysis, and dimension-reduction ideas.

Related problems include boundary reactions, fully nonlinear equations, transmission problems, overdetermined problems, and local or nonlocal elliptic operators. In many of these questions, the goal is to understand how analytic assumptions impose geometric structure on the solutions.

Evolution equations, diffusion, and transport equations

Some of my work concerns time-dependent PDE, including nonlinear diffusion equations, gradient flows, kinetic equations, aggregation models, and transport equations with rough vector fields.

Optimal transport and Wasserstein geometry provide useful tools for studying several of these problems, especially when the equation can be interpreted as a gradient flow in a space of probability measures. In other cases, the main challenge is to understand existence, uniqueness, compactness, stability, and long-time behavior under weak regularity assumptions.

Although these topics are not the main focus of my current research, they remain part of the broader analytical framework that connects PDE, calculus of variations, and geometric methods.

Other directions

Other directions of my work include geometric measure theory, weak KAM theory, Hamilton–Jacobi equations, stochastic analysis, random matrix theory, sub-Riemannian geometry, variational models in fluid mechanics, and selected problems in applied mathematics, especially where analytical, variational, and optimal-transport methods play a role.

Some of these topics arose from specific questions, while others are connected to the broader themes of regularity, stability, optimal transport, and geometric structure. Together, they reflect my general interest in using methods from nonlinear analysis to understand problems where geometry, PDE, variational ideas, and applications interact.