Research and Recent Publications

1a. Potential Theory and Partial Differential Equations

Norbert Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results in XX century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs, and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories. My recent developments precisely characterize the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extend the role of the Wiener test in classical analysis. The Wiener test at ∞ arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at ∞ arises as a thinness criteria at ∞ in fine topology. In a probabilistic context, the Wiener test at ∞ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criteria for the removability of singularities and/or for unique continuation at the finite boundary point. Here are my recent related publications:

1b. Geometric Boundary Regularity Tests for PDEs and Asymptotic Laws for Wiener Processes

One of the oldest problems in the theory of PDEs is the problem of finding geometric conditions on the boundary manifold for the regularity of the solution of the elliptic and parabolic PDEs. There is a deep connection between this problem and the delicate problem of asymptotics of the corresponding Wiener processes. In my paper

I proved a geometric iterated logarithm test for the boundary regularity of the solution to the heat equation. In addition, I proved an exterior hyperbolic paraboloid condition for the boundary regularity, which is the parabolic analogy of the exterior cone condition for the Laplace equation. In fact, for the characteristic top boundary point of the symmetric rotational boundary surfaces, the necessary and sufficient condition for the regularity coincides with the well-known Kolmogorov-Petrovsky test for the local asymptotics of the multi-dimensional Brownian motion trajectories:

In the case when symmetric rotational boundary surfaces extends to t=-∞, the regularity of the point at ∞ in my sense precisely characterize the uniqueness of the bounded solutions. I proved a geometric necessary and sufficient condition for the regularity of ∞. In the probabilistic context the result coincides with the Kolmogorov-Petrovsky test for the asymptotics of the multi-dimensional Brownian motion trajectories at infinity. Here are related publications:

1c. Qualitative Theory of Nonlinear PDEs

PDEs arising in a majority of real world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs, one can observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear degenerate/singular parabolic equation, so called nonlinear diffusion equation. In a series of publications I developed a general theory of the nonlinear degenerate and/or singular parabolic equations in general non-smooth domains under the minimal regularity assumptions on the boundary. This development was motivated with numerous applications, including flow in porous media, heat conduction in a plasma, free boundary problems with singularities, etc. For example, primarily by applying this theory, I solved Barenblatt's problem about the short-time behaviour of interfaces for the reaction-diffusion equations. Here are some of my related publications: