JAY KOVATS

RESEARCH AREAS

A main area of research in the qualitative theory of second-order partial differential equation is the theory of the regularity of solutions. Simply put, regularity theory investigates smoothness properties of solutions inherited as a result of their satisfying a particular PDE. The prototypical second-order parabolic differential equation is the heat equation that describes the diffusion of heat within a given region. In general, second-order linear parabolic equations describe, in physical applications, the time-evolution of the density of some quantity u (e.g., chemical concentration, temperature, electrostatic potential) diffusing within a region.

We are interested in solving these equations, or at the very least, determining whether solutions exist. One way to do this is to suppose a solution to a particular class of equations exists and the deduce properties of this solution that it must a priori possess as a result of satisfying the equation. Parabolic equations have a definite structure and the way this structure affects regularity is of interest. It is of interest because the regularity of solutions is intimately related to the existence of solutions by means of the concept of a priori estimate. If one can obtain a "smoothness" estimate, valid for all possible solutions of a class of equations, then by standard compactness techniques (e.g., Arzela-Ascoli, the method of continuity) one can prove that solutions to this class of equations actually exist.

Thus, solvability is one of the motivations for investigating regularity properties of solutions. Given a particular class of equations, the questions becomes "assuming only minimal continuity conditions on an arbitrary solution, what additional properties does it possess, simply by virtue of being a solution?" We remark that in the nonlinear setting, techniques for obtaining these estimates play an even greater role, as integral representations of solutions are generally not available.

SELECTED PUBLICATIONS

• Kovats, J., A Three-Curves Theorem for Viscosity Subsolutions of Parabolic Equations, Proceedings of the MAS 131 (2003), No. 5, 1509-1514.
• Kovats, J., The Kolmogorov Equation with Time-Measurable Coefficients, Eloctronic Journal of Differential Equations 77 (July 2003), 1-14.
• Kovats, J., On the Regularity of Viscosity Solutions of Fully Nonlinear Elliptic Equations, Proceedings of the 2004 World Congress of Nonlinear Analysts, Orlando, Nonlinear Analysis (2005).
• Kovats, J., Dec. 1997. "Fully Nonlinear Elliptic Equations and the Dini Condition," Communications in PDE, Vol. 22, 11-12, pp. 1911-1927.
  

KEYWORDS

Qualitative Properties of Solutions of Second-order Elliptic and Parabolic Equations

Florida Institute of Technology
Department of Mathematical Sciences
150 West University Blvd.
Melbourne, FL 32901-6975
Fax (321) 674-7412
E-mail: jkovats@fit.edu