Qualitative Properties of Solutions to Second Order Elliptic and Parabolic Differential Equations

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The project deals with properties of solutions to elliptic and parabolic equations of second order, such as estimates for Green's functions and harmonic or caloric measures, uniqueness of blowup solutions to semilinear elliptic equations, regularity of solutions to quasilinear and nonlinear equations under weak structural assumptions. Part 1 of the project is devoted to the behavior of solutions near the boundary. Special attention is paid to the conditions on the coefficients, which guarantee the absolute continuity of harmonic and caloric measures with respect to the surface measure on the boundary. Such Dini-type conditions are known for equations in the divergence form. The study of equations in the non-divergence form needs quite different technique. Part 1 also includes the problem of uniqueness of solutions to semilinear equations, which blow up on the boundary. Our recent results show that positive solutions, which vanish on the boundary, have same rate of decay. We expect blowup solution near a portion of the boundary must have same rate of growth. Part 2 of the project deals with the regularity of solutions of quasilinear equations with minimal smoothness with respect to the gradient of solution, and of fully nonlinear equations without the concavity conditions with respect to the second derivatives. An essential part of the project deals with boundary properties of solutions of partial differential equations with non-smooth coefficients. Such equations appear in the investigation of different processes in composite materials, porous media, chemistry, biology, etc. The boundary properties of solutions are especially important, because they are "responsible" for interaction of a given object with its environment (heat emission, radiation, etc). In many situations, in order to get a desirable effect in a given region, one can only act on its boundary. The study of behavior of solutions near the boundary allows us to make such boundary control more predictable and effective. The blowup solutions of semilinear equations are associated with certain processes in biology, medicine, nuclear engineering, where their control cannot be overestimated. In this project, we also discuss the regularity of solutions of nonlinear equations. Positive results of such sort are interesting not only from the theoretical point of view, they also give the background for numerical solution of equations.