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

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.