Mathematical Sciences: Second Order Elliptic and Parabolic Differential Equations

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ABSTRACT DMS-9623287 PI: SAFONOV UNIVERSITY OF MINNESOTA The project relates to important topics in the theory of Second Order Elliptic and Parabolic Partial Differential Equations, such as properties of solutions of linear equations with measurable coefficients (the backward Harnack inequality, the estimates for fundamental solutions, etc.), the problem of uniqueness of blowup solutions to semilinear elliptic equations, and the problem of regularity of solutions to fully nonlinear equations. Part 1 of the project deals with the estimates of solutions of linear equations which do not depend on the smoothness of coefficients. Special attention is paid to the local boundedness of solutions and to the behavior of positive solutions which blowup at the boundary. Uniqueness of such solutions is investigated under minimal assumptions on the smoothness of coefficients and on the structure of the boundary. In Part 3, the results on the classical solutions of linear equations are extended to the fully nonlinear equations satisfying a Dini condition with respect to independent variables. The estimates of solutions of linear equations with non-smooth coefficients are associated with certain properties of different processes in non-homogeneous environment, with many applications to heat-mass transfer, chemistry, porous media, traffic flow, biology, etc. Such estimates also serve as a background for the theory of nonlinear equations. The investigation of blowup solutions of semilinear equations is important in ecology, combustion theory, chemical and nuclear engineering. Many phenomena in these and other areas can be treated in terms of a superdiffusion process which describes a cloud rising as the limit of a system of independent Brownian particles which die at random times, leaving a random number of offspring. Fully nonlinear equations are closely related to the theory of optimal control of random processes where the optimal strategy can e found by solving of appropriate initial or bou ndary value problem for the corresponding nonlinear equation. This theory has applications to many physical and economical problems.