My main research interest is in partial differential equations (PDE) of parabolic type. Solutions to these equations have representations as expectation values of functions of a random variable. Therefore my work often involves some intuitions drawn from probability theory. Currently there are two particular directions I am concentrating on.
The first of these involves solving some non-linear PDE and trying to understand the large time behavior of the solutions. Solutions of the PDE model the physical phenomenon known as Ostwald ripening, first observed in 1896. It occurs in solid solutions in which small crystals dissolve and redeposit onto larger crystals. There is a hierarchy of models describing Ostwald ripening. In recent years I have been studying the simplest of these models-the Lifschitz-Slyozov-Wagner (LSW) and Becker-Doering (BD) models- and proving some rigorous mathematical results about them.
The second direction I am concentrating on concerns linear elliptic and parabolic PDE in divergence form with random coefficients. It was shown around 1980 that if the environment for the coefficients is ergodic, then the solution of the PDE behaves on large scales like the solution of a constant coefficient PDE, the so called homogenized equation. In 1986 Yurinskii obtained the first results showing that for certain strongly mixing environments one can estimate the error on a large scale between the solution to the random PDE and the solution to the homogenized PDE. I have been working in recent years on extensions of Yurinskii?s results and their connections with understanding correlations in the statistical mechanics of the Coulomb dipole gas.
