ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

10. Partial Differential Equations

1991 MS Classification: 35K Safonov, Mikhail, University of Minnesota, USA Estimates near the boundary for solutions of second order parabolic equations

We consider linear uniformly parabolic equations of second order, with measurable coefficients, in a Lipschitz cylinder Q = Ω × (0, T), in both the divergence form

Lu = ∑_{i,j=1}^{n} D_{i}(a_{ij}D_{j}u) − u_{t} = 0

and the non-divergence form

Lu = ∑_{i,j=1}^{n} a_{ij}D_{i}D_{j}u − u_{t} = 0

where D_{i} = ∂/∂x_{i}. We will discuss different (boundary, backward, etc.) forms of the Harnack principle, the Hölder continuity of the quotient of two positive solutions vanishing on a portion of the parabolic boundary, the doubling property of the L-caloric measure on the boundary, the classical Fatou theorem. A general approach to these problems will be demonstrated, which is based on the “standard” Harnack inequality and an elementary estimate for oscillation of solutions near the boundary.

ICM 1998 Berlin · Plenary and Invited Lectures