19 - Parabolic Harnack inequality and regularity theory

Summary

In this chapter, we will present Moser's version of the Nash–Mosers Harnack inequality for parabolic equations. The elliptic version (also proved by De Giorgi) can be considered as a special case when the solution is time independent. The iteration procedure of Moser was particularly useful in the theory of geometric analysis. We will attempt to cover this in as much generality as possible while keeping explicit account of the dependency of various geometric and analytic constants. In applying this type of argument in the study of geometric partial differential equations, often the explicit geometric dependency is crucial. As a result of these estimates, one derives a mean value inequality for nonnegative subsolutions and a Harnack inequality for positive solutions of a fairly general class of parabolic operators. In particular, it gives a C^α estimate for solutions of any second order parabolic (elliptic) operators of divergence form with only measurable coefficients. This regularity result was the original motivation for the development of this theory. We shall point out that the mean value inequality and the Harnack inequality derived from this argument are applicable to a more general class of equations, while the ones given in earlier chapters yield stronger results but require more smoothness from the operator. Both approaches are important in the theory of geometric analysis, but they are suited to different types of situation. The following account is a slightly modified version of Moser's argument that has been adapted to a more geometrical setting.