A Harnack inequality and Hölder continuity for weak solutions to parabolic operators involving Hörmander vector fields

Details A Harnack inequality and Hölder continuity for weak solutions to parabolic operators involving Hörmander vector fields A Harnack inequality and Hölder continuity for weak solutions to parabolic operators involving Hörmander vector fields

2010-10-07

arxiv.org/abs/1010.1554v1

Mathematics

Analysis of PDEs

Scientific paper

This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with H\"ormander vector fields. Adapting the iteration scheme of J\"urgen Moser for elliptic and parabolic equations in $\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i (a_{ij} X_j u)$ we use this to show H\"older continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parabolic equations of the form $u_t = -X_i^* A_j(X_j u)$. With this we show both a Harnack inequality and H\"older continuity of weak solutions.

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