Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions (Q2915221)

The authors study the following second order parabolic equation NEWLINE\[NEWLINE\sum_{i,j=1}^{m}a_{ij}(x,t)X_{i}X_{j}u-\partial_{t}u=0NEWLINE\]NEWLINE in a special cylinder over a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance. The \(X_{i}\)'s form a system of smooth vector fields satisfying the Hörmander rank condition. There are also some assumptions on the coefficient matrix \((a_{ij})\). In this setting the authors prove three main theorems. NEWLINENEWLINENEWLINENEWLINE The first one is a backward Harnack-type inequality for nonnegative solutions vanishing on the lateral boundary. More specifically one has that NEWLINE\[NEWLINEu(x,t)\leq c u(A_{r}(x_{0},t_{0})),NEWLINE\]NEWLINE where \(c\geq 1\) is a constant depending only on the differential operator, the Carnot-Carathéodory diameter, the height of the cylinder, as well as on other constants which measure the degree of non-tangentiality of the approach to the point \((x_{0},t_{0})\) which lies in the lateral boundary of the cylinder. The point \(A_{r}(x_{0},t_{0})\) is defined by a ``corkscrew'' condition. NEWLINENEWLINENEWLINENEWLINE The second one is the Hölder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a piece of the lateral boundary. Namely, one has that if \(u,v\) are nonnegative solutions vanishing on a subset of the lateral boundary defined as the intersection with the Cartesian product of a Carnot-Carathéodory ball times a specially chosen interval, then the Hölder continuity holds on a subset of the cylinder given by the intersection with a smaller ball times some smaller interval. NEWLINENEWLINENEWLINENEWLINE The final one is the doubling property for the parabolic measure associated with the operator which acts on \(u\) to give the left hand side of the studied equation. The parabolic measure is the (unique) probability measure that can be used to construct the solution of the Dirichlet problem by integrating over a given part of the boundary. Now, if additional assumptions are met, then the measure of the intersection of the lateral boundary with a ball times an interval is estimated above by a suitable constant times the measure of the intersection of the same boundary with a ball of half the above radius times a suitably chosen smaller interval.NEWLINENEWLINENEWLINENEWLINEThis paper somehow generalizes and extends earlier results of Fabes, Safonov and Yuan, where Lipshitz cylinders were considered, see [\textit{E. B. Fabes, M. V. Safonov} and \textit{Y. Yuan}, Trans. Am. Math. Soc. 351, No. 12, 4947--4961 (1999; Zbl 0976.35031)] and [\textit{M. V. Safonov} and \textit{Y. Yuan}, Ann. Math. (2) 150, No. 1, 313--327 (1999; Zbl 1157.35391)].