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

Frentz, Marie¹ ; Garofalo, Nicola²  ; Götmark, Elin³ ; Munive, Isidro² ; Nyström, Kaj⁴

¹ Department of Mathematicsand Mathematical Statistics Umeå University S-90187 Umeå, Sweden
² Department of Mathematics Purdue University West Lafayette IN 47907-1968, USA
³ Department of Mathematics and Mathematical Statistics Umeå University S-90187 Umeå, Sweden
⁴ Department of Mathematics Uppsala University S-751 06 Uppsala, Sweden

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 437-474

Résumé

In a cylinder ${\Omega }_T=\Omega \times (0,T)\subset {ℝ}_{+}^{n+1}$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

$$Hu=\sum _{i,j=1}^m{a}_{ij}(x,t)X_i{X}_{j}u-{\partial }_{t}u=0,(x,t)\in {ℝ}_{+}^{n+1},$$

where $X=\{X_1,...,X_m\}$ is a system of $C^{\infty }$ vector fields in ${ℝ}^n$ satisfying Hörmander’s rank condition (1.2), and $\Omega$ is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance $d$ induced by $X$. Concerning the matrix-valued function $A=\left\{a_{ij}\right\}$, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries $a_{ij}$ are Hölder continuous with respect to the parabolic distance associated with $d$. Our main results are: 1) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Hölder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator $H$ (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20,39]. With one proviso: in those papers the authors assume that the coefficients $a_{ij}$ be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.

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Publié le : 2018-06-21

Classification : 31C05, 35C15, 65N99