Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, pp. 437-474.

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

 $Hu=\sum _{i,j=1}^{m}{a}_{ij}\left(x,t\right){X}_{i}{X}_{j}u-{\partial }_{t}u=0,\phantom{\rule{4pt}{0ex}}\left(x,t\right)\in {ℝ}_{+}^{n+1},$

where $X=\left\{{X}_{1},...,{X}_{m}\right\}$ 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.

Published online:
Classification: 31C05, 35C15, 65N99
Frentz, Marie 1; Garofalo, Nicola 2; Götmark, Elin 3; Munive, Isidro 2; Nyström, Kaj 4

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
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title = {Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Frentz, Marie; Garofalo, Nicola; Götmark, Elin; Munive, Isidro; Nyström, Kaj. Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, pp. 437-474. http://www.numdam.org/item/ASNSP_2012_5_11_2_437_0/

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