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 Ω T =Ω×(0,T) + n+1 we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

Hu= i,j=1 m a ij (x,t)X i X j u- t u=0,(x,t) + n+1 ,

where X={X 1 ,...,X m } is a system of C vector fields in n satisfying Hörmander’s rank condition (1.2), and Ω is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance d induced by X. Concerning the matrix-valued function A={a ij }, 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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     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://archive.numdam.org/item/ASNSP_2012_5_11_2_437_0/

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