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, p. 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=1maij(x,t)XiXju-tu=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 : 2018-06-21
Classification:  31C05,  35C15,  65N99
@article{ASNSP_2012_5_11_2_437_0,
     author = {Frentz, Marie and Garofalo, Nicola and G\"otmark, Elin and Munive, Isidro and Nystr\"om, Kaj},
     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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {2},
     year = {2012},
     pages = {437-474},
     zbl = {1258.31005},
     mrnumber = {3011998},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_437_0}
}
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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