In a cylinder we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form
where is a system of vector fields in satisfying Hörmander’s rank condition (1.2), and is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance induced by . Concerning the matrix-valued function , we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries are Hölder continuous with respect to the parabolic distance associated with . 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 (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 be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.
@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}, pages = {437--474}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {2}, year = {2012}, mrnumber = {3011998}, zbl = {1258.31005}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_437_0/} }
TY - JOUR AU - Frentz, Marie AU - Garofalo, Nicola AU - Götmark, Elin AU - Munive, Isidro AU - Nyström, Kaj TI - Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 437 EP - 474 VL - 11 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2012_5_11_2_437_0/ LA - en ID - ASNSP_2012_5_11_2_437_0 ER -
%0 Journal Article %A Frentz, Marie %A Garofalo, Nicola %A Götmark, Elin %A Munive, Isidro %A Nyström, Kaj %T Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 437-474 %V 11 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2012_5_11_2_437_0/ %G en %F 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, Série 5, Tome 11 (2012) no. 2, pp. 437-474. http://www.numdam.org/item/ASNSP_2012_5_11_2_437_0/
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