In a cylinder ${\Omega}_{T}=\Omega \times (0,T)\subset {\mathbb{R}}_{+}^{n+1}$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form
$$Hu=\sum _{i,j=1}^{m}{a}_{ij}(x,t){X}_{i}{X}_{j}u-{\partial}_{t}u=0,\phantom{\rule{4pt}{0ex}}(x,t)\in {\mathbb{R}}_{+}^{n+1},$$ |
where $X=\{{X}_{1},...,{X}_{m}\}$ is a system of ${C}^{\infty}$ vector fields in ${\mathbb{R}}^{n}$ 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.
@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, 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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