Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions

作者：Frentz Marie^*; Garofalo Nicola; Gotmark Elin; Munive Isidro; Nystrom Kaj

来源：Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2012, 11(2): 437-474.

In a cylinder Omega(T) = Omega x (0, T) subset of R-+(n+1) we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form %26lt;br%26gt;H u = Sigma(m)(i,j=1) a(ij)(x, t)XiX (j)u - partial derivative(t)u = 0, (x, t) is an element of R-+(n+1), %26lt;br%26gt;where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander%26apos;s rank condition (1.2), and Omega is a non-tangentially accessible domain with respect to the Carnot-Caratheodory 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 Holder continuous with respect to the parabolic distance associated with d. Our main results are: I) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder 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 Holder continuity with respect to the intrinsic parabolic distance.

• 出版日期2012