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Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
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2012 (English)In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 11, no 2, 437-474 p.Article in journal (Refereed) Published
Abstract [en]

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

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),

where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander'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.

Place, publisher, year, edition, pages
2012. Vol. 11, no 2, 437-474 p.
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:umu:diva-47921ISI: 000309320600009OAI: oai:DiVA.org:umu-47921DiVA: diva2:445343
Available from: 2011-10-07 Created: 2011-10-03 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Topics on subelliptic parabolic equations structured on Hörmander vector fields
Open this publication in new window or tab >>Topics on subelliptic parabolic equations structured on Hörmander vector fields
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2012. 36 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 51
Keyword
subelliptic, parabolic, obstacle problem, boundary behaviour, weak approximation
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-51604 (URN)978-91-7459-354-9 (ISBN)
Public defence
2012-02-24, Mit-huset, MA121, Umeå Universitet, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2012-02-03 Created: 2012-01-27 Last updated: 2012-01-27Bibliographically approved

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