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Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
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2012 (Engelska)Ingår i: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 11, nr 2, s. 437-474Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
2012. Vol. 11, nr 2, s. 437-474
Nationell ämneskategori
Matematisk analys
Identifikatorer
URN: urn:nbn:se:umu:diva-47921ISI: 000309320600009OAI: oai:DiVA.org:umu-47921DiVA, id: diva2:445343
Tillgänglig från: 2011-10-07 Skapad: 2011-10-03 Senast uppdaterad: 2018-06-08Bibliografiskt granskad
Ingår i avhandling
1. Topics on subelliptic parabolic equations structured on Hörmander vector fields
Öppna denna publikation i ny flik eller fönster >>Topics on subelliptic parabolic equations structured on Hörmander vector fields
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Ort, förlag, år, upplaga, sidor
Umeå: Umeå Universitet, 2012. s. 36
Serie
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 51
Nyckelord
subelliptic, parabolic, obstacle problem, boundary behaviour, weak approximation
Nationell ämneskategori
Matematisk analys
Forskningsämne
matematik
Identifikatorer
urn:nbn:se:umu:diva-51604 (URN)978-91-7459-354-9 (ISBN)
Disputation
2012-02-24, Mit-huset, MA121, Umeå Universitet, Umeå, 13:00 (Engelska)
Opponent
Handledare
Tillgänglig från: 2012-02-03 Skapad: 2012-01-27 Senast uppdaterad: 2018-06-08Bibliografiskt granskad

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Frentz, MarieGötmark, ElinNyström, Kaj

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Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V
Matematisk analys

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