A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields

DOI10.4208/JPDE.V33.N4.3zbMATH Open1474.35355arXiv1705.06278OpenAlexW3047517420WikidataQ114021182 ScholiaQ114021182MaRDI QIDQ4996095

Lingling Hou, Pengcheng Niu

Publication date: 1 July 2021

Published in: Journal of Partial Differential Equations (Search for Journal in Brave)

Abstract: In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hormander's vector fields and establish a Nash type result, i.e., the local Holder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincar'e inequality of Hormander's vector fields and a De Giorgi type Lemma, the Holder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Holder space. As a consequence, we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.

Full work available at URL: https://arxiv.org/abs/1705.06278

zbMATH Keywords

weak solution Harnack inequality Hölder regularity Hörmander's vector fields divergence parabolic equation

Mathematics Subject Classification ID

Smoothness and regularity of solutions to PDEs (35B65) Weak solutions to PDEs (35D30) Second-order parabolic equations (35K10)

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