A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields

2017/05/17 by Lingling Hou, Hou, Lingling, Pengcheng Niu +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1705.06278

openalex publication_date 2017/05/17 · openalex created_date 2017/05/26 · openalex updated_date 2026/07/28

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 Poincaré 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.

Citations

• A new proof of Moser's parabolic harnack inequality using the old ideas of Nash

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