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				\begin{document}$$\varvec{C^{0,0}}$$\end{document} estimates for linear parabolic operators

Hongjie Dong; Luis Escauriaza; Seick Kim

doi:10.1007/s00028-021-00729-8

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Publisher: Springer Journals
Copyright: Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
ISSN: 1424-3199
eISSN: 1424-3202
DOI: 10.1007/s00028-021-00729-8
Publisher site: See Article on Publisher Site

Abstract

We show that weak solutions to parabolic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower-order coefficients verify certain conditions. Analogous results are obtained for non-divergence form parabolic operators and their adjoint operators. Under similar conditions, we also establish a Harnack inequality for nonnegative adjoint solutions, together with upper and lower Gaussian bounds for the global fundamental solution.

Journal

Journal of Evolution Equations – Springer Journals

Published: Dec 1, 2021

Keywords: Parabolic Dini mean oscillation; C1/2,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1/2,1}$$\end{document} estimates; C1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,2}$$\end{document} estimates; C0,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{0,0}$$\end{document} estimates; Primary 35B45; 35B65; Secondary 35K10

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