On C1/2,1 , C1,2 , and C, estimates for linear parabolic operators

Hongjie Dong; Luis Escauriaza; Seick Kim

doi:10.1007/s00028-021-00729-8

On C¹^/²^,¹ , C¹^,² , and C^, estimates for linear parabolic operators

Hongjie Dong, Luis Escauriaza, Seick Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

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.

Original language	English
Pages (from-to)	4641-4702
Number of pages	62
Journal	Journal of Evolution Equations
Volume	21
Issue number	4
DOIs	https://doi.org/10.1007/s00028-021-00729-8
Publication status	Published - 2021 Dec

Bibliographical note

Funding Information:
S. Kim is supported by NRF Grant No. NRF-20151009350 and No. NRF-2019R1A2C2002724.

Funding Information:
L. Escauriaza is supported by Basque Government grant IT1247-19 and MICINN grant PGC2018-094522-B-I00.

Funding Information:
H. Dong was partially supported by the NSF under agreement DMS-1600593.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

All Science Journal Classification (ASJC) codes

Mathematics (miscellaneous)

Access to Document

10.1007/s00028-021-00729-8

Cite this

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title = "On C1/2,1 , C1,2 , and C, estimates for linear parabolic operators",

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.",

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AU - Dong, Hongjie

AU - Escauriaza, Luis

AU - Kim, Seick

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PY - 2021/12

Y1 - 2021/12

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