On C1, C2, and weak type-(1,1) estimates for linear elliptic operators

Hongjie Dong, Seick Kim

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L1-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haïm Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.

Original languageEnglish
Pages (from-to)417-435
Number of pages19
JournalCommunications in Partial Differential Equations
Volume42
Issue number3
DOIs
Publication statusPublished - 2017 Mar 4

Fingerprint

Modulus of Continuity
Elliptic Operator
Linear Operator
Continuously differentiable
Estimate
Elliptic Equations
Weak Solution
Divergence
Coefficient
Form

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

@article{1d005a274fd9439b954534a42b752800,
title = "On C1, C2, and weak type-(1,1) estimates for linear elliptic operators",
abstract = "We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L1-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Ha{\"i}m Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.",
author = "Hongjie Dong and Seick Kim",
year = "2017",
month = "3",
day = "4",
doi = "10.1080/03605302.2017.1278773",
language = "English",
volume = "42",
pages = "417--435",
journal = "Communications in Partial Differential Equations",
issn = "0360-5302",
publisher = "Taylor and Francis Ltd.",
number = "3",

}

On C1, C2, and weak type-(1,1) estimates for linear elliptic operators. / Dong, Hongjie; Kim, Seick.

In: Communications in Partial Differential Equations, Vol. 42, No. 3, 04.03.2017, p. 417-435.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On C1, C2, and weak type-(1,1) estimates for linear elliptic operators

AU - Dong, Hongjie

AU - Kim, Seick

PY - 2017/3/4

Y1 - 2017/3/4

N2 - We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L1-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haïm Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.

AB - We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L1-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haïm Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.

UR - http://www.scopus.com/inward/record.url?scp=85014815170&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014815170&partnerID=8YFLogxK

U2 - 10.1080/03605302.2017.1278773

DO - 10.1080/03605302.2017.1278773

M3 - Article

AN - SCOPUS:85014815170

VL - 42

SP - 417

EP - 435

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 3

ER -