The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

Ky Ho; Yun Ho Kim; Patrick Winkert; Chao Zhang

doi:10.1016/j.jde.2022.01.004

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

Ky Ho
, Yun Ho Kim
, Patrick Winkert^*
, Chao Zhang

^*Corresponding author for this work

University of Economics Ho Chi Minh City
Sangmyung University
Technical University of Berlin
School of Mathematics, Harbin Institute of Technology

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

31 Scopus citations

Abstract

In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao (1999) [12] and Winkert-Zacher (2012) [47] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the C^1,α-regularity follows immediately.

Original language	English
Pages (from-to)	503-532
Number of pages	30
Journal	Journal of Differential Equations
Volume	313
DOIs	https://doi.org/10.1016/j.jde.2022.01.004
State	Published - 15 Mar 2022
Externally published	Yes

Keywords

A-priori bounds
De Giorgi iteration
Hölder continuity
Localization method
Variable exponent Lebesgue and Sobolev spaces
p(⋅)-Laplacian

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10.1016/j.jde.2022.01.004

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title = "The boundedness and H{\"o}lder continuity of weak solutions to elliptic equations involving variable exponents and critical growth",

abstract = "In this paper we prove the boundedness and H{\"o}lder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao (1999) [12] and Winkert-Zacher (2012) [47] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the C1,α-regularity follows immediately.",

keywords = "A-priori bounds, De Giorgi iteration, H{\"o}lder continuity, Localization method, Variable exponent Lebesgue and Sobolev spaces, p(⋅)-Laplacian",

author = "Ky Ho and Kim, \{Yun Ho\} and Patrick Winkert and Chao Zhang",

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2022",

month = mar,

day = "15",

doi = "10.1016/j.jde.2022.01.004",

language = "英语",

volume = "313",

pages = "503--532",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Academic Press Inc.",

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TY - JOUR

T1 - The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

AU - Ho, Ky

AU - Kim, Yun Ho

AU - Winkert, Patrick

AU - Zhang, Chao

PY - 2022/3/15

Y1 - 2022/3/15

N2 - In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao (1999) [12] and Winkert-Zacher (2012) [47] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the C1,α-regularity follows immediately.

AB - In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao (1999) [12] and Winkert-Zacher (2012) [47] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the C1,α-regularity follows immediately.

KW - A-priori bounds

KW - De Giorgi iteration

KW - Hölder continuity

KW - Localization method

KW - Variable exponent Lebesgue and Sobolev spaces

KW - p(⋅)-Laplacian

UR - https://www.scopus.com/pages/publications/85122670291

U2 - 10.1016/j.jde.2022.01.004

DO - 10.1016/j.jde.2022.01.004

M3 - 文章

AN - SCOPUS:85122670291

SN - 0022-0396

VL - 313

SP - 503

EP - 532

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

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Keywords

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