A second-order Sobolev regularity for p(x)-Laplace equations

Yuqing Wang; Chao Zhang; Yizhe Zhu

doi:10.1016/j.jmaa.2023.127328

A second-order Sobolev regularity for p(x)-Laplace equations

Yuqing Wang
, Chao Zhang^*
, Yizhe Zhu

^*Corresponding author for this work

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

2 Scopus citations

Abstract

Given a Hölder regular variable exponent p(x), under the assumption Dp∈L_loc², we establish some quantative interior Sobolev W_loc^1,2-regularity of |Du|^βDu for weak solutions u to the quasi-linear elliptic equations div(|Du|^p(x)−2Du)=0 anddiv(p(x)|Du|^p(x)−2Du)=0 in domains of Euclidean space Rⁿ, where β>−1 in dimension 2 and β depends on p(x) and n in dimension n≥3. If Dp∈L_loc^2+μ for some μ>0, we obtain W_loc^1,2+δ-regularity of |Du|^βDu for some δ>0 depending on n,β,p(x) and μ.

Original language	English
Article number	127328
Journal	Journal of Mathematical Analysis and Applications
Volume	526
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2023.127328
State	Published - 15 Oct 2023
Externally published	Yes

Keywords

Generalized Lebesgue and Sobolev spaces
Second-order regularity
Variable exponents
p(x)-Laplacian

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10.1016/j.jmaa.2023.127328

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@article{53b53a72126a4a129a70d3f833c64b67,

title = "A second-order Sobolev regularity for p(x)-Laplace equations",

abstract = "Given a H{\"o}lder regular variable exponent p(x), under the assumption Dp∈Lloc2, we establish some quantative interior Sobolev Wloc1,2-regularity of |Du|βDu for weak solutions u to the quasi-linear elliptic equations div(|Du|p(x)−2Du)=0 anddiv(p(x)|Du|p(x)−2Du)=0 in domains of Euclidean space Rn, where β>−1 in dimension 2 and β depends on p(x) and n in dimension n≥3. If Dp∈Lloc2+μ for some μ>0, we obtain Wloc1,2+δ-regularity of |Du|βDu for some δ>0 depending on n,β,p(x) and μ.",

keywords = "Generalized Lebesgue and Sobolev spaces, Second-order regularity, Variable exponents, p(x)-Laplacian",

author = "Yuqing Wang and Chao Zhang and Yizhe Zhu",

note = "Publisher Copyright: {\textcopyright} 2023 Elsevier Inc.",

year = "2023",

month = oct,

day = "15",

doi = "10.1016/j.jmaa.2023.127328",

language = "英语",

volume = "526",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Academic Press Inc.",

number = "2",

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TY - JOUR

T1 - A second-order Sobolev regularity for p(x)-Laplace equations

AU - Wang, Yuqing

AU - Zhang, Chao

AU - Zhu, Yizhe

PY - 2023/10/15

Y1 - 2023/10/15

N2 - Given a Hölder regular variable exponent p(x), under the assumption Dp∈Lloc2, we establish some quantative interior Sobolev Wloc1,2-regularity of |Du|βDu for weak solutions u to the quasi-linear elliptic equations div(|Du|p(x)−2Du)=0 anddiv(p(x)|Du|p(x)−2Du)=0 in domains of Euclidean space Rn, where β>−1 in dimension 2 and β depends on p(x) and n in dimension n≥3. If Dp∈Lloc2+μ for some μ>0, we obtain Wloc1,2+δ-regularity of |Du|βDu for some δ>0 depending on n,β,p(x) and μ.

AB - Given a Hölder regular variable exponent p(x), under the assumption Dp∈Lloc2, we establish some quantative interior Sobolev Wloc1,2-regularity of |Du|βDu for weak solutions u to the quasi-linear elliptic equations div(|Du|p(x)−2Du)=0 anddiv(p(x)|Du|p(x)−2Du)=0 in domains of Euclidean space Rn, where β>−1 in dimension 2 and β depends on p(x) and n in dimension n≥3. If Dp∈Lloc2+μ for some μ>0, we obtain Wloc1,2+δ-regularity of |Du|βDu for some δ>0 depending on n,β,p(x) and μ.

KW - Generalized Lebesgue and Sobolev spaces

KW - Second-order regularity

KW - Variable exponents

KW - p(x)-Laplacian

UR - https://www.scopus.com/pages/publications/85162118583

U2 - 10.1016/j.jmaa.2023.127328

DO - 10.1016/j.jmaa.2023.127328

M3 - 文章

AN - SCOPUS:85162118583

SN - 0022-247X

VL - 526

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 127328

ER -

A second-order Sobolev regularity for p(x)-Laplace equations

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