Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth
We establish local Calderón–Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard p(x,t)${p(x,t)}$-growth. More precisely, we will show that the spatial gradient Du${Du}$ of the solution to the obstacle problem is as integrable as the obstacle ψ${\psi...
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Online Access: | https://doi.org/10.1515/anona-2013-0024 |
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doaj-0f034fe4d0a042edac16326d148d7a942021-09-06T19:39:53ZengDe GruyterAdvances in Nonlinear Analysis2191-94962191-950X2014-02-0131154410.1515/anona-2013-0024Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growthErhardt André0Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, GermanyWe establish local Calderón–Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard p(x,t)${p(x,t)}$-growth. More precisely, we will show that the spatial gradient Du${Du}$ of the solution to the obstacle problem is as integrable as the obstacle ψ${\psi }$, i.e. |Dψ|p(·),|∂tψ|γ1'∈L loc q⇒|Du|p(·)∈L loc q,foranyq>1,$ |D\psi |^{p(\,\cdot \,)},|\partial _t\psi |^{\gamma _1^{\prime }}\in L^q_\mathrm {loc}\Rightarrow |Du|^{p(\,\cdot \,)}\in L^q_\mathrm {loc},\quad \text{for any}~q>1, $ where γ1'=γ1γ1-1${\gamma _1^{\prime }=\frac{\gamma _1}{\gamma _1-1}}$ and γ1${\gamma _1}$ is the lower bound for p(·)${p(\,\cdot \,)}$.https://doi.org/10.1515/anona-2013-0024calderón–zygmund estimatesnonlinear parabolic obstacle problemsvariational inequalitynonstandard growthlocalizable solution35k8635b4535b5142b2049n60 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Erhardt André |
spellingShingle |
Erhardt André Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth Advances in Nonlinear Analysis calderón–zygmund estimates nonlinear parabolic obstacle problems variational inequality nonstandard growth localizable solution 35k86 35b45 35b51 42b20 49n60 |
author_facet |
Erhardt André |
author_sort |
Erhardt André |
title |
Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth |
title_short |
Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth |
title_full |
Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth |
title_fullStr |
Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth |
title_full_unstemmed |
Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth |
title_sort |
calderón–zygmund theory for parabolic obstacle problems with nonstandard growth |
publisher |
De Gruyter |
series |
Advances in Nonlinear Analysis |
issn |
2191-9496 2191-950X |
publishDate |
2014-02-01 |
description |
We establish local Calderón–Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard p(x,t)${p(x,t)}$-growth. More precisely, we will show that the spatial gradient Du${Du}$ of the solution to the obstacle problem is as integrable as the obstacle ψ${\psi }$, i.e.
|Dψ|p(·),|∂tψ|γ1'∈L loc q⇒|Du|p(·)∈L loc q,foranyq>1,$
|D\psi |^{p(\,\cdot \,)},|\partial _t\psi |^{\gamma _1^{\prime }}\in L^q_\mathrm {loc}\Rightarrow |Du|^{p(\,\cdot \,)}\in L^q_\mathrm {loc},\quad \text{for any}~q>1,
$
where γ1'=γ1γ1-1${\gamma _1^{\prime }=\frac{\gamma _1}{\gamma _1-1}}$ and γ1${\gamma _1}$ is the lower bound for p(·)${p(\,\cdot \,)}$. |
topic |
calderón–zygmund estimates nonlinear parabolic obstacle problems variational inequality nonstandard growth localizable solution 35k86 35b45 35b51 42b20 49n60 |
url |
https://doi.org/10.1515/anona-2013-0024 |
work_keys_str_mv |
AT erhardtandre calderonzygmundtheoryforparabolicobstacleproblemswithnonstandardgrowth |
_version_ |
1717769862375276544 |