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Volume 35, Issue 4
Global Integrability for Solutions to Obstacle Problems

Yanan Shan & Hongya Gao

DOI: 10.4208/jpde.v35.n4.2

J. Part. Diff. Eq., 35 (2022), pp. 320-330.

Published online: 2022-10

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  • Abstract

Denote  $$ {\cal K} _{\psi, \theta} (\Omega) =\left\{ v\in W^{1,p} (\Omega) : v\ge \psi, \mbox { a.e. and }  v-\theta \in W_0^{1,p} (\Omega) \right\}, $$ where $\psi$ is any function in $\Omega \subset \mathbb R^N$, $N\ge 2$, with values in $\mathbb R \cup \{\pm \infty\}$ and $\theta $ is a measurable function. This paper deals with global integrability for $u \in {\cal K}_{\psi, \theta}$ such that

$$ \int_\Omega \langle {\cal A} (x,\nabla u), \nabla (w-u) \rangle {\rm d}x \ge \int_\Omega \langle F, \nabla (w-u) \rangle {\rm d}x, \ \ \forall\ w \in {\cal K}_{\psi,\theta} (\Omega), $$ with $|{\cal A} (x,\xi)| \approx |\xi| ^{p-1}$, $1<p<N$. Some global integrability results are obtained.

  • Keywords

Global integrability, obstacle problem, A-harmonic equation.

  • AMS Subject Headings

35J25

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

1067382557@qq.com (Yanan Shan)

ghy@hbu.cn (Hongya Gao)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-35-320, author = {Shan , Yanan and Gao , Hongya}, title = {Global Integrability for Solutions to Obstacle Problems}, journal = {Journal of Partial Differential Equations}, year = {2022}, volume = {35}, number = {4}, pages = {320--330}, abstract = {

Denote  $$ {\cal K} _{\psi, \theta} (\Omega) =\left\{ v\in W^{1,p} (\Omega) : v\ge \psi, \mbox { a.e. and }  v-\theta \in W_0^{1,p} (\Omega) \right\}, $$ where $\psi$ is any function in $\Omega \subset \mathbb R^N$, $N\ge 2$, with values in $\mathbb R \cup \{\pm \infty\}$ and $\theta $ is a measurable function. This paper deals with global integrability for $u \in {\cal K}_{\psi, \theta}$ such that

$$ \int_\Omega \langle {\cal A} (x,\nabla u), \nabla (w-u) \rangle {\rm d}x \ge \int_\Omega \langle F, \nabla (w-u) \rangle {\rm d}x, \ \ \forall\ w \in {\cal K}_{\psi,\theta} (\Omega), $$ with $|{\cal A} (x,\xi)| \approx |\xi| ^{p-1}$, $1<p<N$. Some global integrability results are obtained.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v35.n4.2}, url = {http://global-sci.org/intro/article_detail/jpde/21051.html} }
TY - JOUR T1 - Global Integrability for Solutions to Obstacle Problems AU - Shan , Yanan AU - Gao , Hongya JO - Journal of Partial Differential Equations VL - 4 SP - 320 EP - 330 PY - 2022 DA - 2022/10 SN - 35 DO - http://doi.org/10.4208/jpde.v35.n4.2 UR - https://global-sci.org/intro/article_detail/jpde/21051.html KW - Global integrability, obstacle problem, A-harmonic equation. AB -

Denote  $$ {\cal K} _{\psi, \theta} (\Omega) =\left\{ v\in W^{1,p} (\Omega) : v\ge \psi, \mbox { a.e. and }  v-\theta \in W_0^{1,p} (\Omega) \right\}, $$ where $\psi$ is any function in $\Omega \subset \mathbb R^N$, $N\ge 2$, with values in $\mathbb R \cup \{\pm \infty\}$ and $\theta $ is a measurable function. This paper deals with global integrability for $u \in {\cal K}_{\psi, \theta}$ such that

$$ \int_\Omega \langle {\cal A} (x,\nabla u), \nabla (w-u) \rangle {\rm d}x \ge \int_\Omega \langle F, \nabla (w-u) \rangle {\rm d}x, \ \ \forall\ w \in {\cal K}_{\psi,\theta} (\Omega), $$ with $|{\cal A} (x,\xi)| \approx |\xi| ^{p-1}$, $1<p<N$. Some global integrability results are obtained.

Yanan Shan & Hongya Gao. (2022). Global Integrability for Solutions to Obstacle Problems. Journal of Partial Differential Equations. 35 (4). 320-330. doi:10.4208/jpde.v35.n4.2
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