| Title:
|
Regularity results for a class of obstacle problems in Heisenberg groups (English) |
| Author:
|
Bigolin, Francesco |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
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0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
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58 |
| Issue:
|
5 |
| Year:
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2013 |
| Pages:
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531-554 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study regularity results for solutions $u\in H W^{1,p}(\Omega )$ to the obstacle problem $$ \int _{\Omega } \mathcal {A}(x, \nabla _{\mathbb H} u)\nabla _{\mathbb H}(v-u) {\rm d} x \geq 0 \quad \forall v\in \mathcal K_{\psi ,u}(\Omega ) $$ such that $u\geq \psi $ a.e. in $\Omega $, where $\mathcal K_{\psi ,u}(\Omega )= \{v\in HW^{1,p}(\Omega )\colon v-u\in HW_{0}^{1,p}(\Omega ) v\geq \psi \text {\rm a.e. in} \Omega \}$, in Heisenberg groups $\mathbb H^n$. In particular, we obtain weak differentiability in the $T$-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$ \begin{aligned}d T\psi \in HW^{1,p}_{\rm loc}(\Omega )&\Rightarrow Tu\in L^p_{\rm loc}(\Omega ), |\nabla _{\mathbb H}\psi |^p\in L^{q}_{\rm loc}(\Omega )&\Rightarrow |\nabla _{\mathbb H} u|^p \in L^q_{\rm loc}(\Omega ), \end{aligned}d $$ where $2<p<4$, $q>1$. (English) |
| Keyword:
|
obstacle problem |
| Keyword:
|
weak solution |
| Keyword:
|
regularity |
| Keyword:
|
Heisenberg group |
| MSC:
|
35D30 |
| MSC:
|
35J20 |
| idZBL:
|
Zbl 06282095 |
| idMR:
|
MR3104617 |
| DOI:
|
10.1007/s10492-013-0027-1 |
| . |
| Date available:
|
2013-09-14T11:42:41Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143431 |
| . |
| Reference:
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