Geodesic
Regularity results for a class of obstacle problems in Heisenberg groups
Applications of Mathematics, Tome 58 (2013) no. 5, pp. 531-554 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $21$.
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 $2$, $q>1$.
DOI : 10.1007/s10492-013-0027-1
Classification : 35D30, 35J20
Keywords: obstacle problem; weak solution; regularity; Heisenberg group 
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     author = {Bigolin, Francesco},
     title = {Regularity results for a class of obstacle problems in {Heisenberg} groups},
     journal = {Applications of Mathematics},
     pages = {531--554},
     year = {2013},
     volume = {58},
     number = {5},
     doi = {10.1007/s10492-013-0027-1},
     mrnumber = {3104617},
     zbl = {06282095},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0027-1/}
}
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Bigolin, Francesco. Regularity results for a class of obstacle problems in Heisenberg groups. Applications of Mathematics, Tome 58 (2013) no. 5, pp. 531-554. doi: 10.1007/s10492-013-0027-1

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