Regularity results for a class of obstacle problems in Heisenberg groups

Francesco Bigolin

Applications of Mathematics (2013)

  • Volume: 58, Issue: 5, page 531-554
  • ISSN: 0862-7940

Abstract

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We study regularity results for solutions u ∈ H W 1 , p ( Ω ) to the obstacle problem ∫ Ω 𝒜 ( x , ∇ ℍ u ) ∇ ℍ ( v - u ) d x ≥ 0 ∀ v ∈ 𝒦 ψ , u ( Ω ) such that u ≥ ψ a.e. in Ω , where 𝒦 ψ , u ( Ω ) = { v ∈ H W 1 , p ( Ω ) : v - u ∈ H W 0 1 , p ( Ω ) v ≥ ψ a.e. in Ω } , in Heisenberg groups ℍ n . In particular, we obtain weak differentiability in the T -direction and horizontal estimates of Calderon-Zygmund type, i.e. d T ψ ∈ H W loc 1 , p ( Ω ) ⇒ T u ∈ L loc p ( Ω ) , | ∇ ℍ ψ | p ∈ L loc q ( Ω ) ⇒ | ∇ ℍ u | p ∈ L loc q ( Ω ) , d where 2 < p < 4 , q > 1 .

How to cite

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Bigolin, Francesco. "Regularity results for a class of obstacle problems in Heisenberg groups." Applications of Mathematics 58.5 (2013): 531-554. <http://eudml.org/doc/260646>.

@article{Bigolin2013,
abstract = {We study regularity results for solutions $u\in H W^\{1,p\}(\Omega )$ to the obstacle problem \[ \int \_\{\Omega \} \mathcal \{A\}(x, \nabla \_\{\mathbb \{H\}\} u)\nabla \_\{\mathbb \{H\}\}(v-u) \{\rm d\} x \ge 0 \quad \forall v\in \mathcal \{K\}\_\{\psi ,u\}(\Omega ) \] such that $u\ge \psi $ a.e. in $\Omega $, where $\mathcal \{K\}_\{\psi ,u\}(\Omega )= \lbrace v\in HW^\{1,p\}(\Omega )\colon v-u\in HW_\{0\}^\{1,p\}(\Omega ) v\ge \psi \text\{\rm a.e. in\} \Omega \rbrace $, 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 $2<p<4$, $q>1$.},
author = {Bigolin, Francesco},
journal = {Applications of Mathematics},
keywords = {obstacle problem; weak solution; regularity; Heisenberg group; obstacle problem; weak solution; regularity; Heisenberg group},
language = {eng},
number = {5},
pages = {531-554},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Regularity results for a class of obstacle problems in Heisenberg groups},
url = {http://eudml.org/doc/260646},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Bigolin, Francesco
TI - Regularity results for a class of obstacle problems in Heisenberg groups
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 5
SP - 531
EP - 554
AB - We study regularity results for solutions $u\in H W^{1,p}(\Omega )$ to the obstacle problem \[ \int _{\Omega } \mathcal {A}(x, \nabla _{\mathbb {H}} u)\nabla _{\mathbb {H}}(v-u) {\rm d} x \ge 0 \quad \forall v\in \mathcal {K}_{\psi ,u}(\Omega ) \] such that $u\ge \psi $ a.e. in $\Omega $, where $\mathcal {K}_{\psi ,u}(\Omega )= \lbrace v\in HW^{1,p}(\Omega )\colon v-u\in HW_{0}^{1,p}(\Omega ) v\ge \psi \text{\rm a.e. in} \Omega \rbrace $, 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 $2<p<4$, $q>1$.
LA - eng
KW - obstacle problem; weak solution; regularity; Heisenberg group; obstacle problem; weak solution; regularity; Heisenberg group
UR - http://eudml.org/doc/260646
ER -

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