# Regularity results for a class of obstacle problems in Heisenberg groups

created by bigolin on 16 Feb 2012

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Submitted Paper

Inserted: 16 feb 2012
Last Updated: 16 feb 2012

Year: 2012

Abstract:

We study regularity results for solutions $u\in H W^{1,p}(\Omega)$ to the obstacle problem $$\int{\Omega} \mathcal{A}(x, \nabla{\mathbb H} u)\nabla{\mathbb H}(v-u)\,dx\geq0\qquad\forall\, v\in\mathcal K{\psi,u}(\Omega)$$ such that $u\geq\psi$ a.e. in $\Omega$, where $\mathcal K_{\psi,u}(\Omega)=\left\{v\in HW^{1,p}(\Omega):\,v-u\in HW_{0}^{1,p}(\Omega)\,v\geq\psi\,{\rm a.e.\,in}\,\Omega\right\}$ 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. $$T\psi\in HW{1,p}{loc}(\Omega)\Rightarrow Tu\in Lp{loc}(\Omega),$$ $$\nabla{\mathbb H}\psi p\in L{q}{loc}(\Omega)\Rightarrow \nabla{\mathbb H} u p\in Lq{loc}(\Omega),$$ where $2<p<4$, $q>1$.