A Lewy-Stampacchia Estimate for quasilinear variational inequalities in the Heisenberg group

TL;DR

This paper establishes a Lewy-Stampacchia estimate for quasilinear variational inequalities within the Heisenberg group, demonstrating that the operator on the obstacle bounds the operator on the solution pointwise.

Contribution

It extends Lewy-Stampacchia estimates to the sub-Riemannian setting of the Heisenberg group for quasilinear variational inequalities.

Findings

Operator on the obstacle bounds the solution operator pointwise.

The estimate holds for minimizers of a p-Dirichlet type functional.

Results apply to obstacle problems in the Heisenberg group.

Abstract

We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. More explicitly, if $\epsilon\ge0$ and $\bar u_\epsilon$ minimizes the functional $$ \int_\Omega(\epsilon+|\nabla_{\H^n}u|^2)^{p/2}$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle $\psi$, then 0 \le \div_{\H^n}\, \Big((\epsilon+|\nabla_{\H^n}\bar u_\epsilon|^2)^{(p/2)-1} \nabla_{\H^n}\bar u_\epsilon\Big) \qquad \le (\div_{\H^n}\, \Big((\epsilon+|\nabla_{\H^n}\psi|^2)^{(p/2)-1} \nabla_{\H^n}\psi\Big))^+.