Regularity for solutions to anisotropic obstacle problems

Abstract

For Ω a bounded subset of ℝⁿ, n ⩾ 2, ψ any function in Ω with values in ℝ ∪ {±∞} and $\theta \in W^{1,\left( {q_i } \right)} \left( \Omega \right)$, let

$$\mathcal{K}_{\psi ,\theta }^{\left( {q_i } \right)} \left( \Omega \right) = \left\{ {v \in W^{1,\left( {q_i } \right)} \left( \Omega \right):v \geqslant \psi , a.e. and v - \theta \in W_0^{1,\left( {q_i } \right)} \left( \Omega \right)} \right\}.$$

This paper deals with solutions to $\mathcal{K}_{\psi ,\theta }^{\left( {q_i } \right)}$-obstacle problems for the A-harmonic equation

$$- div\mathcal{A}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right) = - div f\left( x \right)$$

as well as the integral functional

$$I\left( {u;\Omega } \right) = \int_\Omega {f\left( {x,u\left( x \right),\nabla u\left( x \right)} \right)dx.}$$

Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.