Capacity and the obstacle problem

Abstract

For the bilinear forma(u, v) = ∫ _Ω a _ij (x)u _xi v _xj dx, whereu, v∈H ¹₀ (Ω), Ω a bounded domain in ℝⁿ, anda _ij (x) bounded and uniformly elliptic coefficients on Ω, theL ^p integrability on Ω forp>2n/(n−2) of the solution to the variational inequality (the unilateral obstacle problem with obstacle

$$a(u,v - u) \geqslant \left\langle {f,v - u} \right\rangle , f \in H^{ - 1} (\Omega ),$$

is studied. Hereψ is an arbitrary function given on Ω andC is the Newtonian conductor capacity relative to Ω. The sufficient conditions that permit such quantitative estimates are given in terms of capacitary integrals on Ω:∫|ψ| ^q dC<∞. Some simple examples show that is not sufficient to assume merely thatψ∈H ¹₀ (Ω)∩L ^q(Ω) for sufficiently largeq. In particular,$\mathbb{K}_\psi \ne \emptyset$ precisely when∫ψ ²₊ dC<∞. An estimate is also given for theL ^p norm of the gradient of such solutions in terms of these integrals.