Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems

Abstract

We prove that the solution operator of the classical unilateral obstacle problem on a nonempty open bounded set $\Omega \subset {\mathbb{R}}^d$, $d\in \mathbb{N}$, is Newton differentiable as a function from $L^p(\Omega )$ to $H_0^1(\Omega )$ whenever $\max (1,2d/(d+2))<p\le \infty$. By exploiting this Newton differentiability property, results on angled subspaces in $H^{-1}(\Omega )$, and a formula for orthogonal projections onto direct sums, we further show that the solution map of the classical bilateral obstacle problem is Newton differentiable as a function from $L^p(\Omega )$ to $H_0^1(\Omega )\cap L^q(\Omega )$ whenever $\max (1,d/2)<p\le \infty$ and $1\le q<\infty$. For both the unilateral and the bilateral case, we provide explicit formulas for the Newton derivative. As a concrete application example for our results, we consider the numerical solution of an optimal control problem with $H_0^1(\Omega )$-controls and box-constraints by means of a semismooth Newton method.