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

TL;DR

This paper establishes Newton differentiability of solution maps for unilateral and bilateral obstacle problems in certain function spaces, providing explicit formulas and applications to numerical optimal control problems.

Contribution

It proves Newton differentiability of obstacle problem solution maps in specific Sobolev spaces and derives explicit formulas for their derivatives.

Findings

Newton differentiability holds for unilateral obstacle problems in L^p to H_0^1 spaces.

Explicit formulas for Newton derivatives are provided.

Applications to semismooth Newton methods for optimal control are demonstrated.

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 $\max(1, 2d/(d+2)) < p \leq \infty$. 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 $\max(1, d/2) < p \leq \infty$ and $1 \leq q <\infty$. 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.