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

2023/08/29 by Christof, Constantin, Wachsmuth, Gerd
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2308.15289

Abstract

We prove that the solution operator of the classical unilateral obstacle problem on a nonempty open bounded set Ω⊂ ℝ^d, d ∈ ℕ, is Newton differentiable as a function from L^p(Ω) to H₀¹(Ω) whenever max(1, 2d/(d+2)) < p ≤ ∞. By exploiting this Newton differentiability property, results on angled subspaces in H^-1(Ω), 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(Ω) to H₀¹(Ω)∩ L^q(Ω) whenever max(1, d/2) < p ≤ ∞ and 1 ≤ q <∞. 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 H₀¹(Ω)-controls and box-constraints by means of a semismooth Newton method.

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