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Subdifferentials and penalty approximations of the obstacle problem

Amal Alphonse, Gerd Wachsmuth

TL;DR

This work develops a unified penalty framework for the obstacle problem and proves that derivatives of penalised solution maps converge, in the weak operator topology, to elements of the strong-weak Bouligand subdifferential of the VI solution map. Using a capacitary-measure approach, the authors characterize the limiting derivatives via capacitary measures μ with μ(I)=0 and μ=∞ on the strictly active set A_s, and show convergence for both smooth and nonsmooth penalty regimes. The analysis yields pointwise and capacity-based limit conditions for the derivatives, connects them to the generalized derivative set ∂_B^{sw} S, and provides rigorous first-order optimality conditions for control problems constrained by obstacle-type VI constraints. The results apply to a broad class of penalty functions and clarify how capacitary structures govern the behavior of penalised derivatives, enabling robust regularisation strategies and principled optimal control formulations.

Abstract

We consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalised problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving for example the positive part function $\max(0,\cdot)$. Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem.

Subdifferentials and penalty approximations of the obstacle problem

TL;DR

This work develops a unified penalty framework for the obstacle problem and proves that derivatives of penalised solution maps converge, in the weak operator topology, to elements of the strong-weak Bouligand subdifferential of the VI solution map. Using a capacitary-measure approach, the authors characterize the limiting derivatives via capacitary measures μ with μ(I)=0 and μ=∞ on the strictly active set A_s, and show convergence for both smooth and nonsmooth penalty regimes. The analysis yields pointwise and capacity-based limit conditions for the derivatives, connects them to the generalized derivative set ∂_B^{sw} S, and provides rigorous first-order optimality conditions for control problems constrained by obstacle-type VI constraints. The results apply to a broad class of penalty functions and clarify how capacitary structures govern the behavior of penalised derivatives, enabling robust regularisation strategies and principled optimal control formulations.

Abstract

We consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalised problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving for example the positive part function . Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem.

Paper Structure

This paper contains 12 sections, 22 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Abstract setup and properties of the penalised problem
  3. Setup
  4. First properties
  5. Convergence results
  6. Pointwise characterisation of the limit of the derivatives
  7. A complementarity condition on alpha
  8. An orthogonality condition on -Delta alpha - d
  9. Generalised derivatives as limits
  10. Existence of limiting elements
  11. Capacitary measures and characterisation of limits
  12. Optimal control of the obstacle problem

Key Result

Theorem 1.4

Let ass:for_Bsw hold. For every $f \in H^{-1}(\Omega)$ with $S(f) \in C_0(\Omega)$, if $f_\rho \to f$ in $H^{-1}(\Omega)$, then there exist maps $L^\mathrm{sm}, L^\mathrm{sc} \in \partial_B^{sw}S(f)$ such that for a subsequence (that we relabel),

Theorems & Definitions (51)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.4
  • Remark 1.5: On the assumption $S(f) \in C_0(\Omega)$
  • Theorem 1.6
  • Remark 1.7: Generalisation to other elliptic operators
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.5
  • ...and 41 more