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Exterior Nonlocal Variational Inequalities

Harbir Antil, Madeline O. Horton, Mahamadi Warma

TL;DR

This work extends obstacle problems to the exterior of the observation domain by formulating a nonlocal variational inequality driven by the fractional Laplacian. It develops a rigorous well-posedness framework, establishing equivalence among a minimization problem, a variational inequality, a slack-variable (Lagrange multiplier) system, and a distributional Euler–Lagrange formulation, including a nonnegative multiplier and complementarity. Two penalty strategies, an $L^2$ Moreau–Yosida penalty and a Sobolev-norm penalty, are analyzed via Mosco convergence, yielding explicit rates: linear convergence of the solution in $W^{s,2}(\,\\Omega,\\ olinebreak[0] \\Sigma_1)$ and quadratic decay of constraint violation with respect to the penalty parameter, and linear convergence in the Sobolev penalty parameter for the corresponding multiplier. These contributions enable exterior obstacle modeling in nonlocal contexts and provide a foundation for numerical schemes and finite element approximations.

Abstract

This paper introduces a new class of variational inequalities where the obstacle is placed in the exterior domain that is disjoint from the observation domain. This is carried out with the help of nonlocal fractional operators. The need for such novel variational inequalities stems from the fact that the classical approach only allows placing the obstacle either inside the observation domain or on the boundary. A complete analysis of the continuous problem is provided. Additionally, perturbation arguments to approximate the problem are discussed.

Exterior Nonlocal Variational Inequalities

TL;DR

This work extends obstacle problems to the exterior of the observation domain by formulating a nonlocal variational inequality driven by the fractional Laplacian. It develops a rigorous well-posedness framework, establishing equivalence among a minimization problem, a variational inequality, a slack-variable (Lagrange multiplier) system, and a distributional Euler–Lagrange formulation, including a nonnegative multiplier and complementarity. Two penalty strategies, an Moreau–Yosida penalty and a Sobolev-norm penalty, are analyzed via Mosco convergence, yielding explicit rates: linear convergence of the solution in and quadratic decay of constraint violation with respect to the penalty parameter, and linear convergence in the Sobolev penalty parameter for the corresponding multiplier. These contributions enable exterior obstacle modeling in nonlocal contexts and provide a foundation for numerical schemes and finite element approximations.

Abstract

This paper introduces a new class of variational inequalities where the obstacle is placed in the exterior domain that is disjoint from the observation domain. This is carried out with the help of nonlocal fractional operators. The need for such novel variational inequalities stems from the fact that the classical approach only allows placing the obstacle either inside the observation domain or on the boundary. A complete analysis of the continuous problem is provided. Additionally, perturbation arguments to approximate the problem are discussed.
Paper Structure (6 sections, 10 theorems, 109 equations)

This paper contains 6 sections, 10 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Fractional order Sobloev spaces and the fractional Laplacian
  4. Useful results from convex analysis
  5. Well-posedness of the variational inequality
  6. Penalization

Key Result

Proposition 2.1

The norm $\|\cdot\|_{W^{s,2}(\Omega,\Sigma_1)}$ given in NORM is equivalent to the one induced by $W^{s,2}({\mathbb{R}}^N)$. As a consequence, $(W^{s,2}(\Omega,\Sigma_1),\|\cdot\|_{W^{s,2}(\Omega,\Sigma_1)})$ is a Hilbert space with the scalar product

Theorems & Definitions (21)

  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4: The integration by parts formula I
  • Remark 2.5
  • Proposition 2.6: The integration by parts formula II
  • proof
  • Lemma 2.7
  • proof
  • Definition 3.2
  • ...and 11 more