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The nonlocal Kelvin principle and the dual approach to nonlocal control in the conduction coefficients

Anton Evgrafov, Jose C. Bellido

Abstract

We explore the dual approach to nonlocal optimal design, specifically for a classical min-max problem which in this study is associated with a nonlocal scalar diffusion equation. We reformulate the optimal design problem utilizing a dual variational principle, which is expressed in terms of nonlocal two-point fluxes. We introduce the proper functional space framework to deal with this formulation, and establish its well-posedness. The key ingredient is the inf-sup (Ladyzhenskaya--Babuska--Brezzi) condition, which holds uniformly with respect to small nonlocal horizons. As a byproduct of this, we are able to prove convergence of nonlocal to local optimal design problems in a straightforward fashion.

The nonlocal Kelvin principle and the dual approach to nonlocal control in the conduction coefficients

Abstract

We explore the dual approach to nonlocal optimal design, specifically for a classical min-max problem which in this study is associated with a nonlocal scalar diffusion equation. We reformulate the optimal design problem utilizing a dual variational principle, which is expressed in terms of nonlocal two-point fluxes. We introduce the proper functional space framework to deal with this formulation, and establish its well-posedness. The key ingredient is the inf-sup (Ladyzhenskaya--Babuska--Brezzi) condition, which holds uniformly with respect to small nonlocal horizons. As a byproduct of this, we are able to prove convergence of nonlocal to local optimal design problems in a straightforward fashion.

Paper Structure

This paper contains 6 sections, 15 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Primal and dual local optimal design problems
  3. Nonlocal Dirichlet principle and the associated optimal design problem
  4. Nonlocal Kelvin principle and the associated optimal design problem
  5. Flux recovery operator and an upper estimate for the local problem
  6. Adjoint flux recovery operator and a lower estimate for the local problem

Key Result

Proposition 2.1

The problem eq:compliance_local_dual admits an optimal solution.

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2: Poincaré inequality for $\mathcal{G}_{\delta}=\mathcal{D}_{\delta}^*$
  • proof
  • ...and 19 more