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A nonlocal transmission problem on a hybrid continuous-discrete domain

Hafida Abbas, Abdelhalim Azzouz

Abstract

We study a quadratic nonlocal variational problem on a hybrid domain formed by a compact interval and finitely many discrete points. The associated energy splits into continuous, discrete, and interface contributions. Our main estimate shows that the interface term yields a coercive coupling between the two phases and provides an equivalent hybrid norm. As a consequence, we prove existence and uniqueness of a minimizer for the corresponding variational problem and characterize it as the unique weak solution of the associated hybrid Euler--Lagrange system. The latter combines a nonlocal integral equation on the continuous component with a finite nonlocal algebraic system on the discrete nodes.

A nonlocal transmission problem on a hybrid continuous-discrete domain

Abstract

We study a quadratic nonlocal variational problem on a hybrid domain formed by a compact interval and finitely many discrete points. The associated energy splits into continuous, discrete, and interface contributions. Our main estimate shows that the interface term yields a coercive coupling between the two phases and provides an equivalent hybrid norm. As a consequence, we prove existence and uniqueness of a minimizer for the corresponding variational problem and characterize it as the unique weak solution of the associated hybrid Euler--Lagrange system. The latter combines a nonlocal integral equation on the continuous component with a finite nonlocal algebraic system on the discrete nodes.
Paper Structure (15 sections, 12 theorems, 96 equations)

This paper contains 15 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The hybrid setting and the main coercivity result
  3. The hybrid domain and the nonlocal energy
  4. Decomposition of the energy and the interface estimate
  5. Interface coercivity and the hybrid energy space
  6. The variational problem and weak solutions
  7. Formulation of the problem
  8. Existence, uniqueness, and variational characterization
  9. The hybrid transmission system
  10. A variational example on a hybrid time scale
  11. The hybrid time scale and the energy space
  12. The variational problem
  13. Weak formulation and continuous-discrete decomposition
  14. A reduced Galerkin approximation
  15. Conclusion and Perspectives

Key Result

Lemma 2.2

There exist constants $c_0,C_0>0$, depending only on $\alpha$ and $N$, such that for every one has Consequently, $\mathcal{H}_\alpha(\mathbb{T})$ is a Hilbert space.

Theorems & Definitions (28)

  • Definition 2.1
  • Lemma 2.2: Equivalence with the product norm
  • proof
  • Proposition 2.3: Decomposition of the energy
  • proof
  • Lemma 2.4: Interface estimate
  • proof
  • Lemma 2.5: Algebraic decomposition around the mean
  • proof
  • Theorem 2.6: Interface coercivity
  • ...and 18 more