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On the classical Reinforcement problem and Optimisation

Emanuele Cristoforoni, Carlo Nitsch, Cristina Trombetti

Abstract

In the present survey, we consider the classical reinforcement problem for elliptic boundary value problems originally studied by Sanchez-Palencia in 1969. We focus on the seminar papers by Brezis, Caffarelli, & Friedman, and by Acerbi & Buttazzo, and discuss the related optimisation problems proposed by Friedman and by Buttazzo.

On the classical Reinforcement problem and Optimisation

Abstract

In the present survey, we consider the classical reinforcement problem for elliptic boundary value problems originally studied by Sanchez-Palencia in 1969. We focus on the seminar papers by Brezis, Caffarelli, & Friedman, and by Acerbi & Buttazzo, and discuss the related optimisation problems proposed by Friedman and by Buttazzo.
Paper Structure (8 sections, 19 theorems, 208 equations, 4 figures)

This paper contains 8 sections, 19 theorems, 208 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A brief history of the problem
  3. The result by Brezis, Caffarelli & Friedman
  4. The result by Acerbi & Buttazzo
  5. Subsequent developments and other boundary conditions
  6. The optimisation point of view
  7. Optimisation of variational energy
  8. Spectral optimisation

Key Result

Lemma 1

Let $u_\varepsilon$ be the solution to BCF. There exists a positive constant $C>0$, such that, for every $\varepsilon$ sufficiently small, the following estimates hold where $D^2 u_\varepsilon$ is the Hessian of $u_\varepsilon$.

Figures (4)

  • Figure 1: Thin metal sheet $\Sigma_\varepsilon$ covered by insulating material $\Omega_\varepsilon=D\setminus\Sigma_\varepsilon$.
  • Figure 2: Reinforcement of $\Omega$ with a thin set of variable thickness $\Sigma_\varepsilon$.
  • Figure 3: Interior reinforcement of $D$ with a thin set of variable thickness $\Sigma_\varepsilon$.
  • Figure 4: Example of non-radial solution to \ref{['minEigaux']} and corresponding configuration of insulating material.

Theorems & Definitions (34)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 3
  • Theorem 2.3
  • Theorem 2.4
  • ...and 24 more