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Avscon the Schur topology

Andreas Buchinger, Marcus Waurick

Abstract

The aim of the course is to lead to an understanding of homogenisation processes in an operator-theoretic sense. In fact, using solely operator-theoretic means not referring to the particular form of the coefficients, we will identify an operator topology on the level of coefficients that will fully capture the convergence involved in the context of homogenisation. One upshot of this perspective will be that we will obtain homogenisation results for time-dependent partial differential equations (almost) for free.

Avscon the Schur topology

Abstract

The aim of the course is to lead to an understanding of homogenisation processes in an operator-theoretic sense. In fact, using solely operator-theoretic means not referring to the particular form of the coefficients, we will identify an operator topology on the level of coefficients that will fully capture the convergence involved in the context of homogenisation. One upshot of this perspective will be that we will obtain homogenisation results for time-dependent partial differential equations (almost) for free.

Paper Structure

This paper contains 25 sections, 52 theorems, 167 equations.

Table of Contents

  1. Lecture 1
  2. Introduction
  3. Lax--Milgram Revisited
  4. The domain of the gradient
  5. $\mathrm{H}$-convergence
  6. Comments
  7. Exercises
  8. Lecture 2
  9. Introduction
  10. The $\operatorname{div}$-$\mathop{\mathrm{curl}}\nolimits$ lemma
  11. Examples for $\mathrm{H}$-convergence
  12. The Schur topology and its relation to $\mathrm{H}$-convergence
  13. Comments
  14. Exercises
  15. Lecture 3
  16. ...and 10 more sections

Key Result

Lemma 1.2.2

Let $A \in \mathcal{L}_{\mathrm{b}}(\mathcal{H}_0,\mathcal{H}_1)$. Then the following statements hold: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (98)

  • Example 1.2.1
  • Lemma 1.2.2
  • proof
  • Proposition 1.2.3
  • proof
  • Lemma 1.2.4
  • proof
  • Theorem 1.2.5
  • proof
  • Theorem 1.2.6
  • ...and 88 more