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Invertibility criteria for the biharmonic single-layer potential

Alexandre Munnier

Abstract

While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems.

Invertibility criteria for the biharmonic single-layer potential

Abstract

While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems.
Paper Structure (7 sections, 9 theorems, 42 equations, 1 figure)

This paper contains 7 sections, 9 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The biharmonic transmission problem
  3. The Robin matrix
  4. Invertibility of the biharmonic single-layer potential
  5. Properties of the Robin matrix
  6. Generalization
  7. Statements and Declarations

Key Result

Theorem 1.1

Figures (1)

  • Figure 1: The multi-connected curve $\varGamma$ can be decomposed into the disjoint union of ${\varGamma_{\!e}}$ and the Jordan curves included in $\varOmega_\varGamma^-$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.1
  • Definition 4.1
  • Theorem 4.1
  • Lemma 4.1
  • Lemma 4.2
  • ...and 8 more