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BGG sequences with weak regularity and applications

Andreas Čap, Kaibo Hu

Abstract

We investigate some Bernstein-Gelfand-Gelfand (BGG) complexes on bounded Lipschitz domains in $\mathbb{R}^n$ consisting of Sobolev spaces. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy-Riemann operator and an additional third order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge-Laplacian problem of a twisted de-Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.

BGG sequences with weak regularity and applications

Abstract

We investigate some Bernstein-Gelfand-Gelfand (BGG) complexes on bounded Lipschitz domains in consisting of Sobolev spaces. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy-Riemann operator and an additional third order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge-Laplacian problem of a twisted de-Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.
Paper Structure (19 sections, 6 theorems, 86 equations, 2 tables)

This paper contains 19 sections, 6 theorems, 86 equations, 2 tables.

Table of Contents

  1. Conformal Korn inequalities.
  2. Linear Cosserat continuum.
  3. Abstract framework
  4. The basic setup
  5. Twisted complex
  6. The BGG construction
  7. Cohomology of the BGG sequence
  8. Examples
  9. The structure of the examples
  10. Conformal Hessian complex
  11. Conformal deformation complex
  12. Higher order generalizations of the Hessian complex
  13. Algebraic and geometric background
  14. Conformal Korn inequalities in 2D
  15. Continuum with microstructures
  16. ...and 4 more sections

Key Result

Theorem 1

Assume SKKS and DKKD and that all the operators $K^{i,j}$ and $S^{i,j}$ are bounded. Then the operators $F^i$ defined above are bounded and induce an isomorphism between the twisted complex twisted-complex and the complex $(Z^{ \bullet},d^{ \bullet})$. Hence for the cohomology of twisted-complex, we

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • ...and 3 more