BGG sequences with weak regularity and applications

TL;DR

This paper explores BGG complexes with low regularity on Lipschitz domains, computes their cohomology, and applies these results to problems in elasticity, conformal geometry, and continuum mechanics.

Contribution

It introduces a novel approach to BGG complexes in Sobolev spaces without algebraic injectivity/surjectivity constraints, enabling new applications in geometry and elasticity.

Findings

Computed cohomology of conformal complexes in Sobolev spaces

Established a conformal Korn inequality in 2D

Linked Cosserat elasticity to Hodge-Laplacian problems

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 M\"obius 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.