BGG sequences with weak regularity and applications
The purpose of the article is the investigation of some Bernstein-Gelfand-Gelfand (BGG) complexes consisting of Sobolev spaces on bounded Lipschitz domains in \(\mathbf{R}^n\), with a view toward applications in numerical analysis. One obtains an explicit construction of the relevant complexes from the de Rham complex of differential forms with values in an appropriate finite-dimensional vector space, which, toogether with known results on the de Rham complex, allow to obtain analytical results which are important for the use of BGG complexes in numerical analysis and applied mathematics.\N\NThe cohomology of some special complexes was left open in [\textit{D. N. Arnold} and \textit{K. Hu}, Found. Comput. Math. 21, No. 6, 1739--1774 (2021; Zbl 1520.58011)]. The same paper shows that analytic results, e.g., various Poincaré or Korn inequalities, are deeply rooted in the algebraic structures. However, the conformal Korn inequality in two space dimensions remains open as the diagram does not fulfill the assumptions in [loc. cit.]; also, the algebraic conditions (injectivity/surjectivity of the connecting maps) in raise challenges for constructing numerical methods.\N\NIn the present paper the authors provide a generalization of the results obtained in the above mentioned article of D. N. Arnold and K. Hu. A construction that is closer to the developments of [\textit{A. Čap} et al., Ann. Math. (2) 154, No. 1, 97--113 (2001; Zbl 1159.58309)] is used. In particular, the generalizations of the framework include allowing more than two rows in the diagram and removing the injectivity/surjectivity conditions in the assumptions. The resulting complexes will have a more complicated form, which can be simplified to the cases treated by D. N. Arnold and K. Hu, when algebraic conditions hold.\N\NAs a result of this generalization, the cohomology of important complexes, e.g., the conformal deformation complex, the conformal Hessian complex and higher-order generalizations of the Hessian complex is computed. The authors establish a conformal Korn inequality in two space dimensions and generalizations of the linear Cosserat elasticity model based on its connections to the twisted de Rham complexes. The above generalizations of the algebraic framework play a critical role in these developments.\N\NThe article is interesting and well written. Further directions are suggested.
scientific article; zbMATH DE number 7900825
scientific article