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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\u2013Riemann operator and an additional third-order operator with a background in M\u00f6bius geometry. We show that the linear Cosserat elasticity model is a Hodge\u2013Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.<\/jats:p>","DOI":"10.1007\/s10208-023-09608-9","type":"journal-article","created":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T17:02:21Z","timestamp":1690477341000},"page":"1145-1184","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["BGG Sequences with Weak Regularity and Applications"],"prefix":"10.1007","volume":"24","author":[{"given":"Andreas","family":"\u010cap","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kaibo","family":"Hu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,27]]},"reference":[{"key":"9608_CR1","doi-asserted-by":"crossref","unstructured":"S.\u00a0Amstutz and N.\u00a0Van\u00a0Goethem, The incompatibility operator: from Riemann\u2019s intrinsic view of geometry to a new model of elasto-plasticity, in Topics in Applied Analysis and Optimisation, Springer, 2019, pp.\u00a033\u201370.","DOI":"10.1007\/978-3-030-33116-0_2"},{"key":"9608_CR2","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s00205-014-0806-1","volume":"216","author":"A Angoshtari","year":"2015","unstructured":"A.\u00a0Angoshtari and A.\u00a0Yavari, Differential complexes in continuum mechanics, Archive for Rational Mechanics and Analysis, 216 (2015), pp.\u00a0193\u2013220.","journal-title":"Archive for Rational Mechanics and Analysis"},{"key":"9608_CR3","unstructured":"J.\u00a0Arf and B.\u00a0Simeon, Structure-preserving discretization of the Hessian complex based on spline spaces, arXiv:2109.05293, (2021)."},{"key":"9608_CR4","doi-asserted-by":"crossref","unstructured":"D.\u00a0N. 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