Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

“…for all P ∈ W 1, p (Curl; Ω, R 3×3 ), cf. [99]. These estimates also hold true in all dimensions n ≥ 2 with an adequate understanding of the matrix Curl, [100].…”

“…The Korn inequalities generalize to many different settings, including the geometrically nonlinear counterpart [55,96], mixed growth conditions [30], incompatible fields (also with dislocations) [109,121,9,99,100,97,60,32,66] and trace-free infinitesimal strain measures [35,79,131,132,59,138,9,97,98]. For trace-free Korn's inequalities in pseudo-Euclidean space see [146] and for trace-free Korn inequalities on manifolds see [35,77].…”

“…cf. [121] for p = 2 and [99] for all p > 1. Furthermore, the version (1.3) generalizes to inf A∈so(3)…”

“…Korn's second inequality provides an L p -estimate of a gradient vector field (modulo a constant) in terms of the symmetric part of the derivative. This can be generalized to general fields P if one adds a term in Curl P on the right hand side [99,100,121]. For recent refined estimates which involve only the deviatoric part of sym P and Curl P , see [97,9,98].…”

Sharp rigidity estimates for incompatible fields as consequence of the Bourgain Brezis div-curl result

“…We remark that the main point in this estimate is that it provides an estimate with the total variation and therefore allows for concentrated incompatibilities, which correspond to concentration of crystal defects. Korn's inequalities with incompatibility involving the L p norm of curl β, with p > 1, instead of the total variation, can be found in [LN19] and the references therein.…”