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

Abstract

In this note we show that a sharp rigidity estimate and a sharp Korn's inequality for matrix-valued fields whose incompatibility is a bounded measure can be obtained as a consequence of a Hodge decomposition with critical integrability due to Bourgain and Brezis.

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

“…which follows from [20]. For the geometrically nonlinear counterpart of (1.10) in a mixed-growth setting in two dimensions we refer the reader to [64] and higher-dimensional analogues can be found in [90,32]. Improvements of the Korn inequalities for incompatible tensor fields (1.8) and (1.9) towards the trace-free cases are also valid.…”

“…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].…”