On Korn-Maxwell-Sobolev Inequalities

Abstract

We establish a family of inequalities that allow one to estimate the L qnorm of a matrix-valued field by the L q -norm of an elliptic part and the L p -norm of the matrix-valued curl. This particularly extends previous work by Neff et al. and, as a main novelty, is applicable in the regime p = 1.

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

“…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].…”

“…For compatible P = Du we get back from (1.8), (1.9), (1.13) and (1.14) the corresponding classical Korn inequalities. Recently, Gmeineder and Spector [66] extended inequality (1.8) to the case where sym P is generalized to any linear operator A(P ) such that A(Du) is a first order elliptic operator, thus including also one result of [97] with dev sym P .…”