Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions

We characterise all linear maps A : R-nxn -> R-nxn such that, for 1 <= p < n, parallel to P parallel to L-p* ((Rn)) <= c(parallel to A [P]parallel to(Lp*) ((Rn)) + parallel to Curl P parallel to (Lp(Rn))) holds for all compactly supported P epsilon C-c(infinity) (R-n; R-nxn), where Curl P displays thematrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different combinations between the ellipticities of A, the integrability p and the underlying space dimensions n, especially requiring a finer analysis in the two-dimensional situation.