On Korn-Maxwell-Sobolev Inequalities

We establish a family of inequalities that allow one to estimate the ${\mathrm{L}}^q$-norm of a matrix-valued field by the ${\mathrm{L}}^q$-norm of an elliptic part and the ${\mathrm{L}}^p$-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 $p=1$.