Quantitative Gaffney and Korn inequalities

We prove a homogeneous, quantitative version of Ehrling’s inequality for the function spaces $H^1(\mathrm{\Omega })\subset \subset L^2(\mathrm{\partial }\mathrm{\Omega })$, $H^1(\mathrm{\Omega })\hookrightarrow L^2(\mathrm{\Omega })$ which reflects geometric properties of a given $C^{1,1}$-domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. We use this result to derive quantitative homogeneous versions of Gaffney’s inequality, of relevance in electromagnetism as well as Korn’s inequality, of relevance in elasticity theory. The main difference to the corresponding classical results is that the constants appearing in our inequalities turn out to be dimensional constants. We provide explicit upper bounds for these constants and show that in the case of the tangential homogeneous Korn inequality our upper bound is asymptotically sharp as $n\to \mathrm{\infty }$. Lastly, we raise the question of the optimal values of these dimensional constants.