Quantitative Gaffney and Korn inequalities
Wadim Gerner
TL;DR
This work develops quantitative, homogeneous versions of Gaffney and Korn inequalities on bounded $C^{1,1}$ domains by proving a homogeneous Ehrling-type trace inequality that is tied to the domain reach $ρ(∂Ω)$. The authors then derive explicit, dimension-dependent bounds for the Gaffney and Korn constants, distinguishing between tangential and normal boundary conditions, and show that the tangential Korn constant is asymptotically sharp as $n$ grows. The results are constructive, providing upper bounds $C^*_G(n)$ and $C^*_K(n)$ that depend only on dimension, and offer insight into optimal dimensional constants and their relation to boundary geometry. Applications to electromagnetism and elasticity are emphasized, where only curl/divergence or symmetric gradient information may be available.
Abstract
We prove a homogeneous, quantitative version of Ehrling's inequality for the function spaces $H^1(Ω)\subset\subset L^2(\partialΩ)$, $H^1(Ω)\hookrightarrow L^2(Ω)$ which reflects geometric properties of a given $C^{1,1}$-domain $Ω\subset\mathbb{R}^n$. 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 $n\rightarrow \infty$. Lastly, we raise the question of the optimal values of these dimensional constants.