On Korn inequalities with lower order trace terms

TL;DR

This work develops a general Korn-type inequality framework for $k$-th order, $\\mathbb{C}$-elliptic operators $\\mathbb{A}$ acting on vector fields, showing that the Sobolev norm $\\|\\mathbf{u}\\|_{W^{k}X}$ can be controlled by the $\\mathbb{A}$-image term plus a seminorm on the kernel $\\ker(\\mathbb{A};\\mathbb{R}^{n})$. It yields an equivalence between the norm on the kernel and the validity of the Korn inequality, with two proofs (direct and Dra\v{z}i\c{c}'s indirect argument) and extends to a wide range of trace settings, including full, partial, and lower-dimensional traces, as well as $\\mu$-traces in interior and boundary contexts. The paper then specialized results to bulk and boundary trace conditions, and to Orlicz-type scales, providing a versatile toolbox for coercive estimates in elasticity, fluid mechanics, and numerical PDE analysis. A key contribution is the incorporation of lower-order trace terms, enabling recovery of classical Korn inequalities as special cases and enabling new inequalities under domain geometries (John, extension domains) and trace-measure conditions. The authors also provide a symbolic verification framework for the norm-compatibility condition on kernels, offering a practical route to ascertain when a proposed trace seminorm yields a norm on the kernel. Overall, the framework broadens the applicability of Korn-type coercivity results to nonstandard operators, spaces, and geometric settings, with potential impact on PDE analysis and numerical methods.

Abstract

We give an elementary estimate that entails and generalises numerous Korn inequalities scattered in the literature. As special instances, we obtain general Korn-type inequalities involving normal or tangential trace components, or lower dimensional trace integrals.

Figures (3)

• Figure 1: Sketch of Maz'ya's shrinking rooms and passages domain from Mazya (not to scale)
• Figure 2: Conditions on domains. (i) reflects Jones' $(\varepsilon,\delta)$-condition \ref{['eq:Jones']}. It is not satisfied in (ii) and (iii); still, the slit domain from (iii) is John. Similarly, (iv) is a typical instance of a polyhedral non-Lipschitz, yet John domain.
• Figure 3: Critical geometric scenarios for the deviatoric gradient; the finely dotted arrows indicating the vector field of kernel element $\alpha x+b$.

Theorems & Definitions (50)

• Theorem 1.1
• Lemma 2.1
• Example 2.2: $\mathbb{C}$-elliptic operators and their null spaces
• proof : Proof of Theorem \ref{['thm:korn-gen']}
• Remark 2.3: On the assumptions of Theorem \ref{['thm:korn-gen']}
• Remark 2.4: Choices of $X$ and $Y$
• Lemma 2.5: Dražić
• proof : Proof of \ref{['eq:KornConclude']}
• proof : Indirect proof of Theorem \ref{['thm:korn-gen']}
• Example 2.6