Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions

TL;DR

This paper characterizes all linear maps that satisfy optimal Korn-Maxwell-Sobolev inequalities in all dimensions, extending classical inequalities to incompatible matrix fields with a comprehensive analysis of ellipticity, integrability, and dimension-specific cases.

Contribution

It provides a complete characterization of linear maps satisfying Korn-Maxwell-Sobolev inequalities in all dimensions, including incompatible fields, with optimal constants and a detailed dimension-dependent analysis.

Findings

Characterization of all linear maps satisfying the inequalities.

Extension of Korn-type inequalities to incompatible matrix fields.

Applicability to all space dimensions with optimal results.

Abstract

We characterise all linear maps $\mathcal{A}\colon\mathbb{R}^{n\times n}\to\mathbb{R}^{n\times n}$ such that, for $1\leq p<n$, \begin{align*} \|P\|_{L^{p^{*}}(\mathbb{R}^{n})}\leq c\,\Big(\|\mathcal{A}[P]\|_{L^{p^{*}}(\mathbb{R}^{n})}+\|\mathrm{Curl} P\|_{L^{p}(\mathbb{R}^{n})} \Big) \end{align*} holds for all compactly supported $P\in C_{c}^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n\times n})$, where $\mathrm{Curl} P$ displays the matrix 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 constellations between the ellipticities of $\mathcal{A}$, the integrability $p$ and the underlying space dimensions $n$, especially requiring a finer analysis in the two-dimensional situation.