Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions

Abstract

We characterise all linear maps $\mathcal{A}:{\mathbb{R}}^{n\times n}\to {\mathbb{R}}^{n\times n}$ such that, for $1\le p<n$, \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.

References in corpus (15)

• A unifying perspective: the relaxed linear micromorphic continuum
• On Rank-One Convex Functions that are homogeneous of Degree One
• Limiting Sobolev inequalities for vector fields and canceling linear differential operators
• Generalized Korn's inequality and conformal Killing vectors
• Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy
• Embeddings for $\mathbb{A}$-weakly differentiable functions on domains
• On the Trace Operator for Functions of Bounded $\mathbb{A}$-Variation
• On limiting trace inequalities for vectorial differential operators
• Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields
• Primal and mixed finite element formulations for the relaxed micromorphic model
• $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative
• $L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions
• A canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin accounting for the Burgers vector
• Remarks on Ornstein's non-inequality in ${\mathbb{R}}^{2\times 2}$
• Matrix representation of a cross product and related curl-based differential operators in all space dimensions

Cited by in corpus (5)

• A computational approach to identify the material parameters of the relaxed micromorphic model
• Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple
• Korn-Maxwell-Sobolev inequalities for general incompatibilities
• A global higher regularity result for the static relaxed micromorphic model on smooth domains
• On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators