Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

Abstract

For $1<p<\infty$ we prove an $L^p$-version of the generalized Korn inequality for incompatible tensor fields $P$ in $W_0^{1,p}(\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})$. More precisely, let $\Omega \subset {\mathbb{R}}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \begin{equation*} \| P\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left( \|\operatorname{sym} P\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \| \operatorname{Curl}P \|_{L^p(Ω, \mathbb{R}^{3\times3})}\right)\end{equation*} holds for all tensor fields $P\in W_0^{1,p}(\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})$, i.e., for all $P\in W^{1,p}(\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$. For compatible $P=Du$ this recovers an $L^p$-version of the classical Korn's first inequality $$\Vert Du{\Vert }_{L^p(\Omega ,{\mathbb{R}}^{3\times 3})}\le c\Vert \mathrm{sym}Du{\Vert }_{L^p(\Omega ,{\mathbb{R}}^{3\times 3})}\text{with }Du\times \nu =0\text{on ∂Ω},$$ and for skew-symmetric $P=A\in \mathfrak{so}(3)$ an $L^p$-version of the Poincaré inequality $$\Vert A{\Vert }_{L^p(\Omega ,\mathfrak{so}(3))}\le c\Vert \mathrm{Curl}A{\Vert }_{L^p(\Omega ,{\mathbb{R}}^{3\times 3})}\text{with }A\times \nu =0\iff A=0\text{on ∂Ω}.$$

References in corpus (2)

• On the optimal constants in Korn's and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions
• The $p$-CurlCurl : Spaces, traces, coercivity and a Helmholtz decomposition in $L^p$

Cited by in corpus (7)

• Primal and mixed finite element formulations for the relaxed micromorphic model
• Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
• $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative
• Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple
• $L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions
• Korn-Maxwell-Sobolev inequalities for general incompatibilities
• A global higher regularity result for the static relaxed micromorphic model on smooth domains