Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be an open and bounded set with Lipschitz boundary and outward unit normal $\nu$. For $1<p<\infty$ we establish an improved version of the generalized $L^p$-Korn inequality for incompatible tensor fields $P$ in the new Banach space $$W_0^{1,p,r}(\mathrm{dev}\mathrm{sym}\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})=\{P\in L^p(\Omega ,{\mathbb{R}}^{3\times 3})\mid \mathrm{dev}\mathrm{sym}\mathrm{Curl}P\in L^r(\Omega ,{\mathbb{R}}^{3\times 3}),\mathrm{dev}\mathrm{sym}(P\times \nu )=0\text{ on ∂Ω}\}$$ where $$r\in [1,\infty ),\frac{1}{r}\le \frac{1}{p}+\frac{1}{3},r>1\text{if p=23.}$$ Specifically, there exists a constant $c=c(p,\Omega ,r)>0$ such that the inequality \[ \|P \|_{L^p}\leq c\,\left(\|\operatorname{sym} P \|_{L^p} + \|\operatorname{dev}\operatorname{sym} \operatorname{Curl} P \|_{L^{r}}\right) \] holds for all tensor fields $P\in W_0^{1,p,r}(\mathrm{dev}\mathrm{sym}\mathrm{Curl})$. Here, $\mathrm{dev}X:=X-\frac{1}{3}\mathrm{tr}(X)1$ denotes the deviatoric (trace-free) part of a $3\times 3$ matrix $X$ and the boundary condition is understood in a suitable weak sense.

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Cited by in corpus (11)

• Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)
• Primal and mixed finite element formulations for the relaxed micromorphic model
• Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
• 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
• Matrix representation of a cross product and related curl-based differential operators in all space dimensions
• A note on local higher regularity in the dynamic linear relaxed micromorphic model
• Thermally driven fluid convection in the incompressible limit regime
• Korn's inequality in anisotropic Sobolev spaces