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

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ be an open and bounded set with Lipschitz boundary and outward unit normal $\hat{I}½$. 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 $$W_0^{1,p,r}(\mathrm{dev}\mathrm{sym}\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})=\{P\in L^p(\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})\mid \mathrm{dev}\mathrm{sym}\mathrm{Curl}P\in L^r(\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3}),\mathrm{dev}\mathrm{sym}(P\times \hat{I}½)=0\text{ on ∂I^c◯}\}$$ 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 $c=c(p,\hat{I}\text{c◯},r)>0$ 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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