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 $$ W^{1,\,p,\,r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl}; \Omega,\mathbb R^{3\times3}) = \{ P \in L^p(\Omega,\mathbb R^{3\times3}) \mid \operatorname{dev} \operatorname{sym} \operatorname{Curl} P \in L^r(\Omega,\mathbb R^{3\times3}),\ \operatorname{dev} \operatorname{sym} (P \times \nu) = 0 \text{ on $\partial \Omega$}\} $$ where $$ r \in [1, \infty), \qquad \frac1r \le \frac1p + \frac13, \qquad r >1 \quad \text{if $p = \frac32$.}$$ 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 $P\in W^{1,\,p, \, r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl})$. Here, $\operatorname{dev} X := X -\frac13 \operatorname{tr}(X)\,\mathbb{1}$ denotes the deviatoric (trace-free) part of a $3 \times 3$ matrix $X$ and the boundary condition is understood in a suitable weak sense.