Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy
Authors: Lewintan Peter , Müller Stefan , Neff Patrizio 
Published: July 2021

Let$$\Omega \subset \mathbb {R}^3$$Ω⊂R3be an open and bounded set with Lipschitz boundary and outward unit normal$$\nu $$ν. For$$1<p<\infty $$1<p<∞we establish an improved version of the generalized$$L^p$$Lp-Korn inequality for incompatible tensor fieldsPin the new Banach space$$\begin{aligned}&W^{1,\,p,\,r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3}) \\&\quad = \{ P \in L^p(\Omega ; \mathbb {R}^{3 \times 3}) \mid {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \in L^r(\Omega ; \mathbb {R}^{3 \times 3}),\ {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}(P \times \nu ) = 0 \text { on }\partial \Omega \} \end{aligned}$$W01,p,r(devsymCurl;Ω,R3×3)={P∈Lp(Ω;R3×3)∣devsymCurlP∈Lr(Ω;R3×3),devsym(P×ν)=0on∂Ω}where$$\begin{aligned} r \in [1, \infty ), \qquad \frac{1}{r} \le \frac{1}{p} + \frac{1}{3}, \qquad r >1 \quad \text {if }p = \frac{3}{2}. \end{aligned}$$r∈[1,∞),1r≤1p+13,r>1ifp=32.Specifically, there exists a constant$$c=c(p,\Omega ,r)>0$$c=c(p,Ω,r)>0such that the inequality$$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})}\le c\,\left( \Vert {{\,\mathrm{sym}\,}}P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})} + \Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^{r}(\Omega ,\mathbb {R}^{3\times 3})}\right) \end{aligned}$$‖P‖Lp(Ω,R3×3)≤c‖symP‖Lp(Ω,R3×3)+‖devsymCurlP‖Lr(Ω,R3×3)holds for all tensor fields$$P\in W^{1,\,p, \, r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3})$$P∈W01,p,r(devsymCurl;Ω,R3×3). Here,$${{\,\mathrm{dev}\,}}X :=X -\frac{1}{3} {{\,\mathrm{tr}\,}}(X)\,{\mathbb {1}}$$devX:=X-13tr(X)1denotes the deviatoric (trace-free) part of a$$3 \times 3$$3×3matrixXand the boundary condition is understood in a suitable weak sense. This estimate also holds true if the boundary condition is only satisfied on a relatively open, non-empty subset$$\Gamma \subset \partial \Omega $$Γ⊂∂Ω. If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space$$K_{S,dSC}$$KS,dSCwhich is determined by the conditions$${{\,\mathrm{sym}\,}}P =0$$symP=0and$${{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P = 0$$devsymCurlP=0. In that case one can replace$$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^r(\Omega ,\mathbb {R}^{3\times 3})} $$‖devsymCurlP‖Lr(Ω,R3×3)by$$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{W^{-1,p}(\Omega ,\mathbb {R}^{3\times 3})}$$‖devsymCurlP‖W-1,p(Ω,R3×3). The new$$L^p$$Lp-estimate implies a classical Korn’s inequality with weak boundary conditions by choosing$$P=\mathrm {D}u$$P=Duand a deviatoric-symmetric generalization of Poincaré’s inequality by choosing$$P=A\in {{\,\mathrm{\mathfrak {so}}\,}}(3)$$P=A∈so(3). The proof relies on a representation of the third derivatives$$\mathrm {D}^3 P$$D3Pin terms of$$\mathrm {D}^2 {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P$$D2devsymCurlPcombined with the Lions lemma and the Nečas estimate. We also discuss applications of the new inequality to the relaxed micromorphic model, to Cosserat models with the weakest form of the curvature energy, to gradient plasticity with plastic spin and to incompatible linear elasticity.