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

Abstract

Let $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 be 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 $$L^p$$ L p -Korn inequality for incompatible tensor fields P in the new Banach space $$\begin{aligned}&W^{1,\,p,\,r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3}) \\&\quad = \{ P \in L^p(\Omega ; \mathbb {R}^{3 \times 3}) \mid {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \in L^r(\Omega ; \mathbb {R}^{3 \times 3}),\ {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}(P \times \nu ) = 0 \text { on }\partial \Omega \} \end{aligned}$$ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) = { P ∈ L p ( Ω ; R 3 × 3 ) ∣ dev sym Curl P ∈ L r ( Ω ; R 3 × 3 ) , dev sym ( P × ν ) = 0 on ∂ 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 , ∞ ) , 1 r ≤ 1 p + 1 3 , r > 1 if p = 3 2 . Specifically, there exists a constant $$c=c(p,\Omega ,r)>0$$ c = c ( p , Ω , r ) > 0 such 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 ‖ L p ( Ω , R 3 × 3 ) ≤ c ‖ sym P ‖ L p ( Ω , R 3 × 3 ) + ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 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 ∈ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) . Here, $${{\,\mathrm{dev}\,}}X :=X -\frac{1}{3} {{\,\mathrm{tr}\,}}(X)\,{\mathbb {1}}$$ dev X : = X - 1 3 tr ( X ) 1 denotes the deviatoric (trace-free) part of a $$3 \times 3$$ 3 × 3 matrix X and 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 $$K_{S,dSC}$$ K S , d S C which is determined by the conditions $${{\,\mathrm{sym}\,}}P =0$$ sym P = 0 and $${{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P = 0$$ dev sym Curl P = 0 . In that case one can replace $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^r(\Omega ,\mathbb {R}^{3\times 3})} $$ ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 3 ) by $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{W^{-1,p}(\Omega ,\mathbb {R}^{3\times 3})}$$ ‖ dev sym Curl P ‖ W - 1 , p ( Ω , R 3 × 3 ) . The new $$L^p$$ L p -estimate implies a classical Korn’s inequality with weak boundary conditions by choosing $$P=\mathrm {D}u$$ P = D u and 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 $$\mathrm {D}^3 P$$ D 3 P in terms of $$\mathrm {D}^2 {{\,\mathrm{dev}\,}}{{\

A serendipity fully discrete div-div complex on polygonal meshes

“…For the sake of clarity, we recall the statement of this key result. It has been observed in [26,27] that the previous result can be seen as a generalisation of both the Poincaré-Wirtinger and Korn's second inequalities. In the following Proposition, we apply Lemma 25 to some particular cases in which the tensor field P is skew-symmetric and assuming p = 2.…”

“…The discrete functional inequalities below hinge on [26,Theorem 3.3], which the authors refer to as incompatible Korn type inequality for L p -regular tensor fields. For the sake of clarity, we recall the statement of this key result.…”

On Korn’s First Inequality in a Hardy-Sobolev Space

“…Finally, we note that there are other similar inequalities, such as a generalized Korn inequality for (so-called) incompatible tensor fields (see, e.g., [19,[29][30][31]) and geometric rigidity (see, e.g., [9,11,18]) that are valid for 1 < p < ∞, but not for p = 1 or p = ∞. We are curious if there are versions of such inequalities that are valid in H 1 and its dual space BMO.…”

Lp-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

“…The estimate (1.14) generalizes the corresponding result in [6] from the -setting to the -setting, whereas the trace-free second type inequality (1.12) is completely new. Generalizations to different right-hand sides and higher dimensions have been obtained in the recent papers [40, 41]. Note however that the estimates (1.12) and (1.14) are restricted to the case of three dimensions since the deviatoric operator acts on square matrices and only in the three-dimensional setting the matrix Curl returns a square matrix.…”

“…They allow to bound the -norm of the gradient in terms of the symmetric gradient, i.e. Korn's first inequality states Generalizations to many different settings have been obtained in the literature, including the geometrically nonlinear counterpart [23, 24, 39], mixed growth conditions [15], incompatible fields (also with dislocations) [6, 40–43, 48, 55–58], as well as the case of non-constant coefficients [37, 50, 59, 62] and on Riemannian manifolds [9]. In this paper we focus on their improvement towards the trace-free case: where denotes the deviatoric (trace-free) part of the square matrix .…”