L^p-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

Abstract

For $1< p<\infty$ we prove an $L^{p}$ -version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \] holds for all tensor fields $P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ , i.e., for all $P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$ and $\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$ denotes the deviatoric (trace-free) part of $P$ . We also show the norm equivalence \begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*} for tensor fields $P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial \Omega$ of the boundary.

On Korn-Maxwell-Sobolev Inequalities

“…The strategy to arrive at Theorem 3.1 is similar to that of Theorem 1.1, where now the global singular integral or Fourier multiplier estimate underyling (2.6) is replaced by the Nečas-Lions lemma. This strategy has also been pursued in [20,21], where we now employ a duality estimate as in [2,36] to deal with the corresponding negative norms.…”

“…We then record from [20,Eq. (42)] and [21,Eq. (3.24)] that there exists a constant c = c(q) > 0 such that 2…”

“…Another relevant variant of (1.2) is given by replacing the symmetric part of F on the right-hand side of (1.2) by its trace-free or deviatoric part F dev := F sym − 1 3 tr(F )½ 3 , where ½ 3 denotes the (3 × 3)-unit matrix. As established in [21], the analogue of (1.2) reads…”

“…Such inequalities have been considered and studied extensively by Neff and coauthors, cf. [20,21,24,25]; one form thereof is given by…”

“…is equivalent to A inducing an elliptic operator A of the form (1.5), see the discussion at the end of Section 2.1. By smooth approximation, this gives us back the corresponding inequalities considered in [20,21].…”