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\(L^p\)-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions. (English) Zbl 1504.35015

Summary: 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 \[ \Vert P \Vert_{L^p (\Omega,\mathbb{R}^{3\times 3})} \leqslant c \left(\Vert \operatorname{dev} \operatorname{sym} P \Vert_{L^p (\Omega,\mathbb{R}^{3\times 3})} + \Vert \operatorname{dev}\operatorname{Curl} P \Vert_{L^p(\Omega,\mathbb{R}^{3\times 3})}\right) \] 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 \(\operatorname{dev} P : = P -\frac{1}{3} \operatorname{tr}(P) \cdot 1\) denotes the deviatoric (trace-free) part of \(P\). We also show the norm equivalence \[ \Vert P\Vert_{L^p (\Omega,\mathbb{R}^{3\times 3})}+\Vert \operatorname{Curl} P \Vert_{L^p (\Omega,\mathbb{R}^{3\times 3})} \leq c\,\left(\Vert P\Vert_{L^p(\Omega,\mathbb{R}^{3\times 3})} + \Vert \operatorname{dev} \operatorname{Curl} P \Vert_{L^p (\Omega,\mathbb{R}^{3\times 3})}\right) \] 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.

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35B45 A priori estimates in context of PDEs
35Q74 PDEs in connection with mechanics of deformable solids
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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