Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields (1912.08447v5)

Abstract: For $1<p<\infty$ we prove an $L^p$-version of the generalized Korn inequality for incompatible tensor fields $P$ in $ W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$. More precisely, let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then there exists a constant $c\>0$ such that \begin{equation*} | P|{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left( |\operatorname{sym} P|{L^p(\Omega,\mathbb{R}^{3\times3})} + | \operatorname{Curl}P |{L^p(\Omega, \mathbb{R}^{3\times3})}\right)\end{equation*} holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$, i.e., for all $P\in W^{1,\,p}(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$ with vanishing tangential trace $ P\times \nu=0 $ on $ \partial\Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial\Omega$. For compatible $P=D u$ this recovers an $L^p$-version of the classical Korn's first inequality $$ |D u |{L^p(\Omega,\mathbb{R}^{3\times 3})} \le c\, |\operatorname{sym}D u|{L^p(\Omega,\mathbb{R}^{3\times3})} \quad \text{with }D u \times \nu = 0 \quad \text{on $\partial \Omega$}, $$ and for skew-symmetric $P=A\in\mathfrak{so}(3)$ an $L^p$-version of the Poincar\'{e} inequality $$ |A|{L^p(\Omega,\mathfrak{so}(3))}\le c\, |\operatorname{Curl} A|_{L^p(\Omega,\mathbb{R}^{3\times3})} \quad \text{with } A \times \nu = 0 \ \Leftrightarrow \ A=0 \quad \text{on $\partial \Omega$}. $$