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

Poincare meets Korn via Maxwell: Extending Korn's First Inequality to Incompatible Tensor Fields

For a bounded three-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first inequality. On the other hand, for skew-symmetric tensor fields our new estimate turns to Poincare's inequality. Therefore, our result may be viewed as a natural common generalization of Korn's first and Poincare's inequality. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditi…

$L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative

For $n\ge2$ and $1<p<\infty$ we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:Ω\to \mathbb{R}^{n\times n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}$ and generalized vanishing tangential trace $P\,τ_l=0$ on $\partial Ω$, denoting by $\{τ_l\}_{l=1,\ldots, n-1}$ a moving tangent frame on $\partialΩ$, more precisely we have: $$\| P \|_{L^p(Ω,\mathbb{R}^{n\times n})}\leq c\,(\| \operatorname{sym} P\|_{L^p(Ω,\mathbb{R}^{n \times n})} + \|\underline{\operatorname{Curl}} P \|_{L^p(Ω,(\mathfrak{so}(n))^n)} ),$$…

On an Extension of Korn's First Inequality to Incompatible Tensor Fields on Domains of Arbitrary Dimensions

For a bounded N-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first inequality. On the other hand, for skew-symmetric tensor fields our new estimate turns to Poincare's inequality. Therefore, our result may be viewed as a natural common generalization of Korn's first and Poincare's inequality. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions …

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

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}; Ω,\mathbb{R}^{3\times3})$. More precisely, let $Ω\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \begin{equation*} \| P\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left( \|\operatorname{sym} P\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \| \operatorname{Curl}P \|_{L^p(Ω, \mathbb{R}^{3\times3})}\right)\end{equation*} holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$, …

任意维度下最优的不相容 Korn-Maxwell-Sobolev 不等式

Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions

本文刻画了适用于任意维度且不相容矩阵场的最优 Korn-Maxwell-Sobolev 不等式，推广了线性弹性中的 Korn 型不等式，并特别对二维情形进行了精细分析。

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

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}; Ω,\mathbb{R}^{3\times3})$. More precisely, let $Ω\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev} \operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}…