We present a Korn-Poincaré-type inequality in a planar setting which is in the spirit of the Poincaré inequality in SBV due to De Giorgi, Carriero, Leaci. We show that for each function in SBD$^2$ one can find a modification which differs from the original displacement field only on a small set such that the distance of the modification from a suitable infinitesimal rigid motion can be controlled by an appropriate combination of the elastic and the surface energy. In particular, the result can be used to obtain compactness estimates for functions of bounded deformation.
We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincaré-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequali…
The Fractional Korn Inequality on Uniform Domains and New Korn Inequalities for Truncated Seminorms
Universidad de Buenos Aires / IMAS, CONICET（布宜诺斯艾利斯大学 / 阿根廷国家科学研究委员会物质研究所）；Centro de Matemática de La Plata, Facultad de Cs. Exactas, Universidad Nacional de La Plata / CONICET（拉普拉塔数学中心，拉普拉塔国立大学理学院 / 阿根廷国家科学研究委员会）；Department of Mathematics and Statistics, California State Polytechnic University Pomona（加州州立理工大学波莫纳分校数学与统计系）
In this short communication, we present a new proof for the Korn inequality in a n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result the standard Poincaré inequality plays a fundamental role.
The validity of Korn's first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn's first inequality holds in the case $ps>1$ for fractional $W^{s,p}_0(Ω)$ Sobolev fields in open and bounded $C^{1}$-regular domains $Ω\subset \mathbb R^n$. Also, in the case $ps<1,$ for any open bounded $C^1$ domain $Ω\subset \mathbb R^n$ we construct counterexamples to the inequality, i.e., Korn's first inequality fails to hold in bounded domains. The proof of the inequality in the case $ps>1$ follows a standard compactness approach adopted …
In this paper we prove a fractional analogue of the classical Korn's first inequality. The inequality makes it possible to show the equivalence of a function space of vector field characterized by a Gagliardo-type seminorm with 'projected difference' with that of a corresponding fractional Sobolev space. As an application, we will use it to obtain a Caccioppoli-type inequality for a nonlinear system of nonlocal equations, which in turn is a key ingredient in applying known results to prove a higher fractional differentiability result for weak solutions of the nonlinear system of nonlocal equat…
The asymptotically sharp Korn interpolation and second inequalities for shells
考虑三维欧几里得空间中具有有界主曲率的壳。证明了Korn插值（或所谓的第一个半\footnote{The inequality first introduced in [6]}）和第二个不等式的那种壳的$\Bu\in W^{1,2}$向量场，不对$\Bu.$施加边界或归一化条件估计中的常数在壳厚度的渐近性方面是最佳的$h,$具有缩放$h$或$O(1).$ Korn插值不等式减少了导出任何线性Korn型的问题壳的估计可以简单地用右侧的对称梯度证明庞加莱类型估计。这尤其适用于壳体的线性几何刚度估计，即，无边界条件的Korn第一不等式。
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 $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$, …
We prove a compactness result in GBD which also provides a new proof of the compactness theorem in GSBD, due to Chambolle and Crismale [5, Theorem 1.1]. Our proof is based on a Fréchet-Kolmogorov compactness criterion and does not rely on Korn or Poincaré-Korn inequalities.
Extension operators and Korn inequality for variable coefficients in perforated domains with applications to homogenization of viscoelastic non-simple materials
The general theory of slender structure buckling by Grabovsky and Truskinovsky [\textit{Cont. Mech. Thermodyn.,} 19(3-4):211-243, 2007], (later extended in [\textit{Journal of Nonlinear Science.,} Vol. 26, Iss. 1, pp. 83--119, 2016] by Grabovsky and the author), predicts that the critical buckling load of a thin shell under dead loading is closely related to the Korn's constant (in Korn's first inequality) of the shell under the Dirichlet boundary conditions resulting from the loading program. It is known that under zero Dirichlet boundary conditions on the thin part of the boundary of positiv…
Let $1 < p < \infty$ and let $Ω$ be an open and bounded set of $\mathbb R^n$. We establish classical Korn inequalities \[ \inf_{\substack{v \in L^p(Ω)\\\mathcal A v = 0}} \|u - v\|_{W^{k,p}(Ω)} \le C \| \mathcal A u\|_{L^p(Ω)} \] for all $k$th order operators $\mathcal A$ satisfying the maximal-rank condition. This new condition is satisfied by the divergence, Laplacian, Laplace-Beltrami, and Wirtinger operators, among others. As such, our estimates generalize Fuchs' estimates for the del-bar operator to maximal-rank operators and to arbitrary domains. For domains with sufficiently regular bou…
In these notes, we present versions of trace theorems for Sobolev spaces over an interval in the real line, and also a one-dimensional version of the well-known Poincare inequality.
In this paper we investigate the fractional Poincaré inequality on unbounded domains. In the local case, Sandeep-Mancini showed that in the class of simply connected domains, Poincaré inequality holds if and only if the domain does not allow balls of arbitrarily large radius (finite ball condition). We prove that such a result can not be true in the `nonlocal/fractional' setting even if finite ball condition is replaced by a related stronger condition. We further provide some sufficient criterions on domains for fractional Poincaré inequality to hold. In the end, asymptotic behaviour of all ei…
