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 inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, on a bounded domain.
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…
We investigate the character of the linear constraints which are needed for Poincaré and Korn type inequalities to hold. We especially analyze constraints which depend on restriction on subsets of positive measure and on the trace on a portion of the boundary.
It is well known that Korn inequality plays a central role in the theory of linear elasticity. In the present work we prove new asymptotically sharp Korn and Korn-like inequalities in thin curved domains with a non-constant thickness. This new results will be useful when studying the buckling of compressed shells, in particular when calculating the critical buckling load.
In this work we derive asymptotically sharp weighted Korn and Korn-like interpolation (or first and a half) inequalities in thin domains with singular weights. The constants $K$ (Korn's constant) in the inequalities depend on the domain thickness $h$ according to a power rule $K=Ch^α,$ where $C>0$ and $α\in R$ are constants independent of $h$ and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics $h^α$ is optimal as $h\to 0.$ The choice of the weights is motivated by several factors, in particular a spacial case occurs when making Cartesian t…
In this paper we prove asymptotically sharp weighted "first-and-a-half" $2D$ Korn and Korn-like inequalities with a singular weight occurring from Cartesian to cylindrical change of variables. We prove some Hardy and the so-called "harmonic function gradient separation" inequalities with the same singular weight. Then we apply the obtained $2D$ inequalities to prove similar inequalities for washers with thickness $h$ subject to vanishing Dirichlet boundary conditions on the inner and outer thin faces of the washer. A washer can be regarded in two ways: As the limit case of a conical shell when…
The celebrated Poincaré and Friedrichs inequalities estimate the $\mathbb{L}_p$-norm of a function by the $\mathbb{L}_p$-norm of the gradient. We prove the Poincaré inequality for a domain $Ω\subset \mathbb{R}^n$ and for a hypersurface $\mathcal{C}\subset\mathbb{R}^n$ based on open mapping theorem of Banach only. For a cylinder which has a hypersurface as a base, is prove stronger inequality, involving only the surface derivatives. Similar inequalities for the uniform $C$-norm are proved as well. We also estimate $\mathbb{H}^m_p$-norm of functions prove inequalities for some generalizations of…
Understanding asymptotics of gradient components in relation to the symmetrized gradient is im- portant for the analysis of buckling of slender structures. For circular cylindrical shells we obtain the exact scaling exponent of the Korn constant as a function of shell's thickness. Equally sharp results are obtained for individual components of the gradient in cylindrical coordinates. We also derive an ana- logue of the Kirchho? ansatz, whose most prominent feature is its singular dependence on the slenderness parameter, in marked contrast with the classical case of plates and rods.
We establish sharp trace- and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $p=1$, a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $p=1$ and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full $k$-th order…
Necessary and sufficient conditions are exhibited for a Korn type inequality to hold between (possibly different) Orlicz norms of the gradient of vector-valued functions and of the deviatoric part of their symmetric gradients. As a byproduct of our approach, a positive answer is given to the question of the necessity of the same sufficient conditions in related Korn type inequalities for the full symmetric gradient, for negative Orlicz-Sobolev norms, and for the gradient of the Bogovskii operator.
We establish a family of inequalities that allow one to estimate the $\mathrm{L}^{q}$-norm of a matrix-valued field by the $\mathrm{L}^{q}$-norm of an elliptic part and the $\mathrm{L}^{p}$-norm of the matrix-valued curl. This particularly extends previous work by Neff et al. and, as a main novelty, is applicable in the regime $p=1$.
We consider shells in three dimensional Euclidean space which have bounded principal curvatures. We prove Korn's interpolation (or the so called first and a half\footnote{The inequality first introduced in [6]}) and second inequalities on that kind of shells for $\Bu\in W^{1,2}$ vector fields, imposing no boundary or normalization conditions on $\Bu.$ The constants in the estimates are optimal in terms of the asymptotics in the shell thickness $h,$ having the scalings $h$ or $O(1).$ The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to si…
We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to …
We consider shells of non-constant thickness in three dimensional Euclidean space around surfaces which have bounded principal curvatures. We derive Korn's interpolation (or the so called first and a half (The inequality first introduced in [Gra.Har.1])) and second inequalities on that kind of domains for $\Bu\in H^1$ vector fields, imposing no boundary or normalization conditions on $\Bu.$ The constants in the estimates are asymptotically optimal in terms of the domain thickness $h,$ with the leading order constant having the scaling $h$ as $h\to 0.$ This is the first work that determines the…
In the present paper we extend the $L^2$ Korn interpolation and second inequalities in thin domains, proven in [\ref{bib:Harutyunyan.4}], to the space $L^p$ for any $1<p<\infty.$ A thin domain in space is roughly speaking a shell with non-constant thickness around a smooth enough two dimensional surface. The inequality that we prove in $L^p$ holds for practically any thin domain $Ω\subset\mathbb R^3$ and any vector field $\Bu\in W^{1,p}(Ω).$ The constants in the estimate are asymptotically optimal in terms of the domain thickness $h.$ This in particular solves the problem of finding the asympt…
This paper deals with solutions of the divergence for domains with external cusps. It is known that the classic results in standard Sobolev spaces, which are basic in the variational analysis of the Stokes equations, are not valid for this class of domains. For some bounded domains $Ω\subset{\mathbb R}^n$ presenting power type cusps of integer dimension $m\le n-2$, we prove the existence of solutions of the equation ${div}{\bf u}=f$ in weighted Sobolev spaces, where the weights are powers of the distance to the cusp. The results obtained are optimal in the sense that the powers cannot be impro…
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: $$\| P \|_{L^p(Ω,\mathbb{R}^{n\times n})}\leq c\,(\| \operatorname{sym} P\|_{L^p(Ω,\mathbb{R}^{n \times n})} + \|\underline{\operatorname{Curl}} P \|_{L^p(Ω,(\mathfrak{so}(n))^n)} ),$$…
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.
This investigation develops basic methods for the multi-scale analysis for problems in thin porous layers. More precisely, we provide tools for the homogenization in case of "tangentially" periodic structures and dimensional reduction letting the layer thickness tend to zero proportional to the scale parameter $ε$. A crucial point is the identification of scale limits of functions $v_ε$ in subsets of function spaces characterized by uniform a priori estimates with respect to $ε$, arising for solutions of differential equations in heterogeneous media with thin layers, e.g., of a Navier-Stokes s…
In this paper, we study the weighted Korn inequality on some irregular domains, e.g., $s$-John domains and domains satisfying quasi-hyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.
We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to proving a fractional version of the classi…
We study the Korn-Poincaré inequality: \|u\|_{W^{1,2}(S^h)} < C_h \|D(u)\|_{L^2(S^h)}, in domains S^h that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in R^n. By D(u) we denote the symmetric part of the gradient \nabla u, and we assume the tangential boundary conditions: u\vec n^h = 0 on \partial S^h. We prove that C_h remains uniformly bounded as h tends to 0, for vector fields u in any family of cones (with angle <π/2, uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We also show that this condition…
Let $Ω\subset \mathbb{R}^3$ be an open and bounded set with Lipschitz boundary and outward unit normal $ν$. For $1<p<\infty$ we establish an improved version of the generalized $L^p$-Korn inequality for incompatible tensor fields $P$ in the new Banach space $$ W^{1,\,p,\,r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl}; Ω,\mathbb R^{3\times3}) = \{ P \in L^p(Ω,\mathbb R^{3\times3}) \mid \operatorname{dev} \operatorname{sym} \operatorname{Curl} P \in L^r(Ω,\mathbb R^{3\times3}),\ \operatorname{dev} \operatorname{sym} (P \times ν) = 0 \text{ on $\partial Ω$}\} $$ where $$ r \in [1, \…
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 …
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.
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}…
A thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $θ_{h}$ of one base is considered; the diameter of $θ_{h}$ is of the same order as the plate relative thickness $h\ll1$. In addition to the standard Kirchhoff model with the Sobolev point condition, a three-dimensional boundary layer is investigated in the vicinity of the support $θ_{h}$, which with the help of the derived weighted inequality of Korn's type, will provide an error estimate with the bound $ch^{1/2}|\ln h|$. Ignoring this boundary layer effect reduces the precision order down to $|\…
We prove a family of improved multipolar Poincaré-Hardy inequalities on Cartan-Hadamard manifolds. For suitable configurations of poles, these inequalities yield an improved multipolar Hardy inequality and an improved multipolar Poincaré inequality such that the critical unipolar singular mass is reached at any pole.
We prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.
We introduce a simple criterion to check coercivity of bilinear forms on subspaces of Hilbert-spaces and Banach-spaces. The presented criterion allows to derive many standard and non-standard variants of Poincaré- and Friedrichs-type inequalities with very little effort.
We extend the results of [5], where we proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.
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…
We derive weighted Sobolev-Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic $A_p$-class, where only the parabolic cubes are involved in the definition.
In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $% A_{p(\cdot ),p^{\ast }(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses, and in this case we obtain the first order weighted Poincaré inequalities with variable exponents. In the case of Carnot groups we also set up…
Let $Ø$ be a bounded domain in $\R^N$ with $0\in\deØ$ and $N\ge 2$. In this paper we study the Hardy-Poincaré inequality for maps in $H^1_0(Ω)$. In particular we give sufficient and some necessary conditions so that the best constant is achieved.
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 $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$, …
Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to id…
We present a piecewise Korn inequality for generalized special functions of bounded deformation ($GSBD^2$) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that $GSBD^2$ functions have bounded variation after subtraction of a piecewise i…
We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.
We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space.
Let $Ω$ be an arbitrary bounded domain of $\R^n$. We study the right invertibility of the divergence on $Ω$ in weighted Lebesgue and Sobolev spaces on $Ω$, and rely this invertibility to a geometric characterization of $Ω$ and to weighted Poincaré inequalities on $Ω$. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when $Ω$ is Lipschitz or, more generally, when $Ω$ is a John domain, and focus on the case of $s$-John domains.
