Fabio Silva Botelho
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.
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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 show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both impro…
It is quite well known that Korn's inequality is true on all John domains. We are interested in the converse implication under assumption of so called separation condition of the domain. Our result implies that in a simply connected planar domain the Korn's inequality holds if and only if the domain is John. In particular, we obtain the equivalence of Korn's inequality, Babu\v ska-Aziz inequality and Friedrich's inequality in simply connected planar domains.
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…
In this paper, we introduce and prove the generalizations of Radon inequality. The proofs in the paper unify and are simpler than those in former work. Meanwhile, we also find mathematical equivalences among the Bernoulli inequality, the weighted AM-GM inequality, the Hölder inequality, the weighted power mean inequality and the Minkowski inequality. Finally, a series of the applications are shown in this note.
J. L. Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2\le p<\infty$ and later N. N. Osipov proved the similar inequality for Walsh functions. In this paper we investigate some properties of Banach spaces $X$ such that the latter inequality holds for $X$-valued functions.
The purpose of this note is to provide a summary of the recent work of the authors on two variations of the pointwise convergence problem for the solutions to the fractional Schrödinger equations; convergence along a tangential line and along a set of lines, as exhibiting some new results in each setting. For the former case, we make a simple observation on a path along a tangential curve of exponential order. We discuss counterexamples for the latter case that show some of the known smooth regularities are essentially optimal.
We study a weighted version of Carleman's inequality via Carleman's original approach. As an application of our result, we prove a conjecture of Bennett.
Recently, Nagy-Toókos and Totik-Varga proved an asymptotically sharp $L^{p}$ Bernstein type inequality on union of finitely many intervals. We extend this inequality to the case when the power $p$ is between $0$ and $1$; such sharp Bernstein type inequality was proved first by Arestov.
We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.
We introduce the multivariate analogue of the well known inequality $1+x\leq \mathrm{e}^x$, for an abstract non negative real number $x$. The result emerges from the study of the blow up time of certain solutions of the Cauchy problem for a particular ODE. It is also closely related to the notion of completely monotone functions and the theory of divided differences.
It was shown by E. Gluskin and V.D. Milman in [GAFA Lecture Notes in Math. 1807, 2003] that the classical arithmetic-geometric mean inequality can be reversed (up to a multiplicative constant) with high probability, when applied to coordinates of a point chosen with respect to the surface unit measure on a high-dimensional Euclidean sphere. We present here two asymptotic refinements of this phenomenon in the more general setting of the surface probability measure on a high-dimensional $\ell_p$-sphere, and also show that sampling the point according to either the cone probability measure on $\e…
The aim of the present paper is to obtain some new fractional integral inequalities for convex functions. Saigo fractional integral operator is used to establish the results.
In this paper we study approximation theorems for $L^2$-space on Damek-Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek-Ricci spaces. We also prove Nikolskii-Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application, we prove equivalence of the $K$-functional and modulus of smoothness for Damek-Ricci spaces.
We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed mean inequality involving the symmetric means.
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