Search results for "Pointwise"
showing 10 items of 47 documents
Accessible parts of boundary for simply connected domains
2018
For a bounded simply connected domain $\Omega\subset\mathbb{R}^2$, any point $z\in\Omega$ and any $0<\alpha<1$, we give a lower bound for the $\alpha$-dimensional Hausdorff content of the set of points in the boundary of $\Omega$ which can be joined to $z$ by a John curve with a suitable John constant depending only on $\alpha$, in terms of the distance of $z$ to $\partial\Omega$. In fact this set in the boundary contains the intersection $\partial\Omega_z\cap\partial\Omega$ of the boundary of a John sub-domain $\Omega_z$ of $\Omega$, centered at $z$, with the boundary of $\Omega$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obta…
Pointwise inequalities for Sobolev functions on generalized cuspidal domains
2022
We establish point wise inequalities for Sobolev functions on a wider class of outward cuspidal domains. It is a generalization of an earlier result by the author and his collaborators
Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings
2011
Abstract In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces B ˙ p , q s and Triebel–Lizorkin spaces F ˙ p , q s for all s ∈ ( 0 , 1 ) and p , q ∈ ( n / ( n + s ) , ∞ ] , both in R n and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve F ˙ n / s , q s on R n for all s ∈ ( 0 , 1 ) and q ∈ ( n / ( n + s ) , ∞ ] . A metric measure space version of the above morphism property is also established.
VARIANTS OF A SELECTION PRINCIPLE FOR SEQUENCES OF REGULATED AND NON-REGULATED FUNCTIONS
2008
Let $T$ be a nonempty subset of $\RB$, $X$ a metric space with metric $d$ and $X^T$ the set of all functions mapping $T$ into $X$. Given $\vep>0$ and $f\in X^T$, we denote by $N(\vep,f,T)$ the least upper bound of those $n\in\NB$, for which there exist numbers $s_1,\dots,s_n,t_1,\dots,t_n$ from $T$ such that $s_1\vep$ for all $i=1,\dots,n$ ($N(\vep,f,T)=0$ if there are no such $n$'s). The following pointwise selection principle is proved: {\em If a sequence of functions\/ $\{f_j\}_{j=1}^\infty\subset X^T$ is such that the closure in $X$ of the sequence\/ $\{f_j(t)\}_{j=1}^\infty$ is compact for each $t\in T$ and\/ $\limsup_{j\to\infty}N(\vep,f_j,T)0$, then\/ $\{f_j\}_{j=1}^\infty$ contains …
A reliable incremental method of computing the limit load in deformation plasticity based on compliance : Continuous and discrete setting
2016
The aim of this paper is to introduce an enhanced incremental procedure that can be used for the numerical evaluation and reliable estimation of the limit load. A conventional incremental method of limit analysis is based on parametrization of the respective variational formulation by the loading parameter ? ? ( 0 , ? l i m ) , where ? l i m is generally unknown. The enhanced incremental procedure is operated in terms of an inverse mapping ? : α ? ? where the parameter α belongs to ( 0 , + ∞ ) and its physical meaning is work of applied forces at the equilibrium state. The function ? is continuous, nondecreasing and its values tend to ? l i m as α ? + ∞ . Reduction of the problem to a finit…
Presentations for the punctured mapping class groups in terms of Artin groups
1999
Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F, and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h: F-->F which pointwise fix the boundary of F and such that h(P) = P. In this paper, we calculate presentations for all punctured mapping class groups. More precisely, we show that these groups are isomorphic with quotients of Artin groups by some relations involving fundamental elements of parabolic subgroups.
Regularity properties for quasiminimizers of a $(p,q)$-Dirichlet integral
2021
Using a variational approach we study interior regularity for quasiminimizers of a $(p,q)$-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincar\'{e} inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally H\"{o}lder continuous and they satisfy Harnack inequality, the strong maximum principle, and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for H\"older continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we cons…
Fourier-transform spectroscopy and potential construction of the (2)1Π state in KCs
2015
The paper presents an empirical pointwise potential energy curve (PEC) of the (2)(1)Π state of the KCs molecule constructed by applying the Inverted Perturbation Approach routine. The experimental term values in the energy range E(v', J') ∈ [15 407; 16 579] cm(-1) involved in the fit were based on Fourier-Transform spectroscopy data obtained with 0.01 cm(-1) accuracy from the laser-induced (2)(1)Π → X(1)Σ(+) fluorescence spectra. Buffer gas Ar was used to facilitate the appearance of rotation relaxation lines in the spectra, thus enlarging the (2)(1)Π data set and allowing determination of the Λ-splitting constants. The data set included vibrational v' ∈ [0, 28] and rotational J' ∈ [7, 274]…
Fourier-transform spectroscopy and description of low-lying energy levels in the B(1)(1)Π state of RbCs.
2013
The Fourier transform spectrometer with resolution of 0.03 cm(-1) was applied to disperse the diode laser induced B(1)(1)Π → X(1)Σ(+) fluorescence spectra of the RbCs molecule in a heat pipe. The presence of buffer gas (Ar) produced in the spectra the dense pattern of collision-induced rotation relaxation lines, thus enlarging the B(1)(1)Π data set, allowing to determine the Λ-splitting constants and to reveal numerous local perturbations. In total, 2664 term values for (85)Rb(133)Cs and (87)Rb(133)Cs in the energy range from 13,770 to 15,200 cm(-1) were obtained with accuracy about 0.01 cm(-1). A pointwise potential energy curve (PEC) based on inverted perturbation approach was constructed…
Weighted pointwise Hardy inequalities
2009
We introduce the concept of a visual boundary of a domain �¶ �¼ Rn and show that the weighted Hardy inequality �¶ |u|pd�¶ �A.p C �¶ |�Þu|pd�¶ �A, where d�¶(x) = dist(x, �Ý�¶), holds for all u �¸ C �� 0 (�¶) with exponents �A < �A0 when the visual boundary of �¶ is sufficiently large. Here �A0 = �A0(p, n, �¶) is explicit, essentially sharp, and may even be greater than p . 1, which is the known bound for smooth domains. For instance, in the case of the usual von Koch snowflake domain the sharp bound is shown to be �A0 = p . 2 + �E, with �E = log 4/ log 3. These results are based on new pointwise Hardy inequalities.