scholarly journals On the generalized Hardy-Rellich inequalities

2019 ◽  
Vol 150 (2) ◽  
pp. 897-919 ◽  
Author(s):  
T.V. Anoop ◽  
Ujjal Das ◽  
Abhishek Sarkar
Keyword(s):  
Fundamental Theorem ◽  
Type Inequality ◽  
Weight Functions ◽  
Lebesgue Spaces ◽  
One Dimensional ◽  
Open Set ◽  
Muckenhoupt Condition ◽  
Zygmund Spaces ◽  
The One ◽  
Image Position

AbstractIn this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: $$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$for some constant C > 0, where Ω is an open set in ℝN with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of ${\cal D}_0^{2,2} $ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

scholarly journals A Note on Generalized Hardy-Sobolev Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
T. V. Anoop
Keyword(s):  
Integral Inequality ◽  
Sobolev Inequality ◽  
Banach Function Space ◽  
Weight Functions ◽  
Sobolev Inequalities ◽  
Best Constant ◽  
One Dimensional ◽  
Muckenhoupt Condition ◽  
Previous Inequality ◽  
The One

We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2,   u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

scholarly journals Fourier multipliers on spaces of distributions

1986 ◽  
Vol 29 (3) ◽  
pp. 309-327 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Banach Spaces ◽  
Growth Conditions ◽  
Fourier Multipliers ◽  
Bounded Operators ◽  
Vertical Strip ◽  
One Dimensional ◽  
The One ◽  
Image Position ◽  
Complex Valued

In [8], Rooney defines a class of complex-valued functions ζ each of which is analytic in a vertical strip α(ζ)< Res < β(ζ) in the complex s-plane and satisfies certain growth conditions as |Im s| →∞ along fixed lines Re s = c lying within this strip. These conditions mean that the functionsfulfil the requirements of the one-dimensional Mihlin-Hörmander theorem (see [6, p. 417]) and so can be regarded as Fourier multipliers for the Banach spaces . Consequently, each function gives rise to a family of bounded operators W[ζ,σ] σ ∈(α(ζ),β(ζ)), on , 1<p<∞.

A limit theorem for certain disordered random systems

1994 ◽  
Vol 26 (04) ◽  
pp. 1022-1043 ◽  
Author(s):  
Xinhong Ding
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Limit Theorem ◽  
Random Systems ◽  
Joint Probability ◽  
Dynamical Behaviour ◽  
One Dimensional ◽  
Linear Stochastic Differential Equation ◽  
The One ◽  
Image Position

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in : where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝ p -valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation where P(t, dx dy) is the joint probability law of ξ and X(t).

A note on the one-dimensional maximal function

1989 ◽  
Vol 111 (3-4) ◽  
pp. 325-328 ◽  
Author(s):  
Antonio Bernal
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Type Inequality ◽  
Best Constant ◽  
Weak Type Inequality ◽  
One Dimensional ◽  
Littlewood Maximal Function ◽  
The One

SynopsisIn this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.

scholarly journals SUCCESSIVE ITERATION AND POSITIVE SOLUTIONS FOR A p-LAPLACIAN MULTIPOINT BOUNDARY VALUE PROBLEM

2008 ◽  
Vol 49 (4) ◽  
pp. 551-560 ◽  
Author(s):  
BO SUN ◽  
XIANGKUI ZHAO ◽  
WEIGAO GE
Keyword(s):  
Positive Solutions ◽  
Monotone Iterative Method ◽  
One Dimensional ◽  
Multipoint Boundary ◽  
Monotone Iterative ◽  
Existence Of Positive Solutions ◽  
The One ◽  
Image Position ◽  
Multipoint Boundary Condition ◽  
Multipoint Boundary Value Problem

AbstractIn this paper, we study the existence of positive solutions for the one-dimensional p-Laplacian differential equation, subject to the multipoint boundary condition by applying a monotone iterative method.

scholarly journals On the number of eigenvalues in the spectral gap of a Dirac system

1986 ◽  
Vol 29 (3) ◽  
pp. 367-378 ◽  
Author(s):  
D. B. Hinton ◽  
A. B. Mingarelli ◽  
T. T. Read ◽  
J. K. Shaw
Keyword(s):  
Differential Operators ◽  
Spectral Gap ◽  
Equivalence Classes ◽  
Value Functions ◽  
Complex Vector ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Integrable Functions ◽  
The One ◽  
Image Position

We consider the one-dimensional operator,on 0<x<∞ with. The coefficientsp,V1andV2are assumed to be real, locally Lebesgue integrable functions;c1andc2are positive numbers. The operatorLacts in the Hilbert spaceHof all equivalence classes of complex vector-value functionssuch that.Lhas domainD(L)consisting of ally∈Hsuch thatyis locally absolutely continuous andLy∈H; thus in the language of differential operatorsLis a maximal operator. Associated withLis the minimal operatorL0defined as the closure ofwhereis the restriction ofLto the functions with compact support in (0,∞).

scholarly journals Γ-limit for the extended Fisher–Kolmogorov equation

2002 ◽  
Vol 132 (1) ◽  
pp. 141-162 ◽  
Author(s):  
D. Hilhorst ◽  
L. A. Peletier ◽  
R. Schätzle
Keyword(s):  
Fourth Order ◽  
Lyapunov Functional ◽  
Transition Energy ◽  
Kolmogorov Equation ◽  
One Dimensional ◽  
Cahn Equation ◽  
Area Functional ◽  
The One ◽  
Image Position

We consider the Lyapunov functional, of the rescaled Extended Fisher-Kolmogorov equation This is a fourth order generalization of the Fisher–Kolmogorov or Allen–Cahn equation. We prove that if ε → 0, then tends to the area functional in the sense of Γ-limits, where the transition energy is given by the one-dimensional kink of the Extended Fisher–Kolmogorov equation.

scholarly journals Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

2021 ◽  
Author(s):  
Tadahiro Oh ◽  
Yuzhao Wang
Keyword(s):  
Normal Form ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Lebesgue Spaces ◽  
One Dimensional ◽  
Form Equation ◽  
The One ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Form Approach

AbstractIn this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ) for $$1 \leq p \leq {3 \over 2}$$ 1 ≤ p ≤ 3 2 .

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (1) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

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