scholarly journals Local Integral Estimates for Quasilinear Equations with Measure Data

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Qiaoyu Tian ◽  
Shengzhi Zhang ◽  
Yonglin Xu ◽  
Jia Mu
Keyword(s):  
Radon Measure ◽  
Integrable Function ◽  
Finite Measure ◽  
Measure Data ◽  
Quasilinear Equations ◽  
Locally Finite ◽  
Positive Radon Measure ◽  
Local Integral ◽  
Hessian Equations ◽  
Locally Integrable Function

Local integral estimates as well as local nonexistence results for a class of quasilinear equations-Δpu=σP(u)+ωforp>1and Hessian equationsFk-u=σP(u)+ωwere established, whereσis a nonnegative locally integrable function or, more generally, a locally finite measure,ωis a positive Radon measure, andP(u)~exp⁡αuβwithα>0andβ≥1orP(u)=up-1.

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

scholarly journals Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

2021 ◽  
Vol 21 (2) ◽  
pp. 261-280
Author(s):  
Marie-Françoise Bidaut-Véron ◽  
Marta Garcia-Huidobro ◽  
Laurent Véron
Keyword(s):  
Dirichlet Problem ◽  
Compact Subset ◽  
Elliptic Equations ◽  
Radon Measure ◽  
Measure Data ◽  
Nonnegative Solutions

Abstract In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒ p , q M ⁢ u := - Δ ⁢ u + u p - M ⁢ | ∇ ⁡ u | q = μ {{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p > 1 {p>1} , q > 1 {q>1} and M ≥ 0 {M\geq 0} . We also give conditions under which nonnegative solutions of ℒ p , q M ⁢ u = 0 {{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω ∖ K {\Omega\setminus K} , where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

WEAK SOLUTIONS FOR WEAK SINGULARITIES

2002 ◽  
Vol 17 (20) ◽  
pp. 2769-2769
Author(s):  
B. C. NOLAN
Keyword(s):  
Existence And Uniqueness ◽  
Finite Measure ◽  
Generalized Solutions ◽  
Functions Of Bounded Variation ◽  
Field Equations ◽  
Finite Radius ◽  
Locally Finite ◽  
Naked Singularities ◽  
First Order ◽  
Bounded Functions

We revisit the problem of the development of singularities in the gravitational collapse of an inhomogeneous dust sphere. As shown by Yodzis et al1, naked singularities may occur at finite radius where shells of dust cross one another. These singularities are gravitationally weak 2, and it has been claimed that at these singularities, the metric may be written in continuous form 2, with locally L∞ connection coefficients 3. We correct these claims, and show how the field equations may be reformulated as a first order, quasi-linear, non-conservative, non-strictly hyperbolic system. We discuss existence and uniqueness of generalized solutions of this system using bounded functions of bounded variation (BV) 4, where the product of a BV function and the derivative of another BV function may be interpreted as a locally finite measure. The solutions obtained provide a dynamical extension to the future of the singularity.

Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

1999 ◽  
Vol 6 (5) ◽  
pp. 401-414
Author(s):  
T. Chantladze ◽  
N. Kandelaki ◽  
A. Lomtatidze
Keyword(s):  
Linear Equation ◽  
Integrable Function ◽  
Second Order ◽  
Order Linear ◽  
Locally Integrable Function ◽  
Oscillation And Nonoscillation Criteria ◽  
Oscillation And Nonoscillation

Abstract New oscillation and nonoscillation criteria are established for the equation 𝑢″ + 𝑝(𝑡)𝑢 = 0, where 𝑝 : ]1, + ∞[ → 𝑅 is the locally integrable function. These criteria generalize and complement the well known criteria of E. Hille, Z. Nehari, A. Wintner, and P. Hartman.

Scattering Length and the Spectrum of –Δ + V

2006 ◽  
Vol 49 (1) ◽  
pp. 144-151 ◽  
Author(s):  
Michael Taylor
Keyword(s):  
Discrete Spectrum ◽  
Integrable Function ◽  
Scattering Length ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

AbstractGiven a non-negative, locally integrable functionV on ℝn, we give a necessary and sufficient condition that –Δ + V have purely discrete spectrum, in terms of the scattering length ofV restricted to boxes.

scholarly journals BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS

2012 ◽  
Vol 54 (3) ◽  
pp. 655-663
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
Borel Measure ◽  
Weak Type ◽  
Integrable Function ◽  
Maximal Operators ◽  
Weak Type Estimates ◽  
Locally Integrable Function ◽  
Image Position ◽  
Strong Maximal Operator ◽  
General Product

AbstractLet μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and cp,q, and Cp,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for cp,q and Cp,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝn with respect to a general product measure.

scholarly journals A new proof of Donoghue's interpolation theorem

2004 ◽  
Vol 2 (3) ◽  
pp. 253-265 ◽  
Author(s):  
Yacin Ameur
Keyword(s):  
Hilbert Space ◽  
Radon Measure ◽  
Interpolation Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Positive Radon Measure ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
New Interpretation ◽  
Necessary And Sufficient

We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert spaceH∗to be exact interpolation with respect to a regular Hilbert coupleH¯it is necessary and sufficient that the norm inH∗be representable in the form‖f‖∗=(∫[0,∞](1+t−1)K2(t,f;H¯)2dρ(t))1/2with some positive Radon measureρon the compactified half-line[0,∞]. The result was re-proved in [1] in the finite-dimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ‘Donoghue's Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw.

scholarly journals An extension of a Hardy-Littlewood-Pólya inequality

1978 ◽  
Vol 21 (1) ◽  
pp. 11-15 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Integrable Function ◽  
Right Hand ◽  
Usual Norm ◽  
Locally Integrable Function ◽  
Image Position ◽  
The Right

The Hardy-Littlewood-Pólya inequality in question can be written in the formHere and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean thatdefines a locally integrable function Kf for which (1.1) holds.

scholarly journals Radon Measure-valued Solutions for Nonlinear Strongly Degenerate Parabolic Equations with Measure Data

2021 ◽  
Vol 14 (1) ◽  
pp. 204-233
Author(s):  
Quincy Stevene Nkombo ◽  
Fengquan Li
Keyword(s):  
Parabolic Equations ◽  
Radon Measure ◽  
Degenerate Parabolic Equations ◽  
Measure Data ◽  
Initial Value ◽  
Parabolic Capacity ◽  
Nonlinear Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Strongly Degenerate ◽  
Nonlinear Degenerate

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time.

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