scholarly journals Estimates for the green function and existence of positive solutions for higher-order elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-22
Author(s):  
Imed Bachar
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Measurable Function ◽  
Elliptic Equations ◽  
Nonlinear Problems ◽  
Borel Measurable Function ◽  
Continuous Solutions ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Borel Measurable

We establish a3G-theorem for the iterated Green function of(−∆)pm, (p≥1,m≥1), on the unit ballBofℝn(n≥1)with boundary conditions(∂/∂ν)j(−∆)kmu=0on∂B, for0≤k≤p−1and0≤j≤m−1. We exploit this result to study a class of potentials𝒥m,n(p). Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems(−∆)pmu=h(‧,u), inD(in the sense of distributions),lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for0≤k≤p−1, whereD=BorB\{0}andhis a Borel measurable function onD×(0,∞)satisfying some appropriate conditions related to𝒥m,n(p).

Estimates on the Green Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations

2003 ◽  
Vol 05 (03) ◽  
pp. 401-434 ◽  
Author(s):  
Imed Bachar ◽  
Habib Maâgli ◽  
Noureddine Zeddini
Keyword(s):  
Elliptic Equations ◽  
Continuous Solution ◽  
Regular Domain ◽  
Kato Class ◽  
New Class ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Borel Measurable ◽  
Compact Boundary

We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in ℝn(n ≥ 3), with compact boundary. We exploit this result to introduce a new class of potentials K(D) that properly contains the classical Kato class [Formula: see text]. Next, we study the existence and the uniqueness of a positive continuous solution u in [Formula: see text] of the following nonlinear singular elliptic problem [Formula: see text] where φ is a nonnegative Borel measurable function in D × (0, ∞), that belongs to a convex cone which contains, in particular, all functions φ(x, t) = q(x)t-σ, σ ≥ 0 with q ∈ K(D). We give also some estimates on the solution u.

The identifiability of mixtures of distributions

1969 ◽  
Vol 6 (02) ◽  
pp. 389-398 ◽  
Author(s):  
G. M. Tallis
Keyword(s):  
Distribution Function ◽  
Measurable Function ◽  
Borel Measurable Function ◽  
Mixtures Of Distributions ◽  
Image Position ◽  
Borel Measurable

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

scholarly journals A biharmonic equation with singular nonlinearity

2011 ◽  
Vol 55 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
The Green Function ◽  
Uniqueness Of A Solution

AbstractWe study the biharmonic equation Δ2u=u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn,n≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

2008 ◽  
Vol 06 (02) ◽  
pp. 121-150 ◽  
Author(s):  
IMED BACHAR ◽  
HABIB MÂAGLI ◽  
NOUREDDINE ZEDDINI
Keyword(s):  
Green Function ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Closed Ball ◽  
Dirichlet Boundary Conditions ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Class Of Functions

Let [Formula: see text] be the Green function of (-Δ)m, m ≥ 1, on the complementary D of the unit closed ball in ℝn, n ≥ 2, with Dirichlet boundary conditions [Formula: see text], 0 ≤ j ≤ m - 1. We establish some estimates on [Formula: see text] including the 3G-Inequality given by (1.3). Next, we introduce a polyharmonic Kato class of functions [Formula: see text] and we exploit the properties of this class to study the existence of positive solutions of some polyharmonic nonlinear elliptic problems.

A global random walk on grid algorithm for second order elliptic equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Dmitrii Smirnov
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Green Function ◽  
Elliptic Equations ◽  
Symmetry Property ◽  
Variable Coefficients ◽  
Second Order ◽  
Grid Algorithm ◽  
The Green Function ◽  
Second Order Elliptic Equations

Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

scholarly journals Existence of Positive Solutions for Some Superlinear Fourth-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Imed Bachar ◽  
Habib Mâagli
Keyword(s):  
Green Function ◽  
Fourth Order ◽  
Integrability Condition ◽  
Order Equation ◽  
Global Behavior ◽  
Fourth Order Equation ◽  
Existence Of Positive Solutions ◽  
Nonnegative Continuous Function ◽  
The Green Function ◽  
Perturbation Arguments

We are concerned with the following superlinear fourth-order equationu4t+utφt,−ut=0,   t∈0, 1;−u0=u1=0,  −u′0=a,  −u′1=-b, wherea,−bare nonnegative constants such thata+b>0andφt,−sis a nonnegative continuous function that is required to satisfy some appropriate conditions related to a classKsatisfying suitable integrability condition. Our purpose is to prove the existence, uniqueness, and global behavior of a classical positive solution to the above problem by using a method based on estimates on the Green function and perturbation arguments.

scholarly journals On the Green function of an orthotropic clamped plate in a half-plane

2021 ◽  
Author(s):  
Norbert Ortner ◽  
Peter Wagner
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Linear Combination ◽  
Half Plane ◽  
Orthotropic Plate ◽  
Dirichlet Boundary ◽  
Fundamental Solutions ◽  
Dirichlet Boundary Conditions ◽  
Mirror Point ◽  
The Green Function

AbstractFirst, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form $$(Q(\partial ')-a^2\partial _n^2)(Q(\partial ')-b^2\partial _n^2)$$ ( Q ( ∂ ′ ) - a 2 ∂ n 2 ) ( Q ( ∂ ′ ) - b 2 ∂ n 2 ) (where $$\partial '=(\partial _1,\dots ,\partial _{n-1})$$ ∂ ′ = ( ∂ 1 , ⋯ , ∂ n - 1 ) and $$a>0,b>0,a\ne b)$$ a > 0 , b > 0 , a ≠ b ) with respect to Dirichlet boundary conditions at $$x_n=0.$$ x n = 0 . The Green function $$G_\xi $$ G ξ is represented by a linear combination of fundamental solutions $$E^c$$ E c of $$Q(\partial ')(Q(\partial ')-c^2\partial _n^2),$$ Q ( ∂ ′ ) ( Q ( ∂ ′ ) - c 2 ∂ n 2 ) , $$c\in \{a,b\},$$ c ∈ { a , b } , that are shifted to the source point $$\xi ,$$ ξ , to the mirror point $$-\xi ,$$ - ξ , and to the two additional points $$-\frac{a}{b}\xi $$ - a b ξ and $$-\frac{b}{a}\xi ,$$ - b a ξ , respectively.

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