scholarly journals Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Qiao
Keyword(s):  
Poisson Integrals ◽  
Nontangential Limits ◽  
Boundary Functions ◽  
Tangential Limits

Our aim in this paper is to deal with non-tangential limits for modified Poisson integrals of boundary functions in a cone, which generalized results obtained by Brundin and Mizuta-Shimomura.

scholarly journals Approach regions for the square root of the Poisson kernel and bounded functions

1997 ◽  
Vol 55 (3) ◽  
pp. 521-527 ◽  
Author(s):  
P. Sjögren
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Monotone Function ◽  
Square Root ◽  
Almost Everywhere Convergence ◽  
Poisson Integrals ◽  
Almost Everywhere ◽  
Approach Region ◽  
Boundary Functions ◽  
Bounded Functions

If the Poisson kernel of the unit disc is replaced by its square root, it is known that normalised Poisson integrals of Lp boundary functions converge almost everywhere at the boundary, along approach regions wider than the ordinary non-tangential cones. The sharp approach region, defined by means of a monotone function, increases with p. We make this picture complete by determining along which approach regions one has almost everywhere convergence for L∞ boundary functions.

scholarly journals Approach regions for $l^{p}$ potentials with respect to the square root of the Poisson kernel

2005 ◽  
Vol 96 (2) ◽  
pp. 243
Author(s):  
Martin Brundin
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Higher Dimensions ◽  
Square Root ◽  
Poisson Integrals ◽  
Boundary Functions

{If} one replaces the Poisson kernel of the unit disc by its square root, then normalised Poisson integrals of $L^{p}$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning ($1\leq p<\infty$) and Sjögren ($p=1$ and $p=\infty$). In this paper we present new and simplified proofs of these results. We also generalise the $L^{\infty}$ result to higher dimensions.

scholarly journals Existence results for first derivative dependent ϕ-Laplacian boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imran Talib ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Existence Results ◽  
Main Concern ◽  
First Derivative ◽  
Boundary Functions ◽  
Carathéodory Function

Abstract Our main concern in this article is to investigate the existence of solution for the boundary-value problem $$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$ ( ϕ ( x ′ ( t ) ) ′ = g 1 ( t , x ( t ) , x ′ ( t ) ) , ∀ t ∈ [ 0 , 1 ] , ϒ 1 ( x ( 0 ) , x ( 1 ) , x ′ ( 0 ) ) = 0 , ϒ 2 ( x ( 0 ) , x ( 1 ) , x ′ ( 1 ) ) = 0 , where $g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ g 1 : [ 0 , 1 ] × R 2 → R is an $L^{1}$ L 1 -Carathéodory function, $\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $ ϒ i : R 3 → R are continuous functions, $i=1,2$ i = 1 , 2 , and $\phi :(-a,a)\rightarrow \mathbb{R}$ ϕ : ( − a , a ) → R is an increasing homeomorphism such that $\phi (0)=0$ ϕ ( 0 ) = 0 , for $0< a< \infty $ 0 < a < ∞ . We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.

Über das Randverhalten des POISSON-Integrals der polyharmonischen Gleichung

1980 ◽  
Vol 95 (1) ◽  
pp. 157-164 ◽  
Author(s):  
Luis Gonzáles ◽  
Eckart Keller ◽  
Günther Wildenhain
Keyword(s):  
Poisson Integrals

Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
D. Zhou ◽  
S. H. Lo
Keyword(s):  
Cross Section ◽  
Chebyshev Polynomial ◽  
Numerical Solutions ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Cylindrical Domain ◽  
Linear Elasticity Theory ◽  
Boundary Functions ◽  
Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

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