Solvability of nonlocal boundary-value problems for the Laplace equation in the ball
type: Item
instance of: scholarly article
zbMATH Open document ID: 1298.35042
author: Makhmud A. Sadybekov
author: Berikbol T. Torebek
author: Batirkhan Kh. Turmetov
publication date: 2014-08-25T00:00:00Z
full work available at URL: abstr.html
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 6333998
zbMATH Keywords: Riemann-Liouville operator
zbMATH Keywords: Caputo operator
zbMATH Keywords: periodic problem
zbMATH Keywords: antiperiodic problem
zbMATH Keywords: nonlocal problem
zbMATH Keywords: Laplace equation
zbMATH Keywords: Poisson kernel
zbMATH Keywords: harmonic function
MaRDI profile type: Publication
title: Solvability of nonlocal boundary-value problems for the Laplace equation in the ball
published in: Electronic Journal of Differential Equations (EJDE)
review text: Let \(0<\alpha \leq 1\). For a harmonic function \(u\) on the unit ball \(B(0;1)\) define \(\displaystyle I^\alpha [u](x)=\Gamma (\alpha )^{-1} \int_0^{|x|} (|x|-\tau)^{\alpha -1}u(\tau ,\varphi )\;d\tau \), \(\displaystyle{}_{RL}D^\alpha [u](x)= d I^{1-\alpha }[u](x)/d|x|\), \(\displaystyle{}_C D^\alpha [u](x)= I^{1-\alpha }[u'](x) \). The uniqueness, solvability and regularity of the following two problems are studied:\newline Find a function \(u\in C(\overline{B(0;1)})\) harmonic in \(B(0;1)\) such that \(\displaystyle |x|D^\alpha [u](x) \in C(\overline{B(0;1) })\), \(u(x)-(-1)^k u(x^*)=f(x)\), \(D^\alpha [u](x) +(-1)^k D^\alpha [u](x^*)=g(x)\) on \(\partial B_+(0;1)\), where \( D^\alpha ={}_{RL}D^\alpha \) or \(D^\alpha ={}_C D^\alpha \). Here \(\partial B_+ (0;1)=\{ x\in \partial B(0;1); x_1 \geq 0\} \), \( x^* =(a_1x_1,a_2 x_2, \dots ,a_n x_n)\), where \(a_1 =-1\), and \(a_j\), \(j=2,\dots ,n\), take one of values \(\pm 1\).
reviewed by: Dagmar Medková
description: scientific article; zbMATH DE number 6333998