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August, 2019 Weighted $L^p$ Boundary Value Problems for Laplace's Equation on (Semi-)Convex Domains
Sibei Yang
Taiwanese J. Math. 23(4): 821-840 (August, 2019). DOI: 10.11650/tjm/181001

Abstract

Let $n \geq 2$ and $\Omega$ be a bounded (semi-)convex domain in $\mathbb{R}^n$. Assume that $p \in (1,\infty)$ and $\omega \in A_p(\partial \Omega)$, where $A_p(\partial \Omega)$ denotes the Muckenhoupt weight class on $\partial \Omega$, the boundary of $\Omega$. In this article, the author proves that the Dirichlet and Neumann problems for Laplace's equation on $\Omega$ with boundary data in the weighted space $L^p_{\omega}(\partial \Omega)$ are uniquely solvable. Moreover, the unique solvability of the Regularity problem for Laplace's equation on $\Omega$ with boundary data in the weighted Sobolev space $\dot{W}^p_{1,\omega}(\partial \Omega)$ is also obtained. Furthermore, the weighted $L^p_{\omega}(\partial \Omega)$-estimates for the Dirichlet, Regularity and Neumann problems are established.

Citation

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Sibei Yang. "Weighted $L^p$ Boundary Value Problems for Laplace's Equation on (Semi-)Convex Domains." Taiwanese J. Math. 23 (4) 821 - 840, August, 2019. https://doi.org/10.11650/tjm/181001

Information

Received: 5 June 2018; Revised: 13 September 2018; Accepted: 30 September 2018; Published: August, 2019
First available in Project Euclid: 18 July 2019

zbMATH: 07088949
MathSciNet: MR3982063
Digital Object Identifier: 10.11650/tjm/181001

Subjects:
Primary: 35J25
Secondary: 35J05 , 42B25

Keywords: (semi-)convex domain , Dirichlet problem , Laplace's equation , Neumann problem , weighted estimate

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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Vol.23 • No. 4 • August, 2019
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