Taiwanese Journal of Mathematics

Weighted $L^p$ Boundary Value Problems for Laplace's Equation on (Semi-)Convex Domains

Sibei Yang

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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.

Article information

Taiwanese J. Math., Volume 23, Number 4 (2019), 821-840.

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

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Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 42B25: Maximal functions, Littlewood-Paley theory

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


Yang, Sibei. Weighted $L^p$ Boundary Value Problems for Laplace's Equation on (Semi-)Convex Domains. Taiwanese J. Math. 23 (2019), no. 4, 821--840. doi:10.11650/tjm/181001. https://projecteuclid.org/euclid.twjm/1563436870

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