Electron. J. Differential Equations, Vol. 2021 (2021), No. 87, pp. 1-15.

Solutions and eigenvalues of Laplace's equation on bounded open sets

Guangchong Yang, Kunquan Lan

Abstract:
We obtain solutions for Laplace's and Poisson's equations on bounded open subsets of Rn, (n≥2), via Hammerstein integral operators involving kernels and Green's functions, respectively. The new solutions are different from the previous ones obtained by the well-known Newtonian potential kernel and the Newtonian potential operator. Our results on eigenvalue problems of Laplace's equation are different from the previous results that use the Newtonian potential operator and require n≥3. As a special case of the eigenvalue problems, we provide a result under an easily verifiable condition on the weight function when n≥3. This result cannot be obtained by using the Newtonian potential operator.

Submitted July 12, 2021. Published October 18, 2021.
Math Subject Classifications: 35J05, 31A05, 31B05, 35J08, 47A75.
Key Words: Eigenvalue; Laplace's equation; Poisson's equation; Green's function; Hammerstein integral operator.
DOI: https://doi.org/10.58997/ejde.2021.87

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Guangchong Yang
College of Applied Mathematics
Chengdu University of Information Technology
Chengdu, Sichuan 610225, China
email: gcyang@cuit.edu.cn
Kunquan Lan
Department of Mathematics
Ryerson University
Toronto, Ontario, Canada M5B 2K3
email: klan@ryerson.ca

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