Solutions and eigenvalues of Laplace's equation on bounded open sets
Texas State University, Department of Mathematics
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.
Eigenvalue, Laplace's equation, Poisson's equation, Green's function, Hammerstein integral operator
Yang, G., & Lan, K. (2021). Solutions and eigenvalues of Laplace's equation on bounded open sets. <i>Electronic Journal of Differential Equations, 2021</i>(87), pp. 1-15.