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

[摘要] We obtain solutions for Laplace's and Poisson's equations onbounded 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.

[学科分类] 数学（综合）

[关键词] Eigenvalue;Laplace's equation;Poisson's equation;Green's function;Hammerstein integral operator.