Existence results of two mixed boundary value elliptic PDEs in ${\mathbb{R}}^n$

Abstract

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^N$ and in ${\mathbb{R}}^N\setminus \Omega$ for $N\ge 3$. The boundary $\partial \Omega$ of $\Omega$ is the decomposition of ${\Gamma }_1,{\Gamma }_2\subset \partial \Omega$ such that $\partial \Omega =\Gamma ={\overline{\Gamma }}_1\cup {\Gamma }_2={\Gamma }_1\cup {\overline{\Gamma }}_2$ and ${\Gamma }_1\cap {\Gamma }_2=\varnothing$. We have shown that if the Neumann data $f_2\in H^{-\frac{1}{2}}({\Gamma }_2)$ and the Dirichlet data $f_1\in H^{\frac{1}{2}}({\Gamma }_1)$ then the Helmholtz problem with mixed boundary data admits a unique solution. We have also shown the existence of a weak solution to a mixed boundary value problem governed by the Poisson equation with a measure data and the Dirichlet, Neumann data belongs to $f_1\in H^{\frac{1}{2}}({\Gamma }_1)$, $f_2\in H^{-\frac{1}{2}}({\Gamma }_2)$ respectively.