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 $Ω\subset\mathbb{R}^N$ and in $\mathbb{R}^N\setminusΩ$ for $N\geq3$. The boundary $\partialΩ$ of $Ω$ is the decomposition of $Γ_1,Γ_2\subset\partialΩ$ such that $\partialΩ=Γ=\overlineΓ_1\cupΓ_2=Γ_1\cup\overlineΓ_2$ and $Γ_1\capΓ_2=\emptyset$. We have shown that if the Neumann data $f_2\in H^{-\frac{1}{2}}(Γ_2)$ and the Dirichlet data $f_1\in H^{\frac{1}{2}}(Γ_1)$ 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 $f_1\in H^{\frac{1}{2}}(Γ_1)$, $f_2\in H^{-\frac{1}{2}}(Γ_2)$ respectively.