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

## TL;DR

This paper proves the existence and uniqueness of solutions for mixed boundary value problems involving Helmholtz and Poisson equations in bounded Lipschitz domains, extending the understanding of such PDEs with complex boundary conditions.

## Contribution

It establishes new existence and uniqueness results for mixed boundary value problems for Helmholtz and Poisson equations in Lipschitz domains with specific boundary data.

## Key findings

- Unique solution for Helmholtz problem with mixed boundary data.
- Existence of weak solutions for Poisson problem with measure data.
- Solutions depend on boundary data in appropriate Sobolev spaces.

## 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 $N\geq3$. 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 $\Gamma_1\cap\Gamma_2=\emptyset$. 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.