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.

中文翻译：

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$$\mathbb {R}^N$$ RN 中两个混合边界值椭圆偏微分方程的存在结果

我们研究了有界 Lipschitz 域 $\Omega\subset\mathbb{R}^N$ 和 $\mathbb{R}^N\setminus\ 中 Helmholtz 和 Poisson 类型方程的混合边值问题的解的存在性$N\geq3$ 的 Omega$。$\Omega$的边界$\partial\Omega$是$\Gamma_1,\Gamma_2\subset\partial\Omega$的分解使得$\partial\Omega=\Gamma=\overline{\Gamma}_1\cup \Gamma_2=\Gamma_1\cup\overline{\Gamma}_2$ 和 $\Gamma_1\cap\Gamma_2=\emptyset$。我们已经证明，如果 Neumann 数据 $f_2\in H^{-\​​frac{1}{2}}(\Gamma_2)$ 和 Dirichlet 数据 $f_1\in H^{\frac{1}{2}} (\Gamma_1)$ 那么具有混合边界数据的亥姆霍兹问题承认唯一的解决方案。我们还展示了由具有测量数据和 Dirichlet 的 Poisson 方程控制的混合边值问题的弱解的存在，