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 $$\Gamma _1\cap \Gamma _2=\emptyset $$. 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.