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For simplicity suppose that $\Omega = (a,b)\times(c,d)$. Than solve laplace equation i.e. $$\Delta u = 0$$ in $\Omega$ with boundary conditions(they are give as if $y$ is time coordinate): $$ u = f \qquad f\in C(\partial \Omega \setminus (a,b)\times \{d\}) $$ $$ \frac{ \partial u}{\partial n} = g \qquad g\in C( [a,b]\times \{c\}$$

So I would like to know if such solution exists for any $f,g$.

I have few variations:

1) when $a=\infty, b=\infty$. And functions $f,g$ are periodic with same period.

2) Exteng this to manifolds. let $M$ be compact manifold in $\mathbb{R}^n$ than suppouse $\Omega = M \times [0,1]$. Dirichlet and neuman type conditions will be given at $ M \times \{0\}$ and none will be given at $ M \times \{1\}$.

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