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The Laplace's equation: $$ \nabla^2\phi=0 $$ subjected to only Neumann's boundary conditions has an infinite number of solution. For a unique solution to exist, we need to have Dirichlet's condition at least at a point on the boundary.

Question: If the Laplace's equation is subjected to purely mixed-boundary condition: $$ \alpha(x,y,z)\phi+\beta(x,y,z)\frac{\partial \phi}{\partial n} =f(x,y,z); \alpha \neq0, \beta\neq0$$ , does a unique solution exist?

A Slow Learner
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  • Relevant: https://math.stackexchange.com/q/1607571/221811 and https://math.stackexchange.com/q/2122410/221811 – Chappers Jul 12 '18 at 01:51
  • @Chappers If the solution is unique up to a constant, then does it mean we still need to have a Dirichlet condition at some point on the boundary to have a unique solution? – A Slow Learner Jul 12 '18 at 02:00
  • The linked question was for a positive constant. Hence, the solution is unique up to a constant if $\frac{\beta}{\alpha} > 0$ everywhere on the boundary. – Dylan Jul 12 '18 at 06:47

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