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Hello all,

Does the following boundary value problem admit unique solutions $q$:

$- \beta \Delta q + \beta q = f$, $x \in \Omega$

$\nabla q \cdot \vec{n} = g$, $x \in \Gamma := \partial \Omega$,

where $\beta > 0$ is reasonably small? I am not clear if the pure Neumann boundary conditions make the solution non-unique; does the inhomogeneity in the volume equation take care of this problem? What are the spaces for $f$ and $g$ such that we have uniqueness?

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# Existence of solution for Poisson problem with pure Neumann BCs

Hello all,

Does the following boundary value problem admit unique solutions $q$:

$- \beta \Delta q + \beta q = f$, $x \in \Omega$

$\nabla q \cdot \vec{n} = g$, $x \in \Gamma := \partial \Omega$,

where $\beta > 0$ is reasonably small? I am not clear if the pure Neumann boundary conditions make the solution non-unique; does the inhomogeneity in the volume equation take care of this problem? What are the spaces for $f$ and $g$ such that we have uniqueness?