# Existence of solution for Poisson problem with pure Neumann BCs

Hello all,

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

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

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

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?

• This is homework. Not suitable for MO. Jan 11, 2011 at 15:30
• This is not homework :) My question stems from a finite-difference discretization of such a PDE problem that seems to have trouble converging to a solution upon mesh refinement. If you cannot give the answer out right, could you point me to a reference that discusses this particular form of the equation? I have not been able to find a discussion of this case in the literature, and my experience with theoretical PDE analysis is limited at best, that is, I am not so familiar with standard PDE solution existence proofs. Jan 11, 2011 at 15:35
• A couple remarks: The β parameter is redundant, as you can divide by it and absorb it into $f$. Or did you not intend the β in front of the Laplacian? Also, this being a linear system, the right hand side has no bearing on the uniqueness of any solution. If the homogeneous system has a non-zero solution, then solutions are not unique (if they exist). Finally, you can gain some insight from looking at the one-dimensional case, with Ω an interval. At least it should tell you what to expect. Jan 11, 2011 at 22:25
• yes the $\beta$ parameter should not be in front of the laplacian. thanks for pointing that out. Jan 15, 2011 at 5:14
• Assuming you are dealing with classical solutions, the difference $q' = q_1 - q_2$ of two solutions solves the homogeneous Neumann problem $-\triangle q' + \beta q' = 0$ with 0 Neumann condition. Now use the strong maximum principle and Hopf lemma. Jan 15, 2011 at 14:06

A weak form of your BVP is $$a(q,v)=\ell(v)$$ where $$a(q,v)=\int_{\Omega}\nabla q\cdot\nabla v\,dx+\int_{\Omega}\beta qv\,dx$$ and $$\ell(v)=\int_{\Omega}fv\,dx+\int_{\partial\Omega}gv\,ds$$ with $$q,v\in H^1(\Omega)$$. If $$\beta>0$$, the bilinear form is coercive and continuous in $$H^1(\Omega)$$. Thus, apply Lax-Milgram and you get existence, uniqueness and stability in $$H^1(\Omega)$$. Stability now depends on the value of $$\beta$$.