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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?

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  • $\begingroup$ This is homework. Not suitable for MO. $\endgroup$
    – Denis Serre
    Commented Jan 11, 2011 at 15:30
  • $\begingroup$ 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. $\endgroup$
    – Mihai
    Commented Jan 11, 2011 at 15:35
  • $\begingroup$ 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. $\endgroup$
    – Harald Hanche-Olsen
    Commented Jan 11, 2011 at 22:25
  • $\begingroup$ yes the $\beta$ parameter should not be in front of the laplacian. thanks for pointing that out. $\endgroup$
    – Mihai
    Commented Jan 15, 2011 at 5:14
  • $\begingroup$ 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. $\endgroup$
    – Willie Wong
    Commented Jan 15, 2011 at 14:06

1 Answer 1

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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 $\beta$.

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