3
$\begingroup$

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?

$\endgroup$
6
  • $\begingroup$ This is homework. Not suitable for MO. $\endgroup$ 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
    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. $\endgroup$ 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
    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. $\endgroup$ Jan 15, 2011 at 14:06

1 Answer 1

3
$\begingroup$

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 $\beta$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.