Does the following PDE with local boundary condition have a unique solution:

\begin{cases} \hfill \Delta u=0 \hfill & \Omega \\ \hfill \frac{\partial u}{\partial n_+}- \frac{\partial u}{\partial n_-}=u\hfill & \partial\Omega, \\ \end{cases} where $\frac{\partial }{\partial n_+}$ and $\frac{\partial }{\partial n_-}$ are the exterior and interior normal derivatives.

  • 1
    $\begingroup$ If $u$ is defined on $\bar\Omega$, how do you define the exterior normal derivative? $\endgroup$ – Julián Aguirre Nov 17 '15 at 11:14
  • $\begingroup$ I agree.let say we know a priori that $u$ is defined in a larger domain containing $\Omega$. $\endgroup$ – BigM Nov 17 '15 at 15:13
  • 1
    $\begingroup$ In that case $n_+=-n_-$, the normal derivatives have oposite signs and the boundary condition becomes a Robin boundary condition. $\endgroup$ – Julián Aguirre Nov 17 '15 at 17:07
  • $\begingroup$ @JuliánAguirre the reason behind my question is whether single layer potentials are equivalent to solutions to this equation (supposedly should be obvious but I don't see why). $\endgroup$ – BigM Nov 17 '15 at 17:14
  • $\begingroup$ If $u$ is a solution, then $t u$ will be also a solution for any $t \neq 0$. $\endgroup$ – Voliar Nov 18 '15 at 8:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.