# Uniqueness of solution to PDE with a local boundary condition

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.

• If $u$ is defined on $\bar\Omega$, how do you define the exterior normal derivative? – Julián Aguirre Nov 17 '15 at 11:14
• I agree.let say we know a priori that $u$ is defined in a larger domain containing $\Omega$. – BigM Nov 17 '15 at 15:13
• In that case $n_+=-n_-$, the normal derivatives have oposite signs and the boundary condition becomes a Robin boundary condition. – Julián Aguirre Nov 17 '15 at 17:07
• @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). – BigM Nov 17 '15 at 17:14
• If $u$ is a solution, then $t u$ will be also a solution for any $t \neq 0$. – Voliar Nov 18 '15 at 8:46