Consider $\Omega\subset\mathbb{R}^n$ a smooth bounded open set. Let $G$ be the Green function for the Dirichlet boundary condition: for any $y\in\Omega$, the map $G(\cdot,y):\Omega\setminus \{y\}\to \mathbb{R}$ is the solution to $$ -\Delta_x G(x,y) = \delta(x-y)\ \text{in }\Omega,\quad G(x,y)=0 \text{ for }x\in \partial\Omega. $$ Then we can show (essentially by integration by parts) that for any $t>0$ and $y\in\Omega$, it holds $$ \int_{\{G(\cdot,y)>t\}} |\nabla G(\cdot,y)|^2 = t. $$ Question: Is there a similar formula for the Green function associated to the Neumann boundary conditions?
Namely, we consider the following function $H$: for any $y\in\Omega$, the map $H(\cdot,y):\Omega\setminus \{y\}\to \mathbb{R}$ is the solution to $$ -\Delta_x H(x,y) = \delta(x-y)-\frac{1}{|\Omega|}\ \text{in }\Omega,\quad \partial_{\nu}H(x,y)=0\text{ for }x\in \partial\Omega. $$ Given $t>0$ and $y\in\Omega$, does there exists a domain $D_t\subset \Omega$ (which is a sort of level-set for $H$) such that the following relation is satisfied? $$ \int_{D_t} |\nabla H(\cdot,y)|^2 = t. $$
$\begingroup$ $\endgroup$
Add a comment |