The Newtonian potential of a domain $\Omega$ is defined by $\Gamma*\chi_{\Omega}$ ($\Gamma$ is the fundamental solution of Laplacian operator $\Delta$), i.e. the convolution of the indicator function of $\Omega$ and the fundamental solution of $\Delta$.
Suppose I have two domains $\Omega_1$, $\Omega_2$ and $\operatorname{int}(\Omega_1\cap\Omega_2)\neq \phi$. Then denote their Newtonian potential function by $U_1$ and $U_2$. If I know $U_1=U_2$ in $\operatorname{int}(\Omega_1\cap\Omega_2)$, can we say $\Omega_1=\Omega_2$?
Or under what kind of condition we can say $\Omega_1=\Omega_2$?
\operatorname{int}(U)
, not int$(U)$int$(U)$
. I have edited accordingly. $\endgroup$