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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 $\Omega_1=\Omega_2$?

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  • $\begingroup$ If we allow $\Omega_2$ disconnected, then the answer is no: a counterexample is $\Omega_1 = B(0, 2)$ and $\Omega_2 = B(0, 1) \cup (B(0, 2+r) \setminus \overline B(0, 2))$ for an appropriate $r$. $\endgroup$
    – Mateusz Kwaśnicki
    Jun 2, 2020 at 19:09
  • $\begingroup$ Use $\operatorname{int}(U)$ \operatorname{int}(U), not int$(U)$ int$(U)$. I have edited accordingly. $\endgroup$
    – LSpice
    Jun 2, 2020 at 19:12

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