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Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose

$\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\varphi \, dx =0$

for all harmonic functions $\varphi$, where $\Delta^{-1}f=\int_{\Omega}f(y)\Phi(x-y) \, dy$ and $\Phi$ is the fundamental solution of the laplacian. Is $f_1=f_2$?

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose

$\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\varphi \, dx =0$

for all harmonic functions $\varphi$, where $\Delta^{-1}f=\int_{\Omega}f(y)\Phi(x-y) \, dy$ and $\Phi$ is the fundamental solution of the Laplacian. Is $f_1=f_2$?

Let $\Omega$ be a bounded domain in $R^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose

$\int_{\Omega}(f_2-f_1)\varphi dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\varphi dx =0$

for all harmonic functions $\varphi$, where $\Delta^{-1}f=\int_{\Omega}f(y)\Phi(x-y)dy$ and $\Phi$ is the fundamental solution of the laplacian. Is $f_1=f_2$?

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose

$\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\varphi \, dx =0$

for all harmonic functions $\varphi$, where $\Delta^{-1}f=\int_{\Omega}f(y)\Phi(x-y) \, dy$ and $\Phi$ is the fundamental solution of the laplacian. Is $f_1=f_2$?