Let $a_0$ and $b_0$ be smooth compactly supported functions in $\Omega \subset R^3$, $f\in C^1(\Omega)$, and define

$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{\Omega}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

$b_n=f\Delta^{-1}(b_{n-1})=-f(x)\int_{\Omega}b_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

where $\Phi$ is the fundamental solution of the laplacian. Suppose

$\int_{\Omega}(a_n-b_n)\varphi=0$ for every harmonic function $\varphi$ and every $n\geq 0$. Is $a_0=b_0$? When $f$ is a constant, the above means that $a_0-b_0$ is orthogonal to all functions $\varphi$ with $\Delta^n \varphi=0$ for some $n\in N$. This would imply $a_0=b_0$. Can this be generalized for a general coefficient $f$?

Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define

$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

$b_n=f\Delta^{-1}(b_{n-1})=-f(x)\int_{B}b_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

where $\Phi$ is the fundamental solution of the laplacian. Suppose

$\int_{B}(a_n-b_n)\varphi=0$ for every harmonic function $\varphi$ and every $n\geq 0$. Is $a_0=b_0$? When $f$ is a constant, the above means that $a_0-b_0$ is orthogonal to all functions $\varphi$ with $\Delta^n \varphi=0$ for some $n\in N$. This would imply $a_0=b_0$. Can this be generalized for a general coefficient $f$?

Let $a_0$ and $b_0$ be smooth compactly supported functions in $\Omega \subset R^3$, $f\in C^1(\Omega)$, and define

$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{\Omega}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

$b_n=f\Delta^{-1}(b_{n-1})=-f(x)\int_{\Omega}b_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

where $\Phi$ is the fundamental solution of the laplacian. Suppose

$\int_{\Omega}(a_n-b_n)\varphi=0$ for every harmonic function $\varphi$ and every $n\geq 0$. Is $a_0=b_0$?

Let $a_0$ and $b_0$ be smooth compactly supported functions in $\Omega \subset R^3$, $f\in C^1(\Omega)$, and define

$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{\Omega}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

$b_n=f\Delta^{-1}(b_{n-1})=-f(x)\int_{\Omega}b_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$

where $\Phi$ is the fundamental solution of the laplacian. Suppose

$\int_{\Omega}(a_n-b_n)\varphi=0$ for every harmonic function $\varphi$ and every $n\geq 0$. Is $a_0=b_0$? When $f$ is a constant, the above means that $a_0-b_0$ is orthogonal to all functions $\varphi$ with $\Delta^n \varphi=0$ for some $n\in N$. This would imply $a_0=b_0$. Can this be generalized for a general coefficient $f$?