# A differential equation with test functions

Hi everyone: Let $B$ be an open ball in $\mathbb{R}^{n}$, $n\geq2$, and let $\phi$ be a test function with compact support in $B$. Can we find a test function $\psi$ with compact support in $B$ such that $$\Delta \psi=\phi?$$ Here $\Delta$ is the laplacian.

Take Fourier transform of both parts of the equation, this will yield $$-|\xi|^2\hat \psi = \hat \phi.$$ The functions $\psi$ and $\phi$ belong to the Schwartz class $\cal S$, hence so do their Fourier transforms. THerefore, you obtain a condition that $$\frac{\hat \phi}{|\xi|^2}\in\cal S(\Bbb R^n).$$
For obvious reasons, this condition implies that $\hat \phi(0)=0$, which is not guaranteed - recall that $$\hat \phi(0)=\int_{\Bbb R^n}\phi(x)dx.$$
We can now finally conclude on the negative - not all $\phi$ allow such a $\psi$. Now a better question - can you deduce from the above reasoning the necessary and sufficient conditions on $\phi$ in order for $\psi$ to exist?
• Thank you very much. Is it that $\hat{\phi}(0)=0?$ Aug 24 '16 at 4:22