A differential equation with test functions

Question

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.

Answer 1

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?