Suppose that we have solution of
$$\delta d f = g$$
on sphere. Where $\delta d$ is Laplace-de Rham operator for functions, $f,g$ are scalar functions and $g$ has support on north hemisphere and it is non-negative there.
Than by Stokes theorem we have(I think that I have signs wrong but that does not solve the problem)
$$ \int_{\text{equator}}*df = \int_{\text{north hemisphere}} d*df = - \int_{\text{south hemisphere}} d*df$$
But $$ \int_{\text{north hemisphere}} d*df = \int_{\text{north hemisphere}} *g \neq 0 $$ $$ \int_{\text{south hemisphere}} d*df = \int_{\text{south hemisphere}} *g = 0$$
What have I done wrong?? I can't really see it.
There is no solution. By Stokes's Theorem, since the sphere $S$ is compact, $$\pm\int_S \star g = \int_S d{\star}df = \int_{\partial S} \star df = 0\,,$$ since $\partial S = \emptyset$.