Maximum principle easy proof

Question

Is there an easy proof of the maximum principle from the variational formulation in $\Bbb R^d$, without using Green functions?

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Variational formulation: $$ \forall v\, \text{ smooth, }\, \int \nabla u \cdot \nabla v = \int fv $$ Maximum principle: if $u$ is a solution of the avriational problem, then $$ f\ge 0 \implies u\ge 0 $$

Answer 1

Hint: Take $v=u^{-}=\max\{-u,0\}$ in the variational formulation.