Lemma 26 ([Maximum of weak subsolutions]). Let O⊂³ be open, let g be locally bounded, and let F(y,t) be measurable in y , continuous in t , and bounded on compact spatial sets and bounded ranges of t . If finitely many functions v_i∈C( …

It suffices to consider two functions. For a nonnegative compactly supported smooth test φ , choose an increasing Lipschitz function θ_η that is zero on (-∞,0] and one on [η,∞) . Test the first inequality with θ_η(v_1-v_2)φ and the second with (1-θ_η(v_1-v_2))φ . These nonnegative compactly supported H^1 tests are legitimate by mollification, since the right sides are bounded on their support. The sum of the left sides is , -∫[θ_η(v_1-v_2)∇v_1 +(1-θ_η(v_1-v_2))∇v_2]·∇φ; , 8mm -∫θ_η'(v_1-v_2)|∇(v_1-v_2)|^2φ. The second term is nonpositive. Deleting it and letting η0 gives the gradient of (v_1,v_2) by the Sobolev positive-part formula. Dominated convergence gives the reaction at the maximum; the reactions agree at ties. Induction proves the finite version. For the last assertion subtract V in each inequality. The singular measures cancel, and the reaction for the offsets becomes F(y,V(y)+a_i(y)) . It is locally bounded: it vanishes near every nuclear point, and V is smooth off suppν . Apply the first assertion to the offsets and then add V back.