= Hopf dichotomy with classical regularity away from the zero set
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Let $v\leq0$ be $C^1$ on a connected domain and $C^2$ away from its zero set, with $\Delta v+v=0$ wherever $v<0$. If negative and zero points coexist, an interior <Euclidean ball> in $\{v<0\}$ touches its zero set. On that <Euclidean ball> $\Delta v=-v>0$, so the <Hopf boundary point lemma> forces a nonzero derivative at an interior maximum, contrary to $Dv=0$. Thus $v$ is identically zero or strictly negative. Mere <Lipschitz continuity> does not suffice: $-\max\{\sin x_1,0\}$ on $(-1,1)^n$ has both negative and zero regions.
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