= Negative-potential domination by endpoint control
{title2=$\int_b^cV_-(x)(c-x)\,dx<1$}
Consider $a(v,v)=\int_b^c[|v'|^2+Vv^2]$ on real $H^1$ functions with $v(c)=0$, where $V_-=\max(-V,0)$. The pointwise estimate in the <Endpoint Poincaré inequality> gives $\int_b^cV_-v^2\leq\theta\|v'\|_2^2$ with $\theta=\int_b^cV_-(x)(c-x)\,dx$. If $\theta<1$, then $a(v,v)\geq(1-\theta)\|v'\|_2^2\geq(1-\theta)[1+(c-b)^2/2]^{-1}\|v\|_{H^1}^2$. Thus the form is a <coercive bilinear form> even though the potential can change sign. For $V(x)=x$ on $(-1,1)$, $\theta=\int_{-1}^0(-x)(1-x)\,dx=5/6$, yielding the explicit constant $1/18$.
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