Negative-potential domination by endpoint control (${\int }_b^c{V}_{-}(x)(c-x)dx<1$)

Consider $a(v,v)={\int }_b^c[|v^{\prime }{|}^2+V{v}^2]$ on real $H^1$ functions with $v(c)=0$, where $V_{-}=\max (-V,0)$. The pointwise estimate in the Endpoint Poincaré inequality gives ${\int }_b^c{V}_{-}{v}^2\le \theta \Vert v^{\prime }{\Vert }_2^2$ with $\theta ={\int }_b^c{V}_{-}(x)(c-x)dx$. If $\theta <1$, then $a(v,v)\ge (1-\theta )\Vert v^{\prime }{\Vert }_2^2\ge (1-\theta )[1+(c-b{)}^2/2{]}^{-1}\Vert v{\Vert }_{H^1}^2$. 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 $1/18$.