= Endpoint Poincaré inequality
{c}
{title2=$\|v\|_{L^2(b,c)}^2\leq\frac{(c-b)^2}{2}\|v'\|_{L^2(b,c)}^2,\quad v(c)=0$}
For the <one-dimensional Sobolev representative> with $v(c)=0$, the <fundamental theorem of calculus> gives $v(x)=-\int_x^cv'(t)\,dt$. The <Cauchy-Schwarz inequality> then gives $|v(x)|^2\leq(c-x)\|v'\|_2^2$. Integrating proves the displayed, generally nonoptimal <Poincaré inequality>. In particular $\|v\|_{H^1}^2\leq[1+(c-b)^2/2]\|v'\|_2^2$. Only one endpoint must be fixed, which is useful for mixed <boundary conditions> and for controlling a negative potential.
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