= Mixed-boundary Airy energy principle
{title2=$J[u]=\tfrac12\int_0^1(u'^2+xu^2)dx$}
For $L=-d^2/dx^2+x$ on $(0,1)$, with homogeneous conditions $v(0)=0$, $v'(1)=0$, <integration by parts> gives $\langle v,Lv\rangle=\int_0^1(v'^2+xv^2)dx>0$ for nonzero $v$. The affine energy $J[u]=\tfrac12\int_0^1(u'^2+xu^2)dx$, minimized over $H^1$ functions with $u(0)=a$, yields $Lu=0$ and the natural <boundary condition> $u'(1)=0$. The inequality $\|v\|_2^2\le\tfrac12\|v'\|_2^2$ on $v(0)=0$ proves that the form is a <coercive bilinear form>; <Lax-Milgram theorem> supplies a unique critical point, and $J[u+v]-J[u]=\tfrac12\int(v'^2+xv^2)$ proves that it is the unique minimum.
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