= Quadratic variational principle for a symmetric positive operator
{title2=$I(u+w)-I(u)=B(w,w)$}
For a symmetric <bounded bilinear form> $B$ that is strictly positive on nonzero vectors, and a bounded <linear functional> $\ell$, set $I(v)=B(v,v)-2\ell(v)$. The <first variation> is $DI(v)[w]=2(B(v,w)-\ell(w))$. A <weak solution> $u$ of $B(u,w)=\ell(w)$ satisfies $I(u+w)-I(u)=B(w,w)$, so it is the unique minimizer. Conversely, any minimizer solves the weak equation. A <coercive bilinear form> gives existence by the <Lax-Milgram theorem>; strict positivity alone does not.
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