= Coercive variable-coefficient Dirichlet form
{title2=$a(u,v)=\int_0^1(pu'v'+quv)\,dx$}
On the <zero-boundary Sobolev space> $H_0^1(0,1)$ with norm $\|u'\|_{L^2}$, assume measurable coefficients $0<p_0\le p\le p_1$ and $0\le q\le q_1$. The <Poincaré inequality> and <Cauchy-Schwarz inequality> give $|a(u,v)|\le(p_1+q_1/\pi^2)\|u'\|_2\|v'\|_2$, while nonnegativity of $q$ gives $a(u,u)\ge p_0\|u'\|_2^2$. Thus the form is a <bounded bilinear form> and a <coercive bilinear form>. For a source in $H^{-1}$, the <Lax-Milgram theorem> yields a unique <weak solution> and every conforming <Galerkin method> approximation. No derivative of the bounded coefficient $p$ is needed for this <weak formulation>.
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