= Weak formulation of a clamped fourth-order equation
{title2=$a(u,v)=\int(pu''v''+qu'v'+ruv)$}
For continuous real coefficients with $p>0$ and $q,r\geq0$ on a bounded interval, use the <clamped second-order Sobolev space> and the displayed <bilinear form>. A forcing in $L^2$ defines $\ell(v)=\int fv$. The <weak solution> satisfies $a(u,v)=\ell(v)$ for every clamped test vector. Compactly supported <test functions> give $(pu'')''-(qu')'+ru=f$ as a <distribution>; density extends that equation to the energy space. Continuous coefficients alone do not justify differentiating them classically. The <clamped interval derivative norm bound> gives coercivity, so <Lax-Milgram theorem> supplies a unique solution and the quadratic energy has that unique minimum.
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