= Dirichlet energy on a Riemannian manifold
{c}
{title2=$E(u)$}
For a smooth scalar <function> $u$ on a compact <Riemannian manifold> $M$, its <Dirichlet energy> is
$$
E(u)=\frac12\int_M|\nabla_g u|_g^2\,d\operatorname{vol}_g.
$$
The <Riemannian gradient> measures local variation and the <Riemannian volume> weights its contribution. With source <function> $f$, consider $J(u)=E(u)-\int_Mfu\,d\operatorname{vol}_g$. On a manifold without boundary, or for variations vanishing at the boundary, the first variation in direction $v$ is
$$
\left.\frac d{dt}J(u+tv)\right|_{t=0}=\int_M\langle\nabla_g u,\nabla_g v\rangle_g\,d\operatorname{vol}_g-\int_Mfv\,d\operatorname{vol}_g
=-\int_M(\Delta_g u+f)v\,d\operatorname{vol}_g.
$$
Here $\Delta_g=\operatorname{div}_g\nabla_g$ is the <Laplace-Beltrami operator>. A critical <function> therefore solves the <Poisson equation> $-\Delta_g u=f$.
On a closed manifold, adding a constant leaves $E$ unchanged. If $\int_M f\,d\operatorname{vol}_g\ne0$, shifting that constant makes $J$ unbounded below. Thus zero integral of the source is a necessary compatibility condition for this unconstrained closed-manifold problem.
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