Dirichlet's Principle for Harmonic Functions/Riemannian Manifold

Theorem

Let $\struct {M, g}$ be a compact connected $n$-dimensional Riemannian manifold with nonempty boundary.

Let $\map {C^\infty} M$ be the smooth function space.

Let $u \in \map {C^\infty} M$ be a smooth real function.

Let $\rd V_g$ be the Riemannian volume form.

Let $\grad$ be the gradient operator.

Let $\size {\, \cdot \,}$ be the Riemannian inner product norm.

Then $u$ is harmonic if and only if $u$ minimizes

$\ds \int_M \size {\grad u}^2 \rd V_g$

among all smooth real function with the same boundary values.

Proof

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Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems