Dirichlet energy

Definition

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be an open set. Let $u \colon \Omega \to \R$ be continuously differentiable, or more generally weakly differentiable with locally integrable gradient $\nabla u$ . The Dirichlet energy of $u$ is $E[u] = \frac{1}{2} \int_\Omega \abs{\nabla u(x)}^2 \, dx \in [0, \infty].$

Remarks

Example: on the unit square $(0, 1)^2$ , $u(x, y) = x^2$ has $\nabla u = (2x, 0)$ , so $E[u] = \frac12 \int_0^1 4x^2 \, dx = \frac23$ . Non-example of finiteness: on the interval $(0, 1)$ , $u(x) = x^{1/4}$ is continuously differentiable, but $E[u] = \frac12 \int_0^1 \frac{1}{16} x^{-3/2} \, dx = \infty$ . The functions in $L^2(\Omega)$ with finite energy form the Sobolev space $H^1(\Omega) = W^{1,2}(\Omega)$ . Some authors omit the factor $\frac12$ ; the minimizers are the same. $E[u] = 0$ for a function on a connected open set if and only if $u$ is constant. The energy is invariant under conformal maps in dimension $2$ .

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Open set concept
Sobolev space concept
Dirichlet energy definition

Tag	`AFQGK1NE0X`
URI	`m2://AFQGK1NE0X`
Kind	definition
Writing	W0 stub
MSC	31B05, 35A15, 49J10
Defines	Dirichlet energy
Uses	Open set, Sobolev space

Dirichlet energy

Definition

Remarks

Reading path

Further reading

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