Dirichlet energy
{{Short description|Mathematical measure of a function's variability}}
In mathematics, the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space {{math|H1}}. The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet.
Definition
Given an open set {{math|Ω ⊆ Rn}} and a differentiable function {{math|u : Ω → R}}, the Dirichlet energy of the function {{math|u}} is the real number
:
where {{math|∇u : Ω → Rn}} denotes the gradient vector field of the function {{math|u}}.
Properties and applications
Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. {{math|E[u] ≥ 0}} for every function {{math|u}}.
Solving Laplace's equation for all , subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function {{math|u}} that satisfies the boundary conditions and has minimal Dirichlet energy.
Such a solution is called a harmonic function and such solutions are the topic of study in potential theory.
In a more general setting, where {{math|Ω ⊆ Rn}} is replaced by any Riemannian manifold {{math|M}}, and {{math|u : Ω → R}} is replaced by {{math|u : M → Φ}} for another (different) Riemannian manifold {{math|Φ}}, the Dirichlet energy is given by the sigma model. The solutions to the Lagrange equations for the sigma model Lagrangian are those functions {{math|u}} that minimize/maximize the Dirichlet energy. Restricting this general case back to the specific case of {{math|u : Ω → R}} just shows that the Lagrange equations (or, equivalently, the Hamilton–Jacobi equations) provide the basic tools for obtaining extremal solutions.
See also
- {{annotated link|Dirichlet's principle}}
- {{annotated link|Dirichlet eigenvalue}}
- {{annotated link|Total variation}}
- {{annotated link|Bounded mean oscillation}}
- {{annotated link|Harmonic map}}
- {{annotated link|Capacity of a set}}
References
- {{cite book | author=Lawrence C. Evans | title=Partial Differential Equations | publisher=American Mathematical Society | year=1998 | isbn=978-0821807729 }}