Dirichlet energy is a concept from the field of mathematics, particularly in the study of variational calculus and partial differential equations. It is associated with the Dirichlet problem and plays a significant role in various applications, including physics, engineering, and image processing. The Dirichlet energy of a function is generally defined as a measure of the "smoothness" of that function.
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The Dirichlet energy of a differentiable scalar function is one half the integral of its squared gradient. On a Euclidean domain it is ; the Dirichlet energy on a Riemannian manifold uses its metric and volume measure. Its stationary functions satisfy the Laplace equation with fixed boundary values.