The Dirichlet energy functional is a mathematical tool used to measure the energy associated with a function defined on a domain, specifically in the context of variational problems. It is defined as the integral of the square of the gradient of the function over that domain, providing insight into the smoothness and behavior of functions. This functional plays a critical role in solving boundary value problems, particularly in formulating the Dirichlet problem, where one seeks to minimize energy subject to given boundary conditions.
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