Energy functional

An energy functional assigns an integral energy to a function. Minimization of a suitable energy is the basis of the Dirichlet principle. For the modified Helmholtz equation, a natural example is

E [u] = \int_{V} (∣\nabla u ∣^{2} + m^{2} u^{2}) d V .

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- p-energy Energy functional

p-energy ( $E_{p} [u] = \frac{1}{p} \int_{Ω} ∣ D u ∣^{p}$ )

The p-energy of a Sobolev function is

E_{p} [u] = \frac{1}{p} \int_{Ω} ∣ D u ∣^{p} d x .

Its stationary points under fixed boundary data are weak solutions of the p-Laplacian equation.

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Energy functional by

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The concept of an **energy functional** arises in various fields, particularly in physics, mathematics, and calculus of variations. An energy functional is generally a mathematical expression that associates a scalar value (energy) with a set of functions or configurations, often representing physical states or solutions to a problem.

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