Energy functional

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

Critical point of an energy functional ( $δ E = 0$ )

A field configuration is critical for an energy functional

E

if its first variation vanishes for every admissible infinitesimal field variation. For

E [ϕ] = \int (\frac{1}{2} ∣\nabla ϕ ∣^{2} + U (ϕ))

, integration by parts against compactly supported variations gives

δ E = \int (- Δ ϕ + U^{'} (ϕ)) δ ϕ

Δ ϕ = U^{'} (ϕ)

. A constrained target instead allows only tangent variations, changing that field equation. Criticality is stationarity, not necessarily energetic stability or a local minimum.

The p-energy of a Sobolev function is

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

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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.