The Fréchet and limiting subdifferentials of integral functionals on the spaces

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain formulae for computing the Fréchet and limiting subdifferentials of the integral functional F ( u ) = ∫ Ω f ( ω , u ( ω ) ) d μ ( ω ) , u ∈ L 1 ( Ω , E ) . Here ( Ω , A , μ ) is a measured space with an atomless σ-finite complete positive measure, E is a separable Banach space, and f : Ω × E → R ¯ . Under some assumptions, it turns out that these subdifferentials coincide with the Fenchel subdifferential of F.