where \(X\) is a Sobolev or BV space, and \(F:X \to \mathbbR\) is a functional. The goal is to find a function \(u \in X\) that minimizes the functional \(F\) .
∣∣ u ∣ ∣ W k , p ( Ω ) = ( ∑ ∣ α ∣ ≤ k ∣∣ D α u ∣ ∣ L p ( Ω ) p ) p 1 where \(X\) is a Sobolev or BV space,
Sobolev spaces are a class of function spaces that play a crucial role in the study of PDEs and optimization problems. These spaces are defined as follows: where \(X\) is a Sobolev or BV space,