Variational Analysis In Sobolev And Bv Spaces Applications To Pdes And Optimization Mps Siam Series On Optimization May 2026

where $|u|_BV(\Omega)$ is the total variation of $u$ defined as:

∣∣ u ∣ ∣ W k , p ( Ω ) = ( ∑ ∣ α ∣ ≤ k ∣∣ D α u ∣ ∣ L p ( Ω ) p ) p 1 where $|u|_BV(\Omega)$ is the total variation of $u$

− Δ u = f in Ω

Variational analysis in Sobolev and BV spaces has several applications in PDEs and optimization. For example, consider the following PDE: where $|u|_BV(\Omega)$ is the total variation of $u$

$$-\Delta u = g \quad \textin \quad \Omega where $|u|_BV(\Omega)$ is the total variation of $u$

subject to the constraint:

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: