Variational Analysis In Sobolev And Bv Spaces Applications To Pdes And Optimization Mps Siam Series On Optimization ✓ ❲INSTANT❳
min u ∈ X F ( u )
∣ u ∣ B V ( Ω ) = sup ∫ Ω u div ϕ d x : ϕ ∈ C c 1 ( Ω ; R n ) , ∣∣ ϕ ∣ ∣ ∞ ≤ 1
∣∣ u ∣ ∣ B V ( Ω ) = ∣∣ u ∣ ∣ L 1 ( Ω ) + ∣ u ∣ B V ( Ω ) < ∞ min u ∈ X F ( u
subject to the constraint:
∣∣ u ∣ ∣ W k , p ( Ω ) = ( ∑ ∣ α ∣ ≤ k ∣∣ D α u ∣ ∣ L p ( Ω ) p ) p 1 This PDE can be rewritten as an optimization problem:
where \(|u|_BV(\Omega)\) is the total variation of \(u\) defined as:
Let \(\Omega\) be a bounded open subset of \(\mathbbR^n\) . The Sobolev space \(W^k,p(\Omega)\) is defined as the space of all functions \(u \in L^p(\Omega)\) such that the distributional derivatives of \(u\) up to order \(k\) are also in \(L^p(\Omega)\) . The norm on \(W^k,p(\Omega)\) is given by: min u ∈ X F ( u
with boundary conditions \(u=0\) on \(\partial \Omega\) . This PDE can be rewritten as an optimization problem:
