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

− Δ u = f in Ω

∣ u ∣ B V ( Ω ) ​ = sup ∫ Ω ​ u div ϕ d x : ϕ ∈ C c 1 ​ ( Ω ; R n ) , ∣∣ ϕ ∣ ∣ ∞ ​ ≤ 1 − Δ u = f in Ω ∣

BV spaces have several important properties that make them useful for studying optimization problems. For example, BV spaces are Banach spaces, and they are also compactly embedded in \(L^1(\Omega)\) . BV spaces are Banach spaces

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 \(F\) . − Δ u = f in Ω ∣