with boundary conditions \(u=0\) on \(\partial \Omega\) . This PDE can be rewritten as an optimization problem:
min u ∈ H 0 1 ( Ω ) 2 1 ∫ Ω ∣∇ u ∣ 2 d x − ∫ Ω f u d x with boundary conditions \(u=0\) on \(\partial \Omega\)
Variational analysis in Sobolev and BV spaces involves the study of optimization problems of the form: Sobolev spaces are Banach spaces
− Δ u = f in Ω
Sobolev spaces have several important properties that make them useful for studying PDEs and optimization problems. For example, Sobolev spaces are Banach spaces, and they are also Hilbert spaces when \(p=2\) . Moreover, Sobolev spaces have the following embedding properties: with boundary conditions \(u=0\) on \(\partial \Omega\)
$$-\Delta u = g \quad \textin \quad \Omega