Variational Methods

and

Thus, (u∗, w∗) is a critical point of E T(u, w) since it fulfills the Euler–Lagrange equations. Because of the convexity of the functional in one variable whenever the other one is fixed, we have that (u∗, w∗) actually is a local minimizer. The stated space inclusions hold due to the estimates shown in Proposition 4.1 (see Remark 4.5).

4.4.1.4.2 T = 0

For E0, as we have discussed before, and as can be seen in [4, Thm. 2], we are lacking information on the compactness of the iterates. Thus, the convergence of the alternating minimization scheme ...