A principle for minimizers of regularized functionals

par Alessandro Scagliotti (Université technique de Munich)

mardi 3 juin 2025, 11:00 → 12:00 Europe/Paris

Salle de conférence (LJAD)

When minimizing a regularized functional—i.e., one of the form H(u) = F(u) + α G(u), where G is a regularization term and α is the regularization parameter—one generally expects multiple minimizers to exist; one might furthermore expect those different minimizers to have different regularization values. We show, however, that for most choices of the regularization parameter, all minimizers of the regularized functional share the same regularization value, and the proof does not require any assumptions on the domain or on smoothness/convexity properties of the involved functionals.

We also prove a stronger result concerning the invariance of the limit of the regularization value along minimizing sequences for the regularized functional. Moreover, we demonstrate how these findings extend to multi-regularized functionals and—when an underlying differentiable structure is present—to critical points.