When minimizing a regularized functional - i.e., one of the form \(H(u) = F(u) + \alpha G(u)\), where \(G\) is a regularization term and \(\alpha\) is the regularization parameter - one generally expects multiple minimizers to exist; one might furthermore expect the term \(G\) to assume different values in correspondence of different minimizers. We show, however, that for most choices of the parameter $\alpha$, all minimizers of the regularized functional share the same value of \(G\). This holds without requiring any assumptions on the domain nor on the smoothness/convexity properties of the involved functionals. We also prove a stronger result concerning the invariance of the limit of \(G\) along minimizing sequences. Moreover, we demonstrate how these findings extend to multi-regularized functionals and - when an underlying differentiable structure is present- to critical points.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006