Trade-off Invariance Principle for minimizers of regularized functionals

In this paper, we consider functionals of the form \(H_\alpha(u)=F(u)+\alpha G(u)\) with \(\alpha\in[0,+\infty)\), where \(u\) varies in a set \(U\neq\emptyset\) (without further structure). We first show that, excluding at most countably many values of \(\alpha\), we have that \(\inf_{H_\alpha^\star}G= \sup_{H_\alpha^\star}G\), where \(H_\alpha^\star := \arg \min_U H_\alpha\), which is assumed to be non-empty. We further prove a stronger result that concerns the {invariance of the} limiting value of the functional \(G\) along minimizing sequences for \(H_\alpha\). This fact in turn implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of \(\alpha\), it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent.