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 $\alpha \in [0,+\mathrm{\infty })$, where $u$ varies in a set $U\mathrm{\ne }\mathrm{\varnothing }$ (without further structure). We first revisit a result stating that, excluding at most countably many values of $\alpha$, we have ${\inf }_{H_{\alpha }^{\star }}G={\sup }_{H_{\alpha }^{\star }}G$, where $H_{\alpha }^{\star }:=\arg {\min }_U{H}_{\alpha }$, which is assumed to be non-empty. Then, we prove a stronger result that concerns the invariance of the limiting value of the functional $G$ along minimizing sequences for $H_{\alpha }$, which extends the above Principle to the case $H_{\alpha }^{\star }=\mathrm{\varnothing }$. Moreover, we show to what extent these findings generalize to multi-regularized functionals and – in the presence of an underlying differentiable structure – to critical points. Finally, the main result 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.