Submitted Paper

Inserted: 1 aug 2026
Last Updated: 1 aug 2026

Pages: 48
Year: 2026

Abstract:

A study of the minimizers of one-dimensional Rudin–Osher–Fatemi-type functionals with linear or nonlinear total variation and fidelity term is undertaken. Conditions on the input datum and the parameters of the functional are found that force the input itself to be the minimizer or not. In the linear setting the discriminating conditions on the parameters are complementary highlighting the sharpness of the results. The nonlinear setting is substantially different: while the non-minimality of the datum is treated in analogy with the linear case, for the minimality of the input only partial answers are found. The results in the nonlinear setting hinge on auxiliary constrained or penalized minimization problems investigating the behavior of optimal transitions with prescribed height. Additionally, they are complemented by some numerical examples.

Keywords: regularity of minimizers, energy minimization, optimality conditions, (nonlinear) total variation

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