On the maximum principle for minimizers of variational integrals
Jacopo Schino, Panayotis Smyrnelis · Discrete and Continuous Dynamical Systems · 2025
We revisit the strong maximum principle for minimizers of variational integrals, whose Lagrangian $ L = \frac{1}{p}| abla u|^p+W(u) $ involves a nondecreasing, lower semicontinuous potential $ W $. When $ W\in C^1 $, minimizers solve the equation $ \Delta_p u = W'(u) $, and then necessary and sufficient conditions for the strong maximum principle to hold are well-known. By deriving a comparison principle for minimizers, we extend this result as well as Hopf's lemma, in the variational setting, for nonsmooth potentials.