Calculus of Variations and Geometric Measure Theory

A. Davini - A. Siconolfi

Existence and regularity of strict critical subsolutions in the stationary ergodic setting

created by davini on 12 Jul 2023
modified on 13 Jul 2023


Published Paper

Inserted: 12 jul 2023
Last Updated: 13 jul 2023

Journal: Ann. Inst. H. Poincaré C Anal. Non Linéaire
Volume: 33
Number: 2
Pages: 243–272
Year: 2016

ArXiv: 1205.3351 PDF


We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class $\CC^{1,1}$ in $\R^N$. The proofs are based on the use of Lax--Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set.