JournalsaihpcVol. 33, No. 2pp. 243–272

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

  • Andrea Davini

    Dip. di Matematica, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy
  • Antonio Siconolfi

    Dip. di Matematica, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy
Existence and regularity of strict critical subsolutions in the stationary ergodic setting cover
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Abstract

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 C1,1\mathrm{C}^{1,1} in RN\mathbb{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.

Cite this article

Andrea Davini, Antonio Siconolfi, Existence and regularity of strict critical subsolutions in the stationary ergodic setting. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 243–272

DOI 10.1016/J.ANIHPC.2014.09.010