Convex Monotone Semigroups on Lattices of Continuous Functions

  • Robert Denk

    Universität Konstanz, Germany
  • Michael Kupper

    University of Konstanz, Germany
  • Max Nendel

    Bielefeld University, Germany
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We consider convex monotone -semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a -Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton–Jacobi–Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow us to define the generators in a weak sense.

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Robert Denk, Michael Kupper, Max Nendel, Convex Monotone Semigroups on Lattices of Continuous Functions. Publ. Res. Inst. Math. Sci. 59 (2023), no. 2, pp. 393–421

DOI 10.4171/PRIMS/59-2-4