Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Abstract

Let \( \mathcal{H} \) be a separable Hilbert space, \( \mathcal{U} \subseteq \mathcal{H} \) an open convex subset, and f: \( \mathcal{U} \to \mathcal{H} \) a smooth map. Let Ω be an open convex set in \( \mathcal{H} \) with \( \bar \Omega \subseteq \mathcal{U} \), where \( \bar \Omega \) denotes the closure of Ω in \( \mathcal{H} \). We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): \( \mathcal{H} \to \mathcal{H} \) is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ \( \mathcal{U} \), where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L2-space under the flow of a class of kinetic transport equations so called BGK model.