Parabolic approaches to curvature equations

Paul Bryan*, Mohammad N. Ivaki, Julian Scheuer

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    1 Citation (Scopus)

    Abstract

    We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate Lp-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the Lp-Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli et al. (2007). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures.

    Original languageEnglish
    Article number112174
    Pages (from-to)1-24
    Number of pages24
    JournalNonlinear Analysis
    Volume203
    DOIs
    Publication statusPublished - Feb 2021

    Keywords

    • Curvature measure problems
    • Dual Minkowski problem
    • Christoffel–Minkowski problem
    • Curvature flow

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