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Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

Shanwei Ding, Guanghan Li

Abstract

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE $G(x,u_K,Du_K)F(D^2u_K+u_KI)=1$ on $\mathbb S^n$ for a convex body $K$, where $D$ is the covariant derivative with respect to the standard metric on $\mathbb S^n$ and $I$ is the unit matrix of order $n$. This result covers many previous known solutions to $L^p$ dual Minkowski problem, $L^p$ dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc.. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015).

Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

Abstract

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE on for a convex body , where is the covariant derivative with respect to the standard metric on and is the unit matrix of order . This result covers many previous known solutions to dual Minkowski problem, dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc.. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015).
Paper Structure (10 sections, 22 theorems, 156 equations)

This paper contains 10 sections, 22 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Intrinsic curvature
  4. Extrinsic curvature
  5. Convex hypersurface parametrized by the inverse Gauss map
  6. Curvature estimates for general cases
  7. Convergence and proof of theorems
  8. The uniqueness results
  9. Orlicz Christoffel-Minkowski type problem
  10. Reference

Key Result

Theorem 1.2

Let $F\in C^2(\Gamma_+)\cap C^0(\partial\Gamma_+)$ satisfy Assumption a1.1, and let $M_0$ be a closed, smooth, uniformly convex hypersurface in $\mathbb{R}^{n+1}$, $n\geqslant2$, enclosing the origin. Suppose $(i)$$\lim\sup_{s\rightarrow+\infty}[\max_{y=sx}G(x,y)s^\beta]<1<\lim\inf_{s\rightarrow0^+}

Theorems & Definitions (33)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.10
  • Corollary 1.11
  • Lemma 2.1
  • ...and 23 more