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The Orlicz-Aleksandrov problem for pseudo-cones

Siqi Lei, Xudong Wang

TL;DR

The paper addresses the Orlicz-Aleksandrov problem for unbounded pseudo-cones by introducing the Orlicz-integral Gauss curvature $J_\phi(K,\eta)$ and developing a direct variational framework. A perturbation-based variational formula links infinitesimal changes in the defining data to $J_\phi$, enabling construction of solutions on compact sets and a cone-extension argument to handle general measures. The authors prove existence—and in fact nonuniqueness—of solutions, showing that for a given nonzero finite measure $\mu$ on $\Omega_C$ there exists $K\in ps(C)$ and a constant $c>0$ with $c\,J_\phi(K,\cdot)=\mu$ on $\Omega_C$. This extends the classical Aleksandrov problem to the Orlicz setting for pseudo-cones, connecting Orlicz curvature measures with Gauss image problems and dual Minkowski theory in a noncompact setting.

Abstract

In this paper, integral Gauss curvature of pseudo-cones is extended to the Orlicz setting and the corresponding Orlicz-Aleksandrov problem is studied. By the direct variational method and a restrictive technology, we obtain infinitely many solutions to the Orlicz-Aleksandrov problem for pseudo-cones.

The Orlicz-Aleksandrov problem for pseudo-cones

TL;DR

The paper addresses the Orlicz-Aleksandrov problem for unbounded pseudo-cones by introducing the Orlicz-integral Gauss curvature and developing a direct variational framework. A perturbation-based variational formula links infinitesimal changes in the defining data to , enabling construction of solutions on compact sets and a cone-extension argument to handle general measures. The authors prove existence—and in fact nonuniqueness—of solutions, showing that for a given nonzero finite measure on there exists and a constant with on . This extends the classical Aleksandrov problem to the Orlicz setting for pseudo-cones, connecting Orlicz curvature measures with Gauss image problems and dual Minkowski theory in a noncompact setting.

Abstract

In this paper, integral Gauss curvature of pseudo-cones is extended to the Orlicz setting and the corresponding Orlicz-Aleksandrov problem is studied. By the direct variational method and a restrictive technology, we obtain infinitely many solutions to the Orlicz-Aleksandrov problem for pseudo-cones.
Paper Structure (9 sections, 12 theorems, 80 equations)

This paper contains 9 sections, 12 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudo-cones
  4. The radial Gauss map of pseudo-cones
  5. Schneider selection theorem
  6. Orlicz-integral Gauss curvature
  7. A Variational formula
  8. Existential results on compact sets
  9. Proof of Theorem \ref{['The-Main-Theorem']}

Key Result

Theorem 1.1

Suppose that $\phi$ is a positive continuous function on $(0,+\infty)$. Let $\mu$ be a nonzero finite Borel measure on $\Omega_C$, then there is a $K\in ps(C)$ and a constant $c>0$, such that Moreover, such $c$ and $K$ are not unique. In fact, there exist infinitely many such pairs.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 2.1: Schneider selection theorem, Schneider-Pseudo_cones
  • Lemma 2.1: Schneider-The_Gauss_image_problem_for_pseudo_cones
  • Corollary 2.1
  • proof
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 13 more