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Non-convexity of level sets for $k$-Hessian equations in convex ring

Zhizhang Wang, Ling Xiao

Abstract

In this paper we construct explicit examples that show the sublevel sets of the solution of a $k$-Hessian equation defined on a convex ring do not have to be convex.

Non-convexity of level sets for $k$-Hessian equations in convex ring

Abstract

In this paper we construct explicit examples that show the sublevel sets of the solution of a -Hessian equation defined on a convex ring do not have to be convex.
Paper Structure (7 sections, 18 theorems, 60 equations)

This paper contains 7 sections, 18 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. solvability of the Dirichlet problem
  3. Choice of $M_1$ and $C^0$ estimates
  4. $C^1$ estimates of $u^\epsilon$
  5. $C^2$ estimates of $u^\epsilon$
  6. Hessian measures
  7. Proof of Theorem \ref{['thm1']}.

Key Result

Theorem 1.3

Let $\Omega_1$ be any smooth bounded convex domain in $\mathbb R^n.$ Then there exits a constant $M_1=M_1(\Omega_1)>0$ such that for all $M>M_1,$ there are some smooth convex rings $\Omega=\Omega_1\setminus\overline\Omega_2$ for which problem k-hessian with $n\geq 2k$ has a unique solution $u$ that

Theorems & Definitions (30)

  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 20 more