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Non-convexity of level sets for solutions to $k$-Hessian equations in exterior domains

Wang Bo, Wang Cong, Wang Zhizhang

Abstract

In this paper, we provide examples to show that for $1 \leq k \leq n/2$, solutions to $k$-Hessian equations $S_k(D^2u)=1$ in the exterior of a strictly convex domain need not be quasiconvex, when prescribing quadratic growth at infinity. Additionally, we give a new proof for the quasiconvexity of harmonic functions in such exterior domains that decay to zero at infinity.

Non-convexity of level sets for solutions to $k$-Hessian equations in exterior domains

Abstract

In this paper, we provide examples to show that for , solutions to -Hessian equations in the exterior of a strictly convex domain need not be quasiconvex, when prescribing quadratic growth at infinity. Additionally, we give a new proof for the quasiconvexity of harmonic functions in such exterior domains that decay to zero at infinity.

Paper Structure

This paper contains 9 sections, 17 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Approximating problems in convex rings
  3. Solvability of approximating problems.
  4. $C^0$ estimates of $u^{\varepsilon,R}$
  5. $C^1$ estimates of $u^{\varepsilon,R}$
  6. $C^2$ estimates of $u^{\varepsilon,R}$
  7. Convergence and Hessian measures
  8. Proofs of Theorems \ref{['thm:main']} and \ref{['thm:main-1']}
  9. Proof of Theorem \ref{['thm:main-H']}

Key Result

Theorem 1.3

Let $n\geq3$ and $2\leq k \le n/2$. For any given $A\in\mathcal{A}_k$, $b\in\mathbb{R}^n$, there exist a smooth strictly convex domain $\Omega_0$ and a constant $c^*$, depending only on $n$, $k$, $A$ and $b$, such that for every $c\geq c^*$, the problem admits a unique $k$-convex solution $u\in C^\infty(\mathbb{R}^n\setminus\Omega_0)$ which is not quasiconvex.

Theorems & Definitions (34)

  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6: Microscopic version proof of Lewis-1977 and Kawohl-1985
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 24 more