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Remarks on the Pogorelov type estimate for the degenerate $k$-Hessian equation

Yasheng Lyu

Abstract

This paper investigates the Pogorelov type estimate for the $k$-Hessian equation under a new condition on the degenerate right-hand side $f$.

Remarks on the Pogorelov type estimate for the degenerate $k$-Hessian equation

Abstract

This paper investigates the Pogorelov type estimate for the -Hessian equation under a new condition on the degenerate right-hand side .
Paper Structure (4 sections, 4 theorems, 44 equations)

This paper contains 4 sections, 4 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Pogorelov type interior estimate
  3. Preliminary
  4. Main results

Key Result

Lemma 2.1

For any $k\in\{2,3,\dots,n\}$, the following inequalities and identity hold for any $n\times n$ real symmetric matrix $A$ with $\lambda(A)\in\Gamma_{k}$: where $C_{1}>0$ and $C_{2}>0$ are universal constants.

Theorems & Definitions (5)

  • Lemma 2.1: Hardy1952Wang2009
  • Lemma 2.2: Section 3 of Wang2001
  • Lemma 2.3: Lemma 2.1 of Lyu2025
  • Theorem 2.4
  • proof