Nonuniqueness and nonexistence results for the Lp-dual Minkowski problem with supercritical exponents

TL;DR

This paper investigates the Lp-dual Minkowski problem, revealing new nonuniqueness and nonexistence results for certain parameter ranges, especially in the supercritical case, using advanced inequalities and identities.

Contribution

It provides novel nonuniqueness and nonexistence results for the Lp-dual Minkowski problem across various parameter regimes, extending previous understanding.

Findings

Nonuniqueness results for p ≤ q - n + 1 and p < q - λ₁(n,k) with f ≡ 1.

Nonexistence results for p ≤ -q, q ≥ n, in all dimensions.

Generalization of Chou-Wang identity for the full (p, q) range.

Abstract

In this paper, the $\mathit{L}_{\mathit{p}}$-dual Minkowski problem of Monge-Amp\`ere type were studied for different $\mathit{p}$ and $\mathit{q}$. Some new nonuniqueness results were obtained for the range $\mathit{p}\le\mathit{q}-\mathit{n}+1$, $\mathit{p}\lt\mathit{q}-\lambda_{1}(\mathit{n},\mathit{k})$ and $\mathit{f}\equiv1$, where $\lambda_{1}(\mathit{n},\mathit{k})$ is the best constant of the Poincar\'e inequality on $\mathbb{S}^{n-1}$ with k-symmetricity. The second part of this paper is devoted to prove some new nonexistence results for the supercritical range $\mathit{p}\leq-\mathit{q}, \mathit{q}\geq\mathit{n}$ on all dimensional spaces. The key ingredient of our proof was based on a generalization of Chou-Wang identity for $\mathit{q}=\mathit{n}$, $\mathit{p}$=$-\mathit{q}$ to a full range of $(\mathit{p},\mathit{q})$.