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Shi-Zhong Du

Publications and source records attributed to Shi-Zhong Du.

At least 19 recordsLinked to original sources
arXiv2026

Uniqueness of Lp Minkowski problem in the supercritical range

The uniqueness of the $L_p$-Minkowski problem has been a long standing problem in convex geometry, which draws back earlier in 1974 the paper (Mathematika, {\bf21}, 1974) by Firey, and later developed by Lutwak, Yang, Zhang (Trans. Am. Math. Soc., {\bf356}, 2004) et al. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical exponents $p\in(-n-1,1]$. In the supercritical range, the uniqueness problem is much more complicated, even on the planar case $n=1$. One of the famous results was shown by Andrews in (J. Amer. Math. Soc., {\bf16}, 2003), where he established that the uniqueness holds in the range $p\in(-7,-2)$ and fails to hold for the other supercritical exponents $p\in(-\infty,-7)$. In this paper, we study the same uniqueness problem in the full supercritical range $p\in(-2n-5,-n-1)$ for all higher dimensional cases $n\geq2$. We will prove that for $p\in(-2n-5,-n-1)$, the unique strongly symmetric solution is given by the unit sphere $§^n$. The uniqueness range $(-2n-5,-n-1)$ is optimal due to our recent preprint (arXiv: 2104.07426), where non-spherical strongly symmetric solutions have been constructed for all $p\in(-\infty,-2n-5)$. When considering general solutions which may not be symmetric, the uniqueness set $Γ$ of $p$ for which the uniqueness holds, is shown to be both relatively open and closed in the full interval $(-2n-5,-n-1)$.

math.AP↗
arXiv2026

Complete Classification of the Euclidean Complete Solutions to a Monge-Ampere Equation

We study a Monge-Ampère equation with power term for some $p\in{\mathbb{R}}$. A solution $u$ is called to be Euclidean complete if it is an entire solution defined over the whole ${\mathbb{R}}^n$ or its graph is a large hypersurface satisfying the large condition $u(x)\to\infty$ as $\mathrm{dist}(x,\partialΩ)\to 0$ in case of $Ω\not={\mathbb{R}}^n$. In this paper, we will give various sharp conditions on $p$ and $Ω$ classifying the Euclidean complete solutions.

math.AP↗
arXiv2025

Complete Classification of the Symmetry Groups of Monge-Ampère Equation and Affine Maximal type Equation

The affine maximal type hypersurface has been a core topic in Affine Geometry. When the hypersurface is presented as a regular graph of a convex function $u$, the statement that the graph is of affine maximal type is equivalent to the statement that $u$ satisfies the fully nonlinear partial differential equation $$ D_{ij}(U^{ij}w)=0, \ \ w\equiv[\det D^2u]^{-θ}, \ \ θ>0, \ \ \forall x\in{\mathbb{R}}^N $$ of fourth order. This equation can be regarded as a generalization of the $N$-dimensional Monge-Ampère equation $$ \det D^2u=1, \ \ \forall x\in{\mathbb{R}}^N $$ of second order, since each solution of Monge-Ampère Equation satisfies affine maximal type equation automatically. In this paper, we will determine the symmetry groups of these two important fully nonlinear equations without asymptotic growth assumption. Our method develops the Lie's theory to fully nonlinear PDEs.

math.AP↗
arXiv2025

Complete Classification of the Symmetry Group of $L_p$-Minkowski Problem on the Sphere

In Convex Geometry, a core topic is the $L_p$-Minkowski problem \begin{equation}\label{e0.1} \det(\nabla^2h+hI)=fh^{p-1}, \ \ \forall X\in{\mathbb{S}}^n, \ \ \forall p\in \mathbb{R} \end{equation} of Monge-Ampère type. By the transformation $u(x)=h(X)\sqrt{1+|x|^2}$ and semi-spherical projection, equation \eqref{e0.1} can be reformulated by the Monge-Ampère type equation \begin{equation}\label{e0.2} \det D^2u=(1+|x|^2)^{-\frac{p+n+1}{2}}u^{p-1}, \ \ \forall x\in{\mathbb{R}}^n, \ \ \forall p\in \mathbb{R} \end{equation} on the Euclidean space. In this paper, we will firstly determine the symmetric groups of $n$-dimensional fully nonlinear equation \eqref{e0.2} without asymptotic growth assumption. After proving several key resolution lemmas, we thus completely classify the symmetric groups of the $L_p$-Minkowski problem. Our method develops the Lie theory to fully nonlinear PDEs in Convex Geometry.

math.AP↗
arXiv2025

Limiting shape of the $L_p$-Minkowski problem

Ben Andrews classified the limiting shape for isotropic curvature flow corresponding to the solutions of the $L_p$-Minkowski problem as $p\to-\infty$ in the planar case. In this paper, we use the group-invariant method to study the asymptotic shape of solutions to the $L_p$-Minkowski problem as $p\to-\infty$ in high dimensions. For any regular polytope $T$, we establish the existence of a solution ${Ω^{(p)}}$ to the $L_p$-Minkowski problem that converges to $T$ as $p\to-\infty$, thereby revealing the intricate geometric structure underlying this limiting behavior. We also extend the result to the dual Minkowski problem.

math.AP↗
arXiv2024

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

In this paper, the $\mathit{L}_{\mathit{p}}$-dual Minkowski problem of Monge-Ampère type were studied for different $\mathit{p}$ and $\mathit{q}$. Some new nonuniqueness results were obtained for the range $\mathit{p}\le\mathit{q}-\mathit{n}+1$, $\mathit{p}\lt\mathit{q}-λ_{1}(\mathit{n},\mathit{k})$ and $\mathit{f}\equiv1$, where $λ_{1}(\mathit{n},\mathit{k})$ is the best constant of the Poincaré 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 $(\mathit{p},\mathit{q})$.

math.AP↗
arXiv2023

Li-Yau Inequality and Liouville Property to a Semilinear Heat Equation on Riemannian Manifolds

This work deals with the Entire solutions of a nonlinear equation. The first part of this paper is devoted to investigation of the Liouville property on compact manifolds, which extends a result by Castorina-Mantegazza [4] for positive f. Secondly, we will turn to non-compact manifolds and prove a Liouville theorem under the assumptions of boundedness of the Ricci curvature from below, diffeomorphism of M with R^N and sub-criticality of p defined below. Finally, we also present simplified proofs of Yau's theorem for harmonic function and Gidas-Spruck's theorem for elliptic semilinear equation. Our proofs are based on Li-Yau type estimation for nonlinear equations.

math.AP↗
arXiv2023

Non-quadratic Euclidean complete affine maximal type hypersurfaces for $θ\in(0,(N-1)/N]$

Bernstein problem for affine maximal type equation \begin{equation}\label{e0.1} u^{ij}D_{ij}w=0, \ \ w\equiv[\det D^2u]^{-θ},\ \ \forall x\inΩ\subset{\mathbb{R}}^N \end{equation} has been a core problem in affine geometry. A conjecture proposed firstly by Chern (Proc. Japan-United States Sem., Tokyo, 1977, 17-30) for entire graph and then extended by Trudinger-Wang (Invent. Math., {\bf140}, 2000, 399-422) to its full generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex $C^4$-hypersurface in ${\mathbb{R}}^{N+1}$ must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension $N=2$ and $θ=3/4$, and later extended by Jia-Li (Results Math., {\bf56} 2009, 109-139) to $N=2, θ\in(3/4,1]$ (see also Zhou (Calc. Var. PDEs., {\bf43} 2012, 25-44) for a different proof). On the past twenty years, much efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension $N=3$. Recently, counter examples were found in \cite{Du2} (J. Differential Equations, {\bf269} (2020), 7429-7469) for $N\geq3$ and $θ\in(1/2,(N-1)/N)$ using a much more complicated argument. In this paper, we will construct explicitly various new Euclidean complete affine maximal type hypersurfaces which are not elliptic paraboloid for the improved range $$N\geq2, \ \ θ\in(0,(N-1)/N].$$

math.DG↗
arXiv2022

Sufficient and Necessary Condition of the Lp-Brunn-Minkowski Inequality Conjecture for p\in[0,1)

The Lp-Brunn-Minkowski inequality palys a central role in the Brunn-Minkowski theory proposed by Firey [13] in 60's and developed by Lutwak [26,27] in 90's, which generalizes the classical Brunn-Minkowski inequality by Lp-sum of convex bodies. The inequality has been established by Firey for p>1 and later been conjectured by Borozky-Lutwak-Yang-Zhang [5] for p\in[0,1). (see also [23,7]) The validity of this conjecture was verified for the planar case in [5], and for the higher dimensional case when p closing to 1 by Chen-Huang-Li-Liu [7]. (see alsoo a local version by Kolesnikov-Milman [23]) In this short note, we give a simple argument clarifying the equivalence between the full conjecture and a lower bound of the third eigenvalue of the Aleksandrov's problem.

math.AP↗
arXiv2021

Finite time blowup and type II rate for harmonic heat flow from Riemannian manifolds

In this paper, we will study the existence of finite time singularity to harmonic heat flow and their formation patterns. After works of Coron-Ghidaglia, Ding and Chen-Ding, one knows blow-up solutions under smallness of initial energy for m>=3. soon later, 2 dimensional blowup solutions were found by Chang-Ding-Ye. The first part of this paper is devoted to construction of new examples of finite time blow-up solutions without smallness conditions for 3<=m<7. In fact, when considering rotational symmetric harmonic heat flow from B_1\subset R^m to S^m\subset R^{m+1}, we will prove that the maximal solution blows up in finite time if b>\vartheta_m, and exists for all time if 0 =7 by Bizon-Wasserman. Finally, we also present result of finite time type I blowup for heat flow from S^m to S^m\subset R^{m+1}, when 3<=m<7 and degree is no less than 2.

math.AP↗
arXiv2021

Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation

The Hessian quotient equatio were studied for k-th symmetric elementary function S_k(D^2u) of eigenvalues of the Hessian matrix D^2u. Two pointwise quadratic growth conditions were found by Bao-Cheng-Guan-Ji ([1], American J. Math., 2003, 125, 301-316) ensuring Bernstein properties of Hessian quotient equation or k-Hessian equation respectively. In this paper, we will drop the point wise quadratic growth condition of [1] and prove three necessary and sufficient conditions to Bernstein property of (0.1) and (0.2), using a reverse isoperimetric type inequality, volume growth or Lp-integrable respectively.Our volume growth or Lp-integrable conditions improve largely various known point wise conditions in [1,6,7,13,18] etc.

math.AP↗
arXiv2021

On the planar Lp-Minkowski problem

In this paper, we study the planar Lp-Minkowski problem for all p, which was introduced by Lutwak [23]. A detailed exploration on solvability and uniqueness will be presented.

math.AP↗
arXiv2021

Bernstein Problem of Affine Maximal Type Hypersurfaces on Dimension N>=3

Bernstein problem for affine maximal type equation has been a core problem in affine geometry. A conjecture proposed firstly by Chern for entire graph and then extended by Trudinger-Wang to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniofrmly convex C^4-hypersurface in R^{N+1} must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N=2 and θ=3/4, and later extended by Jia-Li to N=2, θ\in(3/4,1] (see also [Zhou]). On the past twenty years, much efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N=3. In this paper, we will construct non-quadratic affine maximal type hypersurfaces which are Euclidean complete for N>=3, θ\in(1/2,(N-1)/N).

math.DG↗
arXiv2019

Energy Non-collapsing and Refined Blowup for a Semilinear Heat Equation

Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be type II in critical case. When p>p_S=(N+2)/(N-2) for N>=3, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than N-2-4/(p-1), which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collapsing blowups.

math.AP↗
arXiv2014

A nonexistence result on harmonic diffeomorphisms between punctured spaces

In this paper, we will prove a result of nonexistence on harmonic diffeomorphisms between punctured spaces. In particular, we will given an elementary proof to the nonexistence of rotationally symmetric harmonic diffeomorphisms from the punctured Euclidean space onto the punctured hyperbolic space.

math.DG↗