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Hui-Chun Zhang

Publications and source records attributed to Hui-Chun Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Lipschitz regularity of harmonic map heat flows from $\mathrm{RCD}$ spaces into $\mathrm{CAT}(0)$ spaces

We prove positive-time regularity for harmonic map heat flows from finite-dimensional $\mathrm{RCD}(K,N)$ spaces into complete $\mathrm{CAT}(0)$ spaces, without assuming that either the source or the target is smooth. For bounded-image initial data, the $\mathrm{EVI}$ gradient flow of the Dirichlet energy admits a representative that is locally Lipschitz jointly in space and time. Moreover, its spatial pointwise Lipschitz constant satisfies an Eells--Sampson-type parabolic Bochner inequality. The main difficulty is that the smooth parabolic perturbations used to select contact points are unavailable on an $\mathrm{RCD}$ source. We overcome it by a sliced contact-selection principle that converts the elliptic ABP estimate on $\mathrm{RCD}$ spaces into the space--time contact selection required by the Hamilton--Jacobi argument.

math.MG↗
arXiv2026

One-phase Free Boundary Problems on RCD Metric Measure Spaces

In this paper, we consider a vector-valued one-phase Bernoulli-type free boundary problem on a metric measure space $(X,d,μ)$ with Riemannian curvature-dimension condition $RCD(K,N)$. We first prove the existence and the local Lipschitz regularity of the solutions, provided that the space $X$ is non collapsed, i.e. $μ$ is the $N$-dimensional Hausdorff measure of $X$. And then we show that the free boundary of the solutions is an $(N-1)$-dimensional topological manifold away from a relatively closed subset of Hausdorff dimension $\leqslant N-3$.

math.AP↗
arXiv2026

Lipschitz regularity of harmonic map heat flows into $CAT(0)$ spaces

In 1964, Eells and Sampson proved the celebrated long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. In 1992, Gromov and Schoen initiated the study of harmonic maps into $CAT(0) $ metric spaces. It naturally motivates the study of the harmonic map heat flow into singular metric spaces. In the 1990s, Mayer and Jost independently studied convex functionals on $CAT(0)$ spaces and extended Crandall-Liggett's theory of gradient flows from Banach spaces to $CAT(0) $ spaces to obtain the weak solutions for the harmonic map heat flow into $CAT(0)$ spaces. The weak solutions enjoy the favorable long-time existence, uniqueness and well-established long-time behaviors. It is a long-standing open question to ask if the weak solutions possess the Lipschitz regularity. Very recently, by using elliptic approximation method, Lin, Segatti, Sire, and Wang proved the weak solutions are Lipschitz in space and $1\over 2$-Hölder continuous in time, for a wide class of $CAT(0)$ spaces. In the present paper, we give a complete answer to the question. We show that every weak solution of the harmonic map heat flow into $CAT(0)$ spaces is Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality.

math.DG↗
arXiv2025

Optimal boundary regularity of harmonic maps from $RCD(K,N)$-spaces to $CAT(0)$-spaces

In 1983, Schoen-Uhlenbeck \cite{SU83} established boundary regularity for energy-minimizing maps between smooth manifolds with the Dirichlet boundary condition under the assumption that both the boundary and the data are of $C^{2,α}$. A natural problem is to study the qualitative boundary behavior of harmonic maps with rough boundary and/or non-smooth boundary data. For the special case where $u$ is a harmonic function on a domain $Ω\subset \mathbb R^n$, this problem has been extensively studied (see, for instance, the monograph \cite{Kenig94}, the proceedings of ICM 2010 \cite{Tor10} and the recent work of Mourgoglou-Tolsa \cite{MT24}). The $W^{1,p}$-regularity ($1<p<\infty$) has been well-established when $\partialΩ$ is Lipschitz (or even more general) and the boundary data belongs to $W^{1,p}(\partialΩ)$. However, for the endpoint case where the boundary data is Lipschitz continous, as demonstrated by Hardy-Littlewood's classical examples \cite{HL32}, the gradient $|\nabla u|(x)$ may have logarithmic growth as $x$ approaches the boundary $\partial Ω$ even if the boundary is smooth. In this paper, we first establish a version of the Gauss-Green formula for bounded domains in $RCD(K, N)$ metric measure space. We then apply it to obtain the optimal boundary regularity of harmonic maps from $RCD(K, N)$ metric measure spaces into $CAT(0)$ metric spaces. Our result is new even for harmonic functions on Lipschitz domains of Euclidean spaces.

math.DG↗
arXiv2024

Comments on the regularity of harmonic maps between singular spaces

In this work we are going to establish Hölder continuity of harmonic maps from an open set $Ω$ in an ${\rm RCD}(K,N)$ space valued into a ${\rm CAT}(κ)$ space, with the constraint that the image of $Ω$ via the map is contained in a sufficiently small ball in the target. Building on top of this regularity and assuming a local Lipschitz regularity of the map, we establish a weak version of the Bochner-Eells-Sampson inequality in such a non-smooth setting. Finally we study the boundary regularity of such maps.

math.AP↗
arXiv2022

Weyl's lemma on $RCD(K,N)$ metric measure spaces

In this paper, we extend the classical Weyl's lemma to $RCD(K,N)$ metric measure spaces. As its applications, we show the local regularity of solutions for Poisson equations and a Liouville-type result for $L^1$ very weak harmonic functions on $RCD(K,N)$ spaces. Meanwhile, a byproduct is that we obtain a gradient estimate for solutions to a class of elliptic equations with dis-continuous coefficients.

math.DG↗
arXiv2021

Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds

In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.

math.DG↗
arXiv2019

Partial regularity of harmonic maps from Alexandrov spaces

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in \cite{lin97}. The proof extends the argument of Huang-Wang \cite {hua-w10}.

math.DG↗
arXiv2019

Quantitative gradient estimates for harmonic maps into singular spaces

In this paper, we will show the Yau's gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $κ$, $κ\geq0$, in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.

math.DG↗
arXiv2017

Lipschitz continuity of harmonic maps between Alexandrov spaces

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.

math.DG↗
arXiv2017

Sharp gradient estimate for heat kernels on $RCD^*(K,N)$ metric measure spaces

In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels. These results seem new even for smooth Riemannian manifolds.

math.DG↗
arXiv2016

Local Li-Yau's estimates on $RCD^*(K,N)$ metric measure spaces

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(K,N).$

math.DG↗
arXiv2015

Sharp Spectral Gaps on Metric Measure Spaces

In this paper, we extend the sharp lower bounds of spectal gap, due to Chen- Wang [10, 11], Bakry-Qian [6] and Andrews-Clutterbuck [5], from smooth Riemaniannian manifolds to general metric measure spaces with Riemannian curvature-dimension condition RCD*(K;N).

math.DG↗
arXiv2014

A note on Liouville type theorem of elliptic inequality $Δu+u^σ\leq 0$ on Riemannian manifolds

Let $σ>1$ and let $M$ be a complete Riemannian manifold. In a recent work [9], Grigor$^{\prime}$yan and Sun proved that a pointwise upper bound of volume growth is sufficient for uniqueness of nonnegative solutions of elliptic inequality $$(*)\quad\qquad\qquad\qquad Δu(x)+u^σ(x)\leq 0,\qquad x\in M.\quad\qquad \qquad\qquad $$ In this note, we improve their result to that an \emph{integral condition} on volume growth implies the same uniqueness of ($*$). It is inspired by the well-known Varopoulos-Grigor$^{\prime}$yan's criterion for parabolicity of $M$.

math.DG↗
arXiv2012

Sharp Spectral Gap and Li-Yau's Estimate on Alexandrov Spaces

In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li-Yau's estimate for positve solutions of heat equations on Alexandrov spaces.

math.DG↗
arXiv2012

Yau's gradient estimates on Alexandrov spaces

In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

math.DG↗