arXiv subjects

Zhenghuan Gao

Publications and source records attributed to Zhenghuan Gao.

9 recordsLinked to original sources

arXiv2026

$C^{1,1}$ Regularity Global Estimates for Homogeneous Complex Hessian Equations on Punctured Domains

In this paper, we prove the $C^{1,1}$ regularity of Green functions associated with the complex $k$-Hessian operator for $1\leq k<n$, and give a new proof of the corresponding regularity of pluricomplex Green functions. We establish real Hessian estimates for approximating problems associated with homogeneous complex Hessian equations on punctured domains. The main idea is to introduce new auxiliary functions generated by complex linear vector fields to reduce the global real Hessian estimate to the boundary. We also treat the complex Monge-Ampère case. Similar real Hessian estimates also works for the exterior problem.

math.CV↗

arXiv2026

On the Viscosity Solutions of Parabolic p-Laplacian Equations with Capillary-Type Boundary Conditions

In this paper, we establish the well-posedness and large-time asymptotic behavior of viscosity solutions to singular/degenerate parabolic $p$-Laplacian equations with general capillary-type boundary conditions, including Neumann and prescribed contact angle cases, on strictly convex domains. By establishing a gradient estimate independent of the $C^0$ norm of the solution via the maximum principle, and by analyzing the problem through an approximation procedure together with associated elliptic eigenvalue problems, we prove the existence, uniqueness, and asymptotic behavior of solutions. For the elliptic problem with Neumann boundary conditions, we first focus on flat domains with the zero Neumann condition. By reflecting $u$ across the flat boundary $T_1$ and then using inf- and sup-convolution arguments in the reflected domain, we obtain the $C^{1,α}$ result. For the general elliptic case, we obtain sharp global $C^{1,α}$ regularity by flattening the boundary and employing compactness arguments together with an ``improvement of flatness'' iteration. With an extra condition in the iteration, we can also deal with the singular case $1<p<2$. In the parabolic setting, the spatial Hölder regularity of $Du$ follows from elliptic estimates combined with the Lipschitz continuity of $u$ in time, which in turn yields joint Hölder continuity in $(x,t)$. Extensions to non-convex domains are also discussed by incorporating a suitable forcing term.

math.AP↗

arXiv2025

On the solvability for a p-k-Hessian inequality

In this paper, we discuss the solvability of a p-k-Hessian entire inequality. We prove that the inequality with sub-lower-critical exponent admits no negative solutions. Moreover, the exponent is sharp. The proof is based on choosing suitable test functions and integrating by parts.

math.AP↗

arXiv2024

A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $Ω\times \R$, where, $A$ is a constant, $Ω$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $Ω$, $V$ and $H$ denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary $\partial Ω\times \R$ with prescribed angle $θ(x)$. Under certain assumptions such as $Ω$ is strictly convex and $\|\cosθ\|_{C^2}$ is small, or $Ω$ is not necessarily convex but $|A|$ is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when $I:= A|Ω|+\int_{\partial Ω} \cosθ(x) dσ>0$ (resp. $=0$, $<0$), the solution $u$ converges as $t\to \infty$ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).

math.DG↗

arXiv2023

The Dirichlet problem of homogeneous complex k-Hessian equation in a (k-1)-pesudoconvex domain with isolated singularity

In this paper, we consider the homogeneous complex k-Hessian equation in $Ω\backslash\{0\}$. We prove the existence and uniqueness of the $C^{1,α}$ solution by constructing approximating solutions. The key point for us is to construct the subsolution for approximating problem and establish uniform gradient estimates and complex Hessian estimates which is independent of the approximation.

math.AP↗

arXiv2023

The Dirichlet problem of the homogeneous $k$-Hessian equation in a punctured domain

In this paper, we consider the Dirichlet problem for the homogeneous $k$-Hessian equation with prescribed asymptotic behavior at $0\inΩ$ where $Ω$ is a $(k-1)$-convex bounded domain in the Euclidean space. The prescribed asymptotic behavior at $0$ of the solution is zero if $k>\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and $-|x|^{\frac{2k-n}{n}}+O(1)$ if $k<\frac{n}{2}$. To solve this problem, we consider the Dirichlet problem of the approximating $k$-Hessian equation in $Ω\setminus \overline{B_r(0)}$ with $r$ small. We firstly construct the subsolution of the approximating $k$-Hessian equation. Then we derive the pointwise $C^{2}$-estimates of the approximating equation based on new gradient and second order estimates established previously by the second author and the third author. In addition, we prove a uniform positive lower bound of the gradient if the domain is starshaped with respect to $0$. As an application, we prove an identity along the level set of the approximating solution and obtain a nearly monotonicity formula. In particular, we get a weighted geometric inequality for smoothly and strictly $(k-1)$-convex starshaped closed hypersurface in $\mathbb R^n$ with $\frac{n}{2}\le k<n$.

math.AP↗

arXiv2022

Serrin-type overdetermined problems for Hessian quotient equations

We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely Hessian quotient equations and Hessian quotient curvature equations. Our approach is based on establishing a Rellich-Pohozaev type identity for Hessian quotient equations and using a P function. Our result generalizes the overdetermined problems for k-Hessian equations and k-curvature equations.

math.AP↗

arXiv2022

The exterior Dirichlet Problem for homogeneous complex $k$-Hessian equation

In this paper, we consider the homogeneous complex k-Hessian equation in an exterior domain $\mathbb{C}^n\setminusΩ$. We prove the existence and uniqueness of the $C^{1,1}$ solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second order estimate.

math.AP↗

arXiv2022

Serrin-type Overdetermined problems in $\mathbb H^n$

In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $σ_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a P function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.

math.AP↗