arXiv subjects

Dekai Zhang

Publications and source records attributed to Dekai Zhang.

28 records · Page 2Linked to original sources

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↗

arXiv2023

The Monge-Ampère equation for $(n-1)$-quaternionic PSH functions on a hyperKähler manifold

We prove the existence of unique smooth solutions to the quaternionic Monge-Ampère equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperKähler manifold and thus obtain solutions for the quaternionic form type equation. We derive $C^0$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without assuming the flatness of underlying hyperKähler metric comparing to previous results [14].

math.DG↗

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

A Federated Cox Model with Non-Proportional Hazards

Recent research has shown the potential for neural networks to improve upon classical survival models such as the Cox model, which is widely used in clinical practice. Neural networks, however, typically rely on data that are centrally available, whereas healthcare data are frequently held in secure silos. We present a federated Cox model that accommodates this data setting and also relaxes the proportional hazards assumption, allowing time-varying covariate effects. In this latter respect, our model does not require explicit specification of the time-varying effects, reducing upfront organisational costs compared to previous works. We experiment with publicly available clinical datasets and demonstrate that the federated model is able to perform as well as a standard model.

cs.LG↗

arXiv2020

The Neumann problem of Hessian quotient equations

In this paper, we obtain some important inequalities of Hessian quotient operators, and global $C^2$ estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problem of Hessian quotient equations.

math.AP↗

arXiv2016

Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature

For the minimal graph defined on a convex ring in the space form with nonnegative curvature, we obtain the regularity and the strict convexity about its level sets by the continuity method.

math.AP↗

arXiv2016

Gradient Estimates of Mean Curvature Equation with Neumann Boundary Condition in $ M^n\times \mathbb{R}$

In this note, we study the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifold $ M^n\times \mathbb{R}$. The main goal is to establish the boundary gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

math.DG↗

arXiv2015

Hessian equations on closed Hermitian manifolds

In this paper, using the technical tools in \cite{TW5}, we solve the complex Hessian equation on closed Hermitian manifolds, which generalizes the the Kähler case results in \cite{HMW} and \cite{DK}.

math.DG↗