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Yasheng Lyu

Publications and source records attributed to Yasheng Lyu.

6 recordsLinked to original sources
arXiv2026

Global $W^{2,p}$ and $C^{2,α}$ regularity for the sigma-$2$ equation

In this paper, we prove that continuous $2$-convex solutions of the $σ_{2}$ equation with positive continuous density on bounded strictly mean-convex $C^{3}$ domains with $C^{3}$ boundary values belong to $W^{2,p}$ for every $1<p<\infty$ in every dimension. If, in addition, the density is $C^α$, $0<α<1$, the solutions belong to $C^{2,α}$. Moreover, two counterexamples show that neither the $C^{3}$ assumption on the domain nor that on the boundary values can in general be weakened to $C^{2,1}$.

math.AP↗
arXiv2026

Interior $W^{2,p}$ regularity for the sigma-$2$ equation

We prove that continuous $2$-convex solutions of the $σ_{2}$ equation with a density bounded above and below by positive constants belong to $W^{2,1}_{\mathrm{loc}}$ in every dimension. If, in addition, the density is continuous, the solutions belong to $W^{2,p}_{\mathrm{loc}}$ for every $1<p<\infty$.

math.AP↗
arXiv2026

Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations

It is known that the complex $k$-Hessian equation admits almost $C^{1,1}$ regularity (i.e., $\supΔu<\infty$) and the Christoffel-Minkowski equation admits $C^{1,1}$ regularity under the sharp degenerate condition $f^{1/(k-1)}\in C^{1,1}$ for a nonnegative right-hand side $f$. Assuming instead the alternative sharp degenerate condition $f^{3/(2k-2)}\in C^{2,1}$, we prove almost $C^{1,1}$ regularity for the complex $k$-Hessian equation when $k\geq5$ and $C^{1,1}$ regularity for the Christoffel-Minkowski equation. The argument deeply exploits various concavity properties of the operators under the stronger regularity assumption on $f$.

math.AP↗
arXiv2025

On the Dirichlet problem for the degenerate $k$-Hessian equation

This paper investigates the existence of a global $C^{1,1}$ solution to the Dirichlet problem for the $k$-Hessian equation with a nonnegative right-hand side $f$, focusing on the required conditions for $f$. The conditions $f^{1/(k-1)}\in C^{1,1}(\overline{Ω_{0}})$ and $f^{3/(2k-2)}\in C^{2,1}(\overline{Ω_{0}})$, together with $f\geq0$ in a domain $Ω_{0}\SupsetΩ$, are optimal, as demonstrated by classical counterexamples. For the Monge-Ampère equation ($k=n$), we establish the existence under the optimal condition $f^{3/(2n-2)}\in C^{2,1}(\overline{Ω_{0}})$ together with $f\geq0$ in $Ω_{0}$. For the general $k$-Hessian equation, we establish the existence under the condition $f\geq0$ in $Ω_{0}$ together with one of the following three conditions: \begin{align*} &(1)\quad f^{1/(k-1)}\in C^{1,1}(\overline{Ω_{0}}),\ \ \inf_ΩΔu\geq1,\ \ 2\leq k\leq n-1;\\ &(2)\quad f^{3/(2k-2)}\in C^{2,1}(\overline{Ω_{0}}),\ \ \inf_ΩΔu\geq1,\ \ 5\leq k\leq n-1;\\ &(3)\quad f^{3/(2k)}\in C^{2,1}(\overline{Ω_{0}}),\ \ 2\leq k\leq n-1. \end{align*}

math.AP↗
arXiv2025

A new proof of Hölder estimates for the gradient of quasilinear elliptic equations

In this paper, we give a new proof of Hölder estimates for the gradient of quasilinear elliptic equations, using a covering method inspired by the proof of Evans-Krylov theorem for fully nonlinear elliptic equations. Moreover, Hölder estimates for the gradient of fully nonlinear elliptic equations are also obtained by the same method.

math.AP↗