arXiv subjects

Zhenyu Fan

Publications and source records attributed to Zhenyu Fan.

5 recordsLinked to original sources

arXiv2026

Geometric regularity for semi-convex viscosity solutions to the special Lagrangian equation

We prove that the gradient graph of any $W^{2,1}$ semi-convex viscosity solution to the three-dimensional subcritical special Lagrangian equation is area-minimizing and smooth. Furthermore, we prove the smoothness of $W^{2,1}\cap C^{1,1/3+}$ semi-convex viscosity solutions. This yields progress on the conjecture proposed by Wang--Yuan [WY13]. The corresponding results in higher dimensions are also established.

math.AP↗

arXiv2026

Failure of Interior H\"older and Gradient Estimates for the Subcritical Special Lagrangian Equation

For every $n\ge3$, we show, by a family of explicit entire real-analytic solutions, that, for every subcritical phase and every $0<\alpha\le1$, oscillation-based interior $C^\alpha$ estimates fail for the special Lagrangian equation in $\mathbb{R}^n$. In fact, this family has no common interior modulus of continuity.

math.AP↗

arXiv2026

Interior estimates and regularity for the scalar curvature equation in dimension 4

We establish a priori interior curvature estimates for the scalar curvature equation in dimension 4. Our approach relies on the doubling method introduced by Shankar and Yuan [SY25]. Key to the proof are an a priori doubling inequality under a small gradient assumption and the Alexandrov regularity for viscosity solutions; the latter is obtained via a new iterative rotation argument. Furthermore, we establish interior regularity for $C^1$ viscosity solutions.

math.AP↗

arXiv2025

Hessian Estimates for the Sigma-2 Equation with Variable Right-Hand Side Terms in Dimension 4

We derive a priori interior Hessian estimates and regularity for the sigma-2 Hessian equation $\sigma_{2}(D^2u)=f(x,u,Du)$ with positive $C^{1,1}$ right hand side in dimension 4. In higher dimensions, the same result holds under an additional dynamic semi-convexity condition on solutions. This generalizes Qiu's and Shankar-Yuan's results.

math.AP↗

arXiv2025

A generalization of Savin's small perturbation theorem for fully nonlinear elliptic equations and applications

In this note, we generalize Savin's small perturbation theorem to nonhomogeneous fully nonlinear equations $F(D^2u, Du, u,x)=f$ provided the coefficients and the right-hand side terms are H\"older small perturbations. As an application, we establish a partial regularity result for the sigma-$k$ Hessian equation $\sigma_{k}(D^2u)=f$.

math.AP↗