Zhenyu Fan

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math.AP2026

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

Zhenyu Fan

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. Fur…

math.AP2026

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

Zhenyu Fan, Ravi Shankar

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…

math.AP2025

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

Zhenyu Fan

We derive a priori interior Hessian estimates and regularity for the sigma-2 Hessian equation $σ_{2} (D^{2} u) = f (x, u, D u)$ with positive $C^{1, 1}$ right hand side in dimension 4. In hig…

math.AP2025

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

Zhenyu Fan

In this note, we generalize Savin's small perturbation theorem to nonhomogeneous fully nonlinear equations $F (D^{2} u, D u, u, x) = f$ provided the coefficients and the right-hand side te…