Author

Zhen-Yu Fan

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Preprint Sep 2026

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.

Zhen-Yu Fan, Cai-Yan Li, Zhe-Hui Wang · 2 citations

Preprint Aug 2026

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 soluti...

Zhen-Yu Fan, Ravi Shankar · 3 citations

Preprint Sep 2026

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 propos...

Zhen-Yu Fan · 0 citations