Aug 2026· 1 citation· ⚡ 1 influential· 59 references
Mathematics
Abstract
In this note, we prove a priori interior curvature bounds for strictly convex solutions to elliptic Weingarten curvature quotient equations. The proof does not employ the advanced methods involving integral estimates or compactness arguments. Instead, it relies on a concavity inequality for the equation operator and a novel choice of the auxiliary function to carry out an elementary maximum-principle argument. Interior curvature estimates for strictly convex solutions to the special Lagrangian curvature equation in low dimensions also follow as a consequence.
We resolve the long-standing problem of establishing interior \(C^2\) estimates for admissible solutions of the graphical scalar curvature equation in every dimension \(n\ge 3\). More precisely, we prove interior curvature estimates for admissible solutions to the constant graphical scalar curvature equation. The proof...
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...
We prove a concavity inequality for the Hessian quotient operator $\sigma_k/\sigma_{k-1}$ using a change of basis for symmetric polynomials. This removes an additional structural concavity assumption previously imposed in the interior $C^2$ estimate for convex solutions of the $\sigma_k/\sigma_{k-1}$ equation. The meth...
We establish new global regularity estimates for the Hessian of viscosity solutions to non-convex fully nonlinear elliptic equations with oblique boundary conditions in the setting of weighted Lorentz and Morrey-Lorentz spaces. More precisely, our approach relies on an analysis of the ellipticity coefficients of the go...
Junior da Silva Bessa· Analysis and Applications· 0 citations
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...
Establishing global curvature estimates for $k$-convex solutions of prescribed curvature equations is a longstanding problem in fully nonlinear partial differential equations and geometric analysis. We develop a new approach that combines fractional-linear transformations with the logarithmic concavity of hyperbolic po...
Feng-Rui Yang· 0 citations
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