Interior Hessian estimates for Sigma-2 equations in dimension three

Abstract

We prove a priori interior C 2 estimate for σ 2 = f in R 3 , which generalizes Warren-Yuan's result [19].

Hessian estimate for semiconvex solutions to the sigma-2 equation

“…Note that, in three dimensions, the Jacobi inequality for the log-convex b = ln △u = ln 2 + |λ| 2 (with ε = 1/100) was derived by Qiu [Q,Lemma 3] for solutions to (2.1) along with variable right hand side; and the Jacobi inequality with ε = 1/3 for the log-max b = ln λ max (with ε = 1/3) was derived for solutions to (2.1) in [WY,Lemma 2.2].…”

The Dirichlet Problem of Hessian Equation in Exterior Domains

“…Warren and Yuan [44] established the interior estimate for σ 2 (D 2 u) = 1 when n = 3. For smooth f > 0 and n = 3, the second named author [34,35] proved the interior estimates for 2-Hessian equations and scalar curvature equations. Recently, Shankar and Yuan [38] derived a priori interior Hessian estimate and interior regularity for σ 2 (D 2 u) = 1 in dimension four.…”

“…The question of whether viscosity solutions to (1.1) become smooth in dimensions n ≥ 5 remains a famous open problem in the field. The most crucial ingredient for obtaining the interior estimate is the validity of the Jacobi inequality when n = 3 and 4, as demonstrated in [44,34,38]. But the crucial Jacobi inequality does not hold when n ≥ 5, see [38].…”

Hessian Estimates for Lagrangian mean curvature equation

“…Bao-Chen-Guan-Ji [BCGJ03] obtained pointwise Hessian estimates for strictly convex solutions to quotient equations σ n = σ k in terms of certain integrals of the Hessian. Recently, along the integral way, Qiu [Qiu17] proved Hessian estimates for solutions of the three dimensional quadratic Hessian equation with a C 1,1 variable right hand side. Hessian estimates for convex solutions of general quadratic Hessian equations were obtained via a new pointwise approach by Guan-Qiu [GQ19].…”