Remarks on the quadratic Hessian equation

TL;DR

This paper investigates the properties of viscosity solutions to the quadratic Hessian equation, establishing strict convexity conditions, interior regularity estimates, and implications for counterexample strategies in regularity theory.

Contribution

It proves that viscosity solutions cannot touch harmonic functions on minimal surfaces from below and provides interior $C^2$ estimates based on $W^{2,p}$ norms.

Findings

Viscosity solutions exhibit strict 2-convexity.

Interior $C^2$ estimates are established for solutions.

Certain counterexample strategies are ruled out by these results.

Abstract

We prove that viscosity solutions to the quadratic Hessian equation $$\sigma_2(D^2u) = 1$$ cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict $2$-convexity. We also prove an a priori interior $C^2$ estimate in terms of the $W^{2,\,p}$ norm, for any $p > 2$. Finally, we discuss how these results rule out certain strategies for constructing counterexamples to regularity.