Construction of Twice Differentiable Functions with Continuous Laplacian and Bounded Hessian

TL;DR

This paper constructs examples of twice differentiable functions with continuous Laplacian and bounded Hessian, extending the approach to higher order derivatives and geometric PDEs like the Monge-Ampère and mean curvature equations.

Contribution

It provides explicit constructions of smooth functions with specific regularity properties relevant to differential equations and geometric analysis.

Findings

Examples of twice differentiable functions with continuous Laplacian

Construction methods applicable to higher order derivatives

Extensions to Monge-Ampère and mean curvature equations

Abstract

We construct examples of twice differentiable functions in $\mathbb{R}^n$ with continuous Laplacian and bounded Hessian. The same construction is also applicable to higher order differentiability, the Monge-Amp\`ere equation, and mean curvature equation for hypersurfaces.