Examples of Twice Differentiable Functions in $\mathbb{R}^n$ with Continuous Laplacian and Unbounded Hessian

TL;DR

This paper constructs examples of twice differentiable functions in multi-dimensional space that have a continuous Laplacian but an unbounded Hessian, extending to higher derivatives.

Contribution

It provides explicit constructions of functions with continuous Laplacian and unbounded Hessian, demonstrating new phenomena in higher-order differentiability.

Findings

Existence of twice differentiable functions with continuous Laplacian and unbounded Hessian

Construction methods applicable to higher order derivatives

Insights into the relationship between differentiability and second derivatives

Abstract

We construct examples of twice differentiable functions in $\mathbb{R}^n$ with continuous Laplacian and unbounded Hessian. The same construction is also applicable to higher order differentiability.