Singular solutions of conformal Hessian equation

TL;DR

This paper constructs specific solutions to a conformal Hessian equation in five dimensions that are smooth outside the origin but not more regular, highlighting nuanced regularity properties of such solutions.

Contribution

It demonstrates the existence of solutions with precise regularity characteristics for a conformal Hessian equation in five dimensions, revealing new insights into their regularity behavior.

Findings

Existence of solutions in $C^{1, ext{epsilon}}$ but not in higher regularity classes.

Solutions are analytic outside zero but have limited regularity at the origin.

Highlights the delicate regularity properties of solutions to conformal Hessian equations.

Abstract

We show that for any $\epsilon\in ]0,1[$ there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball $B\subset\R^5$ which belongs to $C^{1,\epsilon} (B)\setminus C^{1,\epsilon+} (B)$.