On the Characteristic Curvature Operator

TL;DR

This paper introduces the Characteristic Curvature, relates it to classical curvatures, and studies the existence, uniqueness, and regularity of viscosity solutions for associated Dirichlet problems, revealing limitations of comparison principles.

Contribution

It defines the new Characteristic Curvature, establishes fundamental properties, and analyzes the solvability and regularity of related PDEs, highlighting novel theoretical insights.

Findings

Existence and uniqueness of viscosity solutions under certain conditions

Lipschitz regularity of solutions with suitable hypotheses

Non-existence of solutions on balls for positive constant curvature

Abstract

We introduce the Characteristic Curvature as the curvature of the trajectories of the hamiltonian vector field with respect to the normal direction to the isoenergetic surfaces and by using the Second Fundamental Form we relate it to the Classical and Levi Mean Curvature. Then we prove existence and uniqueness of viscosity solutions for the related Dirichlet problem and we show the Lipschitz regularity of the solutions under suitable hypotheses. Moreover we prove a non existence result on the balls when the prescribed curvature is a positive constant. At the end we show that neither Strong Comparison Principle nor Hopf Lemma do hold for the Characteristic Curvature Operator.