Existence and Blow-Up Behavior for Solutions of the Generalized Jang Equation

TL;DR

This paper thoroughly investigates the generalized Jang equation, establishing existence, regularity, and blow-up behavior, including asymptotics and non-uniqueness of solutions, to aid in understanding the Penrose inequality in general relativity.

Contribution

It provides new existence, regularity, and blow-up results for the generalized Jang equation, including detailed asymptotics and non-uniqueness analysis.

Findings

Existence and regularity of solutions established.

Precise asymptotics for blow-up behavior derived.

Blow-up solutions are shown to be non-unique.

Abstract

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up behavior are given, and it is shown that blow-up solutions are not unique.