# Generalized ergodic problems: existence and uniqueness structures of solutions

## TL;DR

This paper investigates the existence and uniqueness of solutions to a generalized ergodic Hamilton-Jacobi equation on the torus, revealing conditions for existence and analyzing non-uniqueness in convex cases using the nonlinear adjoint method.

## Contribution

It establishes existence results for the generalized ergodic problem and analyzes the solution structure and non-uniqueness in convex settings with the nonlinear adjoint method.

## Key findings

- Existence of solutions under broad assumptions.
- Examples demonstrating non-uniqueness of solutions.
- Uniqueness structures characterized in convex cases.

## Abstract

We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.