# Connecting orbits in Hilbert spaces and applications to P.D.E

## TL;DR

This paper establishes a general theorem on heteroclinic orbits in Hilbert spaces and applies it to prove existence results for certain PDE solutions, including phase transition models and fourth-order PDEs.

## Contribution

It introduces a new general theorem on heteroclinic orbits in Hilbert spaces and demonstrates its application to PDE problems, extending previous results.

## Key findings

- New proof of heteroclinic double layers in a more general setting
- Existence of solutions to a fourth-order PDE with specific boundary conditions
- Method to reduce PDE solutions to heteroclinic orbits in Hilbert spaces

## Abstract

We prove a general theorem on the existence of heteroclinic orbits in Hilbert spaces, and present a method to reduce the solutions of some P.D.E. problems to such orbits. In our first application, we give a new proof in a slightly more general setting of the heteroclinic double layers (initially constructed by Schatzman), since this result is particularly relevant for phase transition systems. In our second application, we obtain a solution of a fouth order P.D.E. satisfying similar boundary conditions.