Exotic PDE's

TL;DR

This paper introduces exotic differential equations within algebraic topology of PDEs, proving existence, stability, and notably the smooth 4-dimensional Poincaré conjecture, advancing understanding of exotic spheres in differential equations.

Contribution

It establishes the existence and stability of exotic PDEs and proves the smooth 4-dimensional Poincaré conjecture within this framework, completing previous theoretical results.

Findings

Proved the smooth 4-dimensional Poincaré conjecture.

Established local and global existence theorems for exotic PDEs.

Demonstrated stability theorems for equations involving exotic spheres.

Abstract

In the framework of the PDE's algebraic topology, previously introduced by A. Pr\'astaro, are considered {\em exotic differential equations}, i.e., differential equations admitting Cauchy manifolds $N$ identifiable with exotic spheres, or such that their boundaries $\partial N$ are exotic spheres. For such equations are obtained local and global existence theorems and stability theorems. In particular the smooth ($4$-dimensional) Poincar\'e conjecture is proved. This allows to complete the previous Theorem 4.59 in \cite{PRA17} also for the case $n=4$.