Generalised Solutions for Fully Nonlinear PDE Systems and Existence-Uniqueness Theorems

TL;DR

This paper develops a new theory of generalized solutions for fully nonlinear PDE systems that accommodates merely measurable maps, avoiding traditional distribution-based methods and employing probabilistic and Young measure techniques.

Contribution

It introduces a novel framework for solving fully nonlinear PDE systems using probabilistic derivatives and Young measures, extending solution concepts to measurable maps and proving key existence and uniqueness results.

Findings

Established existence and uniqueness for solutions to fully nonlinear degenerate elliptic systems.

Proved partial regularity of solutions to certain PDE systems.

Demonstrated existence of solutions to the vectorial infinity-Laplace system.

Abstract

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of Distributions to PDEs and is not based on either integration by parts or on the maximum principle. Instead, our starting point builds on the probabilistic representation of derivatives via limits of difference quotients in the Young measures over a toric compactification of the space of jets. After developing some basic theory, as a first application we consider the Dirichlet problem and we prove existence-uniqueness-partial regularity of solutions to fully nonlinear degenerate elliptic 2nd order systems and also existence of solutions to the $\infty$-Laplace system of vectorial Calculus of Variations in $L^\infty$.