# On a vector-valued generalisation of viscosity solutions for general PDE systems

## TL;DR

This paper introduces a new theory of viscosity solutions for fully nonlinear PDE systems, extending scalar concepts to vector-valued functions with a novel extremum notion.

## Contribution

It develops a vector-valued viscosity solution framework using a new extremum concept, enabling analysis of nonlinear PDE systems beyond scalar cases.

## Key findings

- Introduces a notion of extremum for vector maps.
- Supports stability and convergence results similar to scalar viscosity solutions.
- Lays groundwork for future applications in PDE systems.

## Abstract

We propose a theory of non-differentiable solutions which applies to fully nonlinear PDE systems and extends the theory of viscosity solutions of Crandall-Ishii-Lions to the vectorial case. Our key ingredient is the discovery of a notion of extremum for maps which extends min-max and allows "nonlinear passage of derivatives" to test maps. This new PDE approach supports certain stability and convergence results, preserving some basic features of the scalar viscosity counterpart. In this first part of our two-part work we introduce and study the rudiments of this theory, leaving applications for the second part.