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

2018/12/25 by Nikos Katzourakis, Katzourakis, Nikos
Mathematics · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Analysis #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1812.10069

openalex publication_date 2018/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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