2015/01/25 by Nikos Katzourakis, Katzourakis, Nikos · 2 citations
Mathematics · #35G50 #Advanced Topology and Set Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Analysis #Primary 35D99 #Secondary 35J70 #advanced mathematical theories #math.AP #msc:35D99 #msc:35G50 #msc:35J70
paper · pdf · doi:10.48550/arxiv.1501.06164
39 pages (Journal of Differential Equations) ; companion paper of arXiv:1502.01179 (Calculus of Variations and PDE)
openalex publication_date 2015/01/25 · arxiv created 2017/02/20 · arxiv updated 2017/02/21 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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 ∞-Laplace system of vectorial Calculus of Variations in L^∞.