Generalised Solutions for Fully Nonlinear PDE Systems and Existence Theorems

Nikos Katzourakis

Published 2015-01-25Version 1

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows the interpretation of merely measurable maps as solutions without any further a priori regularity requirements. This approach bypasses the standard problems arising by the application of Distributions to PDEs and is not based on either duality or on integration by parts. Instead, the starting point builds on the probabilistic interpretation of limits of difference quotients via Young (parameterised) measures over compactifications of the "state space". After developing some basic theory, as a first application we prove existence of solution to the Dirichlet problem for the $\infty$-Laplace system of vectorial Calculus of Varations in $L^\infty$ and also for fully nonlinear degenerate elliptic 2nd order hessian systems.