  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.
