  Let F be nonnegative, convex and smooth off a compact set K. We prove that
continuous local minimisers of convex functionals are "very weak" viscosity
solutions in the sense of Juutinen-Lindqvist of the highly singular
Euler-Lagrange PDE expanded. The hypotheses on F do not guarrantee existence of
minimising weak solutions and include the singular p-Laplacian for 1<p<2. A
much deeper converse is also true, if K={0} and extra natural assumptions are
satisfied. Our main advance is that we introduce systematic "flat"
sup-convolution regularisations which apply to general singular nonlinear PDEs
in order to cancel the strong singularity of F. As an application we extend a
classical theorem of Calculus of Variations regarding existence for the
Dirichlet problem. These results extends previous work of Julin-Juutinen and
Juutinen-Lindqvist-Manfredi.
