Nonsmooth Convex Functionals and Feeble Viscosity Solutions of singular Euler-Lagrange Equations

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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.