# Intuition on viscosity solutions [duplicate]

I would tou like to investigate two things:

I) What is the motivation for the study of viscosity solutions of elliptic linear (fully non-linear) Pde's

II) The relationship between viscosity solutions and distributional solutions.

For example, distributional solutions are defined a.e and contains the space of classical solutions. The viscosity solutions also enjoys the same property and with good coefficients are also classical solutions. There exists relation of inclusion between the set of viscosity and weak solutions ? $$Lu=A_{ij}D_{ij}u$$?