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The classical concept was that a PDE
: H(x,u,Du) = 0
over a domain x\in\Omega has a solution if we can find a Function ''u''(''x'') continuous and differentiable over the entire domain such that ''x'', ''u'' and ''Du'' (the differential of ''u'') satisfy the above equation at every point.

Under the viscosity solution concept, ''u'' need not be everywhere differentiable. There may be points where ''Du'' does not exist, i.e. there could be a kink in ''u'' and yet ''u'' satisfies the equation in an appropriate sense. Although ''Du'' may not exist at some point, the ''superdifferential'' D^+ u and the ''subdifferential'' D^- u, to be defined below, are used in its place.