ASmooth test functions touching the solution from above (subsolution) or below (supersolution)
BIntegration against test functions (as in weak solutions)
CLimits of smooth approximate solutions
DCharacteristic curves of the PDE
The viscosity definition replaces derivatives of u (which may not exist) with derivatives of smooth test functions that touch u tangentially. If φ ≥ u near x₀ with φ(x₀) = u(x₀), then Dφ and D²φ at x₀ serve as 'proxy derivatives' for u.
Question 2 True / False
Viscosity solutions require the solution to be differentiable.
TTrue
FFalse
Answer: False
Viscosity solutions need only be continuous—no derivatives of any order are required. The definition cleverly extracts differential information from smooth functions that touch u, bypassing the need for u itself to be differentiable.
Question 3 Short Answer
For which class of equations are viscosity solutions particularly important?
Think about your answer, then reveal below.
Model answer: Fully nonlinear PDEs (like Hamilton-Jacobi equations and Bellman equations) that lack divergence structure
The weak formulation (integration by parts) requires the PDE to be in divergence form. Fully nonlinear equations F(x, u, Du, D²u) = 0 generally are not, so Sobolev-space weak solutions are not available. Viscosity solutions fill this gap.
Question 4 Multiple Choice
The comparison principle for viscosity solutions states that if u is a subsolution and v is a supersolution with u ≤ v on the boundary, then:
Au ≤ v throughout the domain
Bu = v throughout the domain
Cu and v differ by a constant
Du + v is a solution
The comparison principle is the fundamental uniqueness tool for viscosity solutions. It says subsolutions lie below supersolutions, which immediately gives uniqueness (if u is both a sub- and supersolution with the same boundary data, comparison gives u ≤ u, i.e., uniqueness) and stability.