On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations

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Hölder estimates for degenerate elliptic equations with coercive Hamiltonians (2010, background): “This fundamental diﬀerence between the cases p ≤ 2 and p > 2 reﬂects a similar feature occurring in the study of the exit-time problem (see [4]); namely, if 1 < p ≤ 2, then the Dirichlet problem can be solved (in the classical sense) for any boundary datum ϕ , while if p > 2, this is no longer true…”
On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton–Jacobi equations (2009, methodology): “It is standard to show that this problem has a unique solution u : × [0,+∞) → R and, as long as one does not need regularity properties, proofs of this fact are easy by using viscosity solutions theory (see Barles and Da Lio [ 4 ] and references therein).”
Viscosity solutions of general viscous Hamilton–Jacobi equations (2013, background): “See also Barles and Da Lio [6] for results on state constraints problems as well as problems with more general boundary conditions.”
Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains (2010, background): “Existence, uniqueness and regularity of weak solutions of problem (1.5)‐(1.7) with irregular initial data are studied in [1,8,11,22] and references therein. Viscosity solutions are considered in [ 6 ].”