Convex viscosity solutions and state constraints

Concavity maximum principle for viscosity solutions of singular equations

“…Observe that it is required that u is a supersolution in the closure Ω; this is a much stronger condition than u being a supersolution in Ω. The assumptions on F in [1] differ slightly from ours. For example, they do not allow for singular equations and, instead of the (strict) monotonicity of F with respect to u, assume that the comparison principle holds; see e.g.…”

“…Remark 3.6. In [1], the convexity of solutions with state constraints boundary conditions is proved under the assumption that (x, r, P ) → F (x, r, ξ, P −1 ) is concave; (3.18) in view of the convexity of P → P −1 (see [1]), this condition is weaker than the concavity of (x, r, P ) → F (x, r, ξ, P ). It is natural to ask whether our method could be extended to cover this case as well.…”

“…This idea was used already by Brascamp and Lieb [2], but from our point of view the papers [13] and [14] of Korevaar are the most influential ones. Korevaar proved, roughly speaking, that the "concavity function" and the fact that if u is a viscosity supersolution to F (x, u, Du, D 2 u) = 0 in Ω, then the convex envelope of u (i.e., the largest convex function below u) is also a supersolution in Ω (see [1,Prop. 3]).…”

$W^{2,\infty }$ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

“…From Lemma 1, p. 268 in [1], this inequality implies that u(t, x) + 1 2 C(t)x 2 is x-convex, which again is equivalent to the distributional inequality in the proposition.…”

Long-time behavior of solutions to Hamilton–Jacobi equations with quadratic gradient term

“…Hence, by Proposition 3 of [1], the convex envelope w * * of w is a viscosity supersolution of (4.8). Hence, w * * (x)−K |x| 2 is a viscosity supersolution of (3.4).…”