A microscopic convexity principle for nonlinear partial differential equations

Parabolic quasi‐concavity for solutions to parabolic problems in convex rings

“…The study of geometric properties (with special regard to concavity and quasi-concavity) of solutions of partial differential equations is a classical subject, especially in the field of elliptic equations, where it is hardly possible to compile a complete bibliography; then we just recall the classical monograph [19] and very recent related results like [1] and [2], and we refer to these for more references. For parabolic equations, fewer results are disposable, as far as we know, even if it is well-known (see for instance [5] and [12]) that log-concavity is preserved by the heat flow and it also seems to rule the large time behavior of solutions to parabolic equations in R N (see [25]).…”

A uniqueness theorem in Kähler geometry

“…The method we adopt to establish the main theorem is the constant theorem for fully nonlinear elliptic partial differential equations. This type of arguments was used in the real cases in recent papers [3,4]. We remark that the curvature conditions imposed on (M, g) in Theorem 1 can be weakened.…”

Convexity Preserving for Fully Nonlinear Parabolic Integro-Differential Equations

“…On the other hand, convexity of solution has important applications in the financial market completeness and super-replication ( [15], [18]). We refer [11,5,4] for applications of convexity principles in differential geometry.…”

“…By continuity, σ l (u ij (x, t)) > 0 in a neighborhood of (x 0 , t 0 ). As in [4], we pick an open neighborhood O ⊂ Ω × [0, T ) of (x 0 , t 0 ), for any (x, t) ∈ O, let G = {n − l + 1, n − l + 2, ..., n} and B = {1, ..., n − l} be the "good" and "bad" sets of indices for eigenvalues of ∇ 2 u(x, t) respectively.…”

“…Again, as in [4], we will avoid to deal with σ l+1 (W ) = 0 by considering for W ǫ for ǫ > 0 sufficient small, with W ǫ = W + ǫI, G ǫ = (λ n−l+1 + ǫ, ..., λ n + ǫ), B ǫ = (λ 1 + ǫ, ..., λ n−1 + ǫ). We note that W ǫ is the Hessian of function u ǫ (x, t) = u(x, t) + ǫ 2 |x| 2 .…”