Quasiconcave solutions to elliptic problems in convex rings

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

“…Then, by (4.1), the point z is also a minimum point for the function u(z, t ) − u * (z), hence, by [2] and [27], there holds…”

“…What we have to do is just to give conditions which exclude that Π becomes negative on the "boundary" of D 2 × (0, ∞). In particular, to avoid that Π become negative as t → ∞, we need to link the large time behavior of u to the solution of the associated stationary problem, whose quasi-concavity has been studied in [2]. Moreover it is useful to introduce the following condition:…”

“…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]).…”

On the convexity of some free boundaries

“…When we presented our results in Firenze on Dec. 12, 2008, Paolo Salani kindly provided us with a copy of [5] and pointed out that our Proposition 3 could also be derived from Prop. 2.2 in [5] in a way similar to the derivation of their Theorem 3.1. For the reader's convenience we now give our original proof.…”

The Concavity of the Gaussian Curvature of the Convex Level Sets of $$p$$ p -Harmonic Functions with Respect to the Height

“…A survey of this subject is given by Kawohl [12]. For more recent related extensions, please see the papers by Bian-Guan-Ma-Xu [2] and Bianchini-Longinetti-Salani [3]. Now, we turn to the curvature estimates of level sets of the solutions of elliptic partial differential equations.…”