Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems

The Brunn–Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators

“…The above theorem can be read as an extension to viscosity solutions of general fully nonlinear operators of the result proved by Sakaguchi in [64] for the p-Laplacian (see also [57]) and by Bianchini and Salani in [10] for a general class of operators including the ones considered here. Part (i) of the statement is obtained essentially via the convex envelope method of Alvarez-Lasry-Lions, whereas, for part (ii), we use our afore mentioned existence result (Theorem 19), which involves an approximation argument with smooth domains.…”

Concave power solutions of the dominative p-Laplace equation

“…Proof. Since u is, in particular a viscosity supersolution to −Δ N p u = 1, Lemma 3.2 in [12] gives the result.…”

“…We mention some results in [12] obtained for the normalized p-Laplace equation, which we will use together with the relationship between the normalized p-Laplace equation and the Dominative p-Laplace equation. The following Lemma will be applied in the proof of the concavity, and it relies on the fact that the mapping (q, A) → q, A −1 q is convex in S + for each q ∈ R n .…”

Concavity properties for solutions to $p$-Laplace equations with concave nonlinearities

“…Apart from Korevaar-Kennington approach, another fruitful method is to consider the concave envelope, as done in the seminal work [1] in the framework of viscosity solutions of fully nonlinear equations. This technique has been applied in [11,12,18,20,21,27]. Other close approaches are the quasi-concave envelope method and its modifications [4,10,13] and the microscopic concavity principle initiated in [8] and developed in [3,26].…”