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

Abstract

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in R n satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) p-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope method by Alvarez-Lasry-Lions. We also deal with the existence and log-concavity of positive viscosity eigenfunctions.

Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations

“…Let us emphasize that it is indeed possible to investigate spatial convexity of solutions in the framework of viscosity theory; we refer to [17,19,1,33,40] for viscosity techniques in different contexts and to [37,38,35,6,25,26,23,24] etc for related results for classical solutions. Our current work provides new results on parabolic power concavity of viscosity solutions, which are not considered in the aforementioned references (but let us point out that, right after completing this work, we have learnt also about [14], where viscosity solutions have been now considered to study Brunn-Minkowski type inequalities for the eigenvalues of fully nonlinear homogeneous elliptic operators).…”

Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

“…Such a supersolution preserving property enables us to obtain the convexity of the solution immediately if the comparison principle for the equation is known to hold. We refer the reader also to [23] and recent work [21,11] for more applications of this method in the Euclidean space.…”

Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

“…Such a supersolution preserving property enables us to obtain the convexity of the solution immediately if the comparison principle for the equation is known to hold. We refer the reader also to [22] and recent work [20,10] for more applications of this method in the Euclidean space.…”