Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

Abstract

In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator Δpu=∑i∂∂xi|∇u|p−2-0.16em∂u∂xi, that is the p‐Laplacian, and trueΔ̃pu=∑i∂∂xi|∂u∂xi|p−2∂u∂xi, namely the pseudo‐p‐Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic p‐torsional rigidity.

Characterization of ellipsoids through an overdetermined boundary value problem of Monge–Ampère type

“…Secondly the problem admits both radially and non radially symmetric solutions. Such two features can be found in literature for instance in [24,30,32,25,33,3,12,2,16,4,17], where they rarely occur simultaneously and, to our knowledge, not for all dimensions.…”

A sharp weighted anisotropic Poincaré inequality for convex domains

“…We refer the reader, for example, to [CS, FFK] for remarkable examples of convex not even functions in H (R n ). On the other hand, in [VS] some results on isoperimetric and optimal Hardy-Sobolev inequalities for a general function H ∈ H (R n ) have been proved, by using a generalizazion of the so called convex symmetrization introduced in [AFLT] (see also [DG1,DG2,DG3]). …”

On the first Dirichlet Laplacian eigenvalue of regular polygons

“…Without claiming to be exhaustive we remind for instance that in [13,16,18] the author provide upper and lower bounds for convex sets in terms of area and perimeter. The same was done more recently also in [4,6,7,14]. Different classical estimates may also include diameter and inradius like in [17,19] while a different approach consists in restricting the class of sets.…”