Francesco Maggi

The sharp Sobolev inequality in quantitative form

A quantitative version of the sharp Sobolev inequality in W (R), 1 < p < n, is established with a remainder term involving the distance from extremals.

Some methods for studying stability in isoperimetric type problems

We review the method of quantitative symmetrization inequalities introduced in Fusco, Maggi and Pratelli, “The sharp quantitative isoperimetric inequality”, Ann. of Math. Introduction Our aim is to introduce certain ideas related to the quantitative study of isoperimetric type inequalities. The main efforts are devoted to the proof of a sharp quantitative version of the Euclidean isoperimetric inequality, first proved in [FMP1] improving on previous work by Hall, Hayman and Weitsman [HHW, Ha]. The general scheme of proof remains the same, but at different steps it has been possible to simplify the original arguments. Moreover, we have occasionally opted for a rather informal style of presentation, hoping to provide a broadly and easily accessible account of these techniques. There are many relevant variational problems [PS, Ka] intimately related to the same symmetrization inequalities studied here from a quantitative point of view, and these ideas have already been shown to be flexible enough to be employed in the study of quantitative forms of other variational problems, like sharp Sobolev, Faber-Krahn, isocapacitary, and Cheeger inequalities [FMP2, CFMP1, FMP3]. An even more surprising application is found in the quantitative study of the Gaussian isoperimetric inequality [CFMP2], where natural variants of these arguments still play a prominent role. Starting from the seminal paper by Brezis and Lieb [BL], the reader will find various other contributions on quantitative geometric-functional inequalities based on similar or related tools [BE, Ci1, Ci2, CEFT, EFT, AFN]. This work covers the material taught in the first part of a Ph.D. course held by the author at the Università di Napoli “Federico II” during May 2007. In the second part of that course we introduced a different approach to the quantitative isoperimetric inequality, where symmetrizations were dropped out in favor of a heavier use of transportation maps. This method allows us to attack problems where minimizers have no particular symmetry, such as in the case of the anisotropic isoperimetric inequality (4.4). This point of view is developed in the forthcoming paper [FiMP], and therefore it is not presented in here. Received by the editors August 29, 2007. 2000 Mathematics Subject Classification. Primary 49Q20. c ©2008 American Mathematical Society Reverts to public domain 28 years from publication

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝn. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].

A note on Cheeger sets

Starting from the quantitative isoperimetric inequality, we prove a sharp quantitative version of the Cheeger inequality.

Rigorous derivation of Föppl's theory for clamped elastic membranes leads to relaxation

We consider the nonlinear elastic energy of a thin membrane whose boundary is kept fixed, and assume that the energy per unit volume scales as

Stability for the Brunn-Minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry

h^{\beta}

Isoperimetry with upper mean curvature bounds and sharp stability estimates

, with h the film thickness and

Rigidity of equality cases in Steiner's perimeter inequality

\beta \in (0,4)

Bubbling with L2-Almost Constant Mean Curvature and an Alexandrov-Type Theorem for Crystals

. We derive, by means of Γ convergence, a limiting theory for the scaled displacements, which takes a form similar to the one proposed by Foppl in 1907. Our variational approach fully incorporates the possibility of buckling already observed during the derivation of the reduced two‐dimensional theory. At variance with Foppl’s, our limiting model is lower semicontinuous and has an energetics that vanishes on all contractions. Therefore buckling does not need to be explicitly resolved when computing with the reduced theory. If forces normal to the membrane are included, then our result predicts that the normal displacement scales as the cube root of the force. This scaling depends crucially on the clamped boundary conditions. Indeed, if the boundary is left free, then a much softer response is obtained, as w...

A geometric approach to correlation inequalities in the plane

We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.