Containing the extended notes from the Paris lectures held at the Emile Brel Centre of the Henri Poincare Institute, April through July 2002, this volume contains 14 chapters representing recent studies on heat kernels. The studies include research in various situations and using related geometric a

Lectures on isoperimetric and capacitary inequalities in the theory of Sobolev spaces

We consider the problem of maximizing the first non trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock--Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (J. of Inequal. & Appl. 4: 215--240, 1999): we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.

Spectral optimization for the Stekloff--Laplacian: the stability issue

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $$\alpha $$
 , under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $$\gamma $$
 . We show that for a wide class of density functions the energy admits a minimizer for any value of $$\gamma $$
 . Moreover these minimizers are bounded. For monomial densities of the form $$|x|^p$$
 we prove that when $$\gamma $$
 is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the $$\gamma \rightarrow 0$$
 limit corresponds, under a suitable rescaling, to a small mass $$m=|\Omega |\rightarrow 0$$
 limit when $$p<d-\alpha +1$$
 , but to a large mass $$m\rightarrow \infty $$
 for powers $$p>d-\alpha +1$$
 .

A nonlocal isoperimetric problem with density perimeter

We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 0$). The implicit unjustified assumption is that the generating curve is convex.

Balls Isoperimetric in $\mathbb{R}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$

In this paper the author studies the isoperimetric problem in $\re^n$ with perimeter density $|x|^p$ and volume density $1.$ We settle completely the case $n=2,$ completing a previous work by the author: we characterize the case of equality if $0\leq p\leq 1$ and deal with the case $-\infty<p<-1$ (with the additional assumption $0\in\Omega$). In the case $n\geq 3$ we deal mainly with the case $-\infty<p<0,$ showing among others that the results in $2$ dimensions do not generalize for the range $-n+1<p<0.$

On the isoperimetric problem with perimeter density r^p

We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial \Omega } |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Szego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.

A. Alvino

Papers

Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $