Gaussian estimates with best constants for higher-order Schr\"odinger operators with Kato potentials

We establish Gaussian estimates on the heat kernel of a higher-order uniformly elliptic Schr\"odinger operator with variable highest order coefficients and with a Kato class potential. The estimates involve the sharp constant in the Gaussian exponent.Comment: 9 pages; a mistaken example has been removed; to appear in Proc. Amer. Math. So

Higher order linear parabolic equations

We first highlight the main differences between second order and higher order linear parabolic equations. Then we survey existing results for the latter, in particular by analyzing the behavior of the convolution kernels. We illustrate the updated state of art and we suggest several open problems.Comment: Dedicated to Patrizia Pucci. To appear in the the Contemporary Mathematics series of the AM

Series expansion for L^p Hardy inequalities

We consider a general class of sharp $L^p$ Hardy inequalities in $\R^N$ involving distance from a surface of general codimension $1\leq k\leq N$. We show that we can succesively improve them by adding to the right hand side a lower order term with optimal weight and best constant. This leads to an infinite series improvement of $L^p$ Hardy inequalities.Comment: 16 pages, to appear in the Indiana Univ. Math.

Shape sensitivity analysis of the Hardy constant

We consider the Hardy constant associated with a domain in the $n$-dimensional Euclidean space and we study its variation upon perturbation of the domain. We prove a Fr\'{e}chet differentiability result and establish a Hadamard-type formula for the corresponding derivatives. We also prove a stability result for the minimizers of the Hardy quotient. Finally, we prove stability estimates in terms of the Lebesgue measure of the symmetric difference of domains.Comment: 23 pages; showkeys command remove