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

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

Monotonicity, continuity and differentiability results for the LpL^p Hardy constant

We consider the $L^p$ Hardy inequality involving the distance to the boundary for a domain in the $n$-dimensional Euclidean space. We study the dependence on $p$ of the corresponding best constant and we prove monotonicity, continuity and differentiability results. The focus is on non-convex domains in which case such constant is in general not explicitly known.Comment: 12 pages; to appear in the Israel Journal of Mathematic

The Hardy constant: a review

We present a review of results that have been obtained in the past twenty-five years concerning the $L^p$-Hardy inequality with distance to the boundary. We concentrate on results where the best Hardy constant is either computed exactly or estimated from below.Comment: 7 pages. To appear in the volume "Modern Problems in PDEs and Applications - Extended Abstracts of the 2023 Ghent Analysis and PDE Center Summer School", Research Perspectives Ghent Analysis and PDE Center, Trends in Mathematics, Springer. Editors: M. Chatzakou, J. Restrepo, M. Ruzhansky, B. Torebek, K. Van Bocksta

Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega$ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega)$ of $\Omega$ obtained by means of a locally Lipschitz homeomorphism $\phi$ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $\phi$. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $\phi$. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains.Comment: 25 page