Research

My research interests are in Mathematical Analysis, in particular Partial Differential Equations (PDEs), Function Spaces, Functional Analysis, Spectral Theory, Asymptotic Analysis, Control Theory. I am mainly interested in spectral perturbation problems for elliptic operators.

I wrote my bachelor thesis, "Sulla dipendenza di una membrana vibrante dalla densità di massa" in 2009 under the supervision of prof. Pier Domenico Lamberti.

I wrote my master thesis, "Eigenvalues of harmonic and poly-harmonic operators subject to mass density perturbations" in 2012, again under the supervision of prof. Pier Domenico Lamberti.

I wrote my Ph.D thesis, "On mass distribution and concentration phenomena for linear elliptic partial differential operators" in 2012, under the supervision of prof. Pier Domenico Lamberti and Matteo Dalla Riva.

We characterized critical points of the elementary symmetric functions of the eigenvalues of general elliptic operators of even order and homogeneous boundary conditions under mass density perturbations preserving the total mass. In some cases we obtained results of non-existence of such critical densities under the sole fixed mass constraint.

We investigated the asymptotic behavior of the eigenvalues of the Neumann problem for the Laplace operator when the density concentrates in a neighborhood of the boundary and found explicit formulas for the topological derivative.

We introduced a new Steklov-type eigenvalue problem for the biharmonic operator and established a quantitative isoperimetric inequality for the first positive eigenvalue. We aim at showing that the exponent in the quantitative inequality is sharp.

We investigated the behavior of the eigenvalues of the Laplacian with Dirichlet, Neumann and Steklov boundary conditions when the density concentrates near points or hyperplanes.

We aim at addressing these issues for non-linear equations and Grushin's equations as well as considering non-definite or degenerate mass densities. We also shall consider perturbation problems for higher order operators in the framework of Riemannian geometry.