## Research

My research interests are in Mathematical Analysis, in particular Partial Differential Equations (PDEs), Function Spaces, Functional Analysis, Spectral Theory, Asymptotic Analysis, Spectral Geometry.

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 (with P.D. Lamberti).

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. (with M. Dalla Riva). We obtained more precise information on the asymptotic behavior of the Neumann eigenvalues near their limiting Steklov eigenvalues proving that locally the Steklov eigenvalues minimize the Neumann eigenvalues (with P.D. Lamberti)

We introduced a new Steklov-type eigenvalue problem for the biharmonic operator and established a quantitative isoperimetric inequality for the first positive eigenvalue (with D. Buoso). Moreover we proved that such inequality and the corresponding one for the first positive biharmonic Neumann eigenvalue are sharp (with D. Buoso and L. M. Chasman).

We investigated the behavior of the Neumann eigenvalues for the biharmonic operator and Poisson's coefficient σ when σ approaches the singular value -1.

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 proved uniform upper bounds for the Neumann eigenvalues of the polyharmonic operators (operators of order 2m) and variable mass density (with mass constraint) in dimension N≥2m. Such upper bounds present the right growth predicted by the Weyl's law. We showed counterexamples in the case N<2m. We considered also a non-linear natural constraint and proved that Weyl-type upper bounds exists for N=2m while they do not exist for N>2m under such constraint. An important open question is then whether Weyl-type bounds exist with N<2m. We aim at extending the results to smooth N-dimensional Riemannian manifolds (with B. Colbois).

We proved uniform upper bounds for the Steklov eigenvalues of the Laplace operator (with perimeter constraint) which respect the Weyl's law in dimension N≥2 (with J. Stubbe) for sufficiently smooth domains. We aim at considering the same problem in the case of piecewise smooth boundaries, in particular in the case of polygonal or poyhedral domains (with J. Stubbe).

We consider complementary asymptotically sharp bounds for the eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions by means of averaged variational principles (with E. Harrell II and J. Stubbe).