My research interests range in the field of Partial Differential Equations, Calculus of Variations and Inverse Problems. I list here some topics engaging my attention in the present.

#### Non-Linear Eigenvalues

A topic that pervades my work is that of non-linear eigenvalues. This term refers to parameters for which a certain non-linear partial differential equation admits a non-trivial solution, under given conditions. A paradigma is provided by the critical values of the $p$-Dirichlet integral $$\int_\Omega |\nabla u|^p\,dx$$ on functions $u$ satisfying homogeneous Dirichlet conditions on the boundary of $\Omega$, under a $L^p(\Omega)$-constraint. If $p=2$ this leads to the classical Helmoltz equation, whereas if $p\in(1,\infty)$ is different from $2$ the first order condition is a quasilinear (singular or degenerate) elliptic PDE.

A variation on this theme is given by considering the critical points subject to an $L^q(\Omega)$ with $q$ ranging between $1$ and the Sobolev conjugate of $p$, that in $N$ dimensions is $Np/(N-p)$ if $p and $+\infty$ otherwise. If $q$ and $p$ are different numbers, a non-local term appears; nevertheless, after a normalisation in $L^q(\Omega)$ the equation bears itself as a homogeneous condition. This double behaviour is responsibile for phaenomena that cannot happen if $p=q$ (for example, in general, the least eigenvalue is not isolated).

All of this is the object of a long-standing spectral investigation, including collaborations with Lorenzo Brasco, Pier Domenico Lamberti, Peter Lindqvist, Giampiero Palatucci, Berardo Ruffini.

#### Isoperimetric inequalities

Another recurrent research interest of mine is related to the problem of minimising the surface area (the “perimeter”) under a volume constraint $$\min \Big\{ {\rm Per}(E)\mathbin{\colon} {\rm vol}(E)=v\Big\}\,.$$ This question belongs to a very distant past in the history of mathematics but offers a prototypical example of problem in the area of Calculus of Variations, and for this reason it has been the object of a thorough examination in the literature. Nonetheless, in case the surface area and the volume are defined with respect to a non-homogeneous weight in the $N$-dimensional euclidean space, say $${\rm Per}(E)=\int_{\partial E} f(x)\,d S\,,$$$${\rm vol}(E) = \int_E f(x)\,d{\rm vol}\,,$$ the interest in proving the existence of solutions arose just quite recently. Depending on the assumptions on the density $f$ the difficulty in takling this issue can range from trivial to very hard. This topic is the subject of a collaboration with other authors, such as Aldo Pratelli and Guido De Philippis.

The structure of the isoperimetric problem with densities is that of a shape optimisation problem. Other shape optimisation problems leading to isoperimetric-type inequalities that drew my attention are related to spectral functionals. Spectral optimisation in the context of non-linear eigenvalues has been the tough central part of my PhD Thesis, it was the object of a fruitful collaboration with Lorenzo Brasco, and continues to stimulate my research.

#### perforated domains and homogeneisation

This terminology refers to an immense literature in the field of Mathematical Analysis with applications to Physics and Material Sciences. My contribution is related to a very specific problem, related to the appearance of measure-theoretic transmission conditions accross an interface in the homogeneisation of quasilinear elliptic equations of variational type with Neumann conditions on perforations concentrating about the interface. In a paper with Gianni Dal Maso and Davide Zucco, we characterised the general form of the limit energy.

#### Eddy currents and inverse problems

When consider electromagnetic signals propagating in a region filled with a stratified medium at rest, with a low loss structure, it makes sense to neglect displacement currents. This leads one to the eddy currents, that are described by a parabolic system of partial differential equations in non-divergence form. My interest in this diffusive mathematical model comes from the inverse problem of recovering the source in the equations from boundary measurements of the magnetic field. This is an ongoing project, and includes a collaboration with Elisa Francini and Sergio Vessella, and the INGV.