Research interests
My research mainly focuses on Nonlinear Partial Differential Equations
of elliptic type. I have treated the following problems: nonlinear
elliptic equations arising in differential geometry, elliptic equations
with singular potentials, strongly competing systems. My scientific
contributions mainly concern existence and qualitative properties
of solutions, which I have investigated with variational and
topological methods. The classical versions of such methods have often
turned out to be not sufficient to face critical situations (e.g. the
critical growth which characterizes elliptic equations of conformal
geometry, singular potentials with critical homogeneity, etc.) thus
requiring non trivial adaptations and new original approaches. I
briefly describe below my main contributions.
- Nonlinear elliptic equations
arising in differential geometry. I studied the problem of
prescribing some conformal invariants in Riemannian geometry, which is
equivalent to solve some elliptic equations characterized by a lack of compactness, due to
nonlinear growths which are critical in the Sobolev sense. By
perturbative methods combined with blow-up analysis, I proved
existence and a-priori estimates for the Q-curvature problem on
the sphere and (in collaboration with M. Ould Ahmedou) for the problem
of prescribing the boundary mean curvature in compact manifolds with
umbilic boundary.
- Elliptic equations with singular
potentials. I focused my attention on homogeneous singular
potentials which, having the same homogeneity as the operator (e.g.
inverse square potential in the case of the laplacian), make the
problem invariant by scaling. Both existence and qualitative analysis
were considered. My first contributions date back to my Ph.D. studies:
in collaboration with M. Schneider, we proved existence, a-priori
estimates, and regularity for a class of degenerate elliptic equations
with Hardy potential and critical nonlinearity related to the
Caffarelli-Kohn-Nirenberg inequality. Remarkably enough, the study of
nondegeneracy properties needed to perform a finite-dimensional
reduction produced a symmetry
breaking result on minimizers of the Caffarelli-Kohn-Nirenberg
inequality, which improved a previous theorem by Catrina and Wang. We
explicitly found a curve in the parameter space below which simmetry
breaking occurs. I obtained several existence results (in the
semilinear case, for the p-Laplace
operator, for systems) on elliptic equations with Hardy potentials in
collaboration with I. Peral and B. Abdellaoui. Recently, I drew my
attention to qualitative aspects, more precisely to the asymptotic
behavior of solutions at singularities. The first result in this
direction (with M. Schneider) established Hölder continuity of
solutions to degenerate elliptic equations with singular weights, thus
allowing the evaluation of the exact singularity rate of solutions to
Schrödinger equations with Hardy potentials at the pole. In a
series of papers with E.M. Marchini, S. Terracini, and A. Ferrero, we
proved some extensions to the case of Schrödinger equations with
dipole-type potentials and with singular homogeneous electromagnetic
potentials, by developing a new approach based on the Almgren frequency
and obtaining a sharp and very explicit description of the local
behaviour near the singularity.
- Strongly competing systems.
In collaboration with M. Conti, I studied strongly competing
multispecies systems of Lotka-Volterra type, proving, for a class of
nonconvex domains composed by balls connected with thin corridors,
coexistence and spatial segregation of all the species under Dirichlet
boundary conditions, as the competition grows indefinitely. For
such special domains we also proved coexistence, under Neumann boundary
conditions, of limit configurations of variational competitive systems,
which are obtained as local minimizers of the associated free energy.