My main research interests are: existence, multiplicity, and qualitative properties of solutions to semilinear elliptic equations and systems; unique continuation principles for elliptic and parabolic equations; time decay estimates for Schrödinger equations with singular potentials; spectral theory for elliptic PDEs. My most recent research focuses on singular elliptic equations: asymptotic analysis, unique continuation, and spectral stability for singularly perturbed problems.
I study elliptic stability for elliptic operators, looking for sharp eigenvalue estimates for the following singularly perturbed problems:
In these problems the sharp asymptotic behaviour of eigenvalues with respect to the perturbation parameter is expected to depend strongly on the vanishing order of the limit eigenfunction. The problem of the evaluation of the exact rate of convergence of eigenvalues of the perturbed problem to the eigenvalues of the limit problem can be investigated by an Almgren type monotonicity formula combined with a fine blow-up analysis: indeed an Almgren type monotonicity argument allows obtaining quite precise energetic estimates near the singularity, which can be applied to the blow-up analysis of scaled eigenfunctions.
Some references
I am interested in the local asymptotic behavior of solutions to linear or nonlinear elliptic equations with applications to unique continuation principles. In particular I have recently studied fractional elliptic equations: exploiting the Caffarelli-Silvestre characterization of fractional laplacian as the Dirichlet-to-Neumann operator, a monotonicity formula for fractional elliptic equations can be combined with a sharp blow-up analysis to obtain unique continuation properties and asymptotic estimates for solutions.
Some references