Convegno
“ASK 2026 CONFERENCE IN GEOMETRIC ANALYSIS AND PDES ”

In this talk we discuss spectral perturbation theory from an abstract perspective. More precisely, we consider a one-parameter family of varying bilinear forms, each of them defined on a (possibly) different Hilbert space. Assuming the stability of the corresponding spectra, in the first part of the talk we discuss a quantification of the rate of convergence. A key feature is the explicit variational characterization of the first term in the asymptotic expansion of the perturbed eigenvalues, which depends on the limit eigenspace and of the magnitude of the perturbation through the reso- lution of a minimization problem. In the second part of the talk, we explore some concrete applications of our abstract results. First, we consider eigenvalue problems for the Laplace-Beltrami operator with varying measure weights; secondly, we in- vestigate the Neumann approximation of the Steklov eigenvalues of the Laplacian; finally, we focus on how the spectrum of the Laplace-Beltrami operator on a Rie- mannian manifold changes when a second small manifold is glued on a small portion of it. This is a joint project with Andrea Bisterzo, Roberto Ognibene and Prasun Roychowdhury.

We will consider the coupled system of damped $\sigma$-equation \begin{equation*}\begin{cases} \partial_t^2U + (-\Delta)^\sigma U + D \partial_t U = F(U), & t>0, \ x\in\R^n, \\ (U,\partial_tU)(0,x)=(0,\Phi(x)), & x\in\R^n, \end{cases} \end{equation*} where $\sigma>1$,~$U=(u_1,u_2)\tm$,~$\Phi=(\varphi_1,\varphi_2)\tm$ and \begin{equation*} D U = \begin{pmatrix} a_1(-\Delta)^{\frac{\theta_1}2} & 0 \\ 0 & a_2 (-\Delta)^{\frac{\theta_2}2} \end{pmatrix},\quad \theta_1,\theta_2\in[\sigma,2\sigma], \ a_1,a_2\geq0, \end{equation*} is a damping term in the noneffective case. We will analyze the various scenarios of existence of global-in-time solution that arise due to the semilinear term $F(U)=(f_1(U),f_2(U))\tm$, in the case of a general semilinear coupling \[ F(U)=\begin{pmatrix} c_{11}|u_1|^{p_{11}} + c_{12}|u_2|^{p_{12}}\\ c_{21}|u_1|^{p_{21}} + c_{22}|u_2|^{p_{22}} \end{pmatrix}, \] % or in the case of weak coupling of derivative type \[ F(U)=\begin{pmatrix} |\partial_t u_2|^{p_{2}}\\ |\partial_t u_1|^{p_{1}} \end{pmatrix}. \] Sharp $L^p-L^q$ estimates for the corresponding linear scalar equation and a suitable application of the test function method are crucial for determining the so-called critical curves for the existence of small data solutions.\\ Based on a joint work with Marcello D'Abbicco (University of Bari).

The purpose of this talk is to present some recent advances in the phase space analysis of quadratic differential operators modeled by the Weyl quantization of a complex-valued quadratic symbol. The first part will focus on metaplectic and Schroedinger propagators, where Gabor wave packet decompositions lead to a structured representation which cap- tures insightful phenomena (such as sparsity, dispersion, spreading, confinement). In the second part, we will discuss how these methods provide a unified framework also for non-selfadjoint quadratic models, where heat-type dissipative effects play a crucial role, emphasizing how a phase space perspective effectively connects all the relevant features of the problem.

We consider the NLS equation with combined power-type nonlinearities in the mass-subcritical regime, where a focusing leading term is perturbed by a lower order defocusing term, and we prove small data scattering. The proof relies on the pseudo-conformal transformation in conjunction with a general variational argument used to obtain the positivity of certain modified energies. The smallness assumption is only on the mass of the initial datum, and not on the whole Σ-norm. This is a joint work with J. Bellazzini and V. Georgiev.

I will present a work in progress that I am developping jointly with S. Ciani and I. Skrypnik, about the degenerate parabolic equation of Trudinger type in anisotropic form. After recalling the fundamental solutions of Barenblatt type in the anisotropic regime, and the associated sub-potential estimates, I will present a weak Harnack inequality that is adapted to the intrinsic geometry of the problem, and via this inequality I will prove some rigidity results for the non-negative solutions of the equation. This approach opens to a possible extension of the De Giorgi-Nash-Moser theory in non-isotropic setting.

Segregation problems arise when competing species tend to occupy disjoint regions, leading naturally to a free boundary separating them. In this talk, we will discuss a parabolic model in which the diffusion is governed by fully nonlinear operators. We will describe the main ideas behind the existence of Lipschitz solutions, the free boundary condition, and the regularity of the interface near nondegenerate points. This is ongoing joint work with Emily Casey (U. Minnesota), Cornelia Mihaila (Saint Michael’s College), and Clara Torres-Latorre (ICMAT).

In this talk, we discuss how the time-dependent coefficients in a semilinear wave equation influence the critical exponent, i.e., the threshold value for the power of the nonlinear term that separates the blow-up region from the global existence region.

In this talk, we discuss about the regularity properties of fractional logarithmic p-Laplacian. We prove the Harnack inequality (with tails) and local Hölder regularity for the fractional logarithmic p-Laplace operator. For a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{{\rm d}}{{\rm d}t}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order s\in (0, 1). The kernel of this operator involves a logarithmic factor and it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional p-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in our setting. We also present an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear case p=2.

The Boltzmann equation is one of the central equations in statistical mechanics and models the evolution of a gas through particle interactions. In recent years, groundbreaking work by Imbert and Silvestre has led to a conditional regularity theory for periodic solutions of the Boltzmann equation. They established that any possible singularity of a periodic solution to the Boltzmann equation must be visible macroscopically. A major open challenge is whether such a theory can be extended to bounded domains with physically relevant boundary conditions. In this talk, I will first give an accessible overview of the conditional regularity program by Imbert and Silvestre, highlighting its main ideas and implications. As a first step toward understanding the boundary case, I will then discuss the smooth- ness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection and in-flow condition. While the interior regularity of such equations is well understood, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov’s equation. Finally, I will report on recent joint works with Xavier Ros-Oton and Kyeongbae Kim, in which we establish sharp boundary regularity results for this class of equations.