Monotonicity formulas and blow-up methods for the study of spectral stability and fractional obstacle problems

The present dissertation is essentially divided into two parts. In the first part, we investigate questions of spectral stability for eigenvalue problems driven by the Laplace operator, under certain specific kinds of singular perturbations. More precisely, we start by considering the spectrum of the Laplacian on a fixed, bounded domain with prescribed homogeneous boundary conditions (of pure Dirichlet or Neumann type); then, we introduce a singular perturbation of the problem, which gives rise to a perturbed sequence of eigenvalues. Our goal is to understand the asymptotic behavior of the perturbed spectrum as long as the perturbation tends to disappear. In particular, we consider two different types of singular perturbation. On one hand, in the case of homogeneous Dirichlet boundary conditions, we consider a perturbation of the domain, which consists in attaching a thin cylindrical tube to the fixed limit domain and let its section shrink to a point. In this framework, we combine energy estimates coming from a tailor-made Almgren type monotonicity formula with the Courant-Fischer min-max characterization and then we perform a careful blow-up analysis for scaled eigenfunctions; with these ingredients, we identify the sharp rate of convergence of a perturbed eigenvalue in the case in which it is approaching a simple eigenvalue of the limit problem. On the other hand, we deal with a perturbation of the boundary conditions. More specifically, we start with the homogeneous Neumann eigenvalue problem for the Laplacian and we perturb it by prescribing zero Dirichlet boundary conditions on a small subset of the boundary. In this context, we describe the sharp asymptotic behavior of a perturbed eigenvalue when it is converging to a simple eigenvalue of the limit Neumann problem. In particular, the first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. We also provide a more ‘explicit’ expression for the eigenvalue variation in the particular case of Dirichlet boundary conditions imposed on a subset which is scaling to a point. In the second part of this thesis, we deal with two problems, both governed by the fractional Laplace operator, i.e. the power of order between 0 and 1 of the classical (negative) Laplacian. First, we address the question of positivity of a nonlocal Schrödinger operator, driven by the fractional Laplacian and with singular multipolar Hardy-type potentials. Namely, we provide necessary and sufficient conditions on the coefficients of the potential for the existence of a configuration of poles that ensures the positivity of the corresponding Schrödinger operator. This result is based, in turn, on a criterion in the spirit of the Agmon-Allegretto-Piepenbrink principle and on a tool fitting in the theory of localization of binding. The second topic we investigate in this part concerns geometric properties of the free boundary of solutions of a two-phase penalized obstacle-type problem for the fractional Laplacian. In view of the Caffarelli-Silvestre extension, we can interpret it as a thin obstacle-type problem driven by a second-order differential operator living in one dimension more and with a Muckenhoupt weight, that can be either singular or degenerate on the thin space. Working in this framework, by means of Almgren and Monneau type monotonicity formulas and blow-up analysis, we first prove a classification of the possible vanishing orders on the thin space and, as a consequence, the boundary strong unique continuation principle. We finally establish a stratification result for the nodal set (which coincides with the free boundary) on the thin space and we provide sharp estimates on the Hausdorff dimension of its regular and singular part