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We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov–Bohm operators with two colliding poles moving on an axis of symmetry of the domain.

Siano p un numero primo, e S un p-gruppo finito. Un fusion system su S è una categoria, il cui obiettivo è assiomatizzare e generalizzare le nozioni e le proprietà fondamentali del coniugio in gruppi finiti. Per studiare in modo efficace la fusione di gruppi attraverso i fusion systems, la semplice definizione di fusion system non è sufficiente: aggiungendo delle condizioni tecniche, si ottiene la definizione di fusion system saturo. Un fusion system ridotto è un particolare fusion system saturo, che corrisponde “in qualche modo" ai gruppi semplici. Attualmente, la ricerca sui fusion systems riguarda principalmente la classificazione dei fusion systems saturi su p-gruppi “piccoli". L'obiettivo di questa Tesi è di contribuire a questa classificazione, classificando tutti i fusion systems ridotti su un 3-sottogruppo di Sylow S del gruppo sporadico di McLaughlin. Inoltre dimostriamo che esistono dei fusion systems esotici su S (dove un fusion system saturo è esotico se i suoi morfismi non “provengono" da coniugi con elementi di un gruppo finito che contiene S). Inoltre dimostriamo che i fusion systems che non sono esotici “provengono" da certi gruppi finiti quasi semplici che contengono S come 3-sottogruppo di Sylow, e determiniamo tali gruppi., Let p be a prime, and let S be a finite p-group. A fusion system on S is a category whose aim is to axiomatize and generalize the notions and fundamental properties of conjugation in finite groups. In order to study effectively fusion of groups via fusion systems, the “plain" definition of fusion system is not enough: adding some techincal conditions, one obtains the definiton of a saturated fusion system. A reduced fusion system is a particular saturated fusion system, with correspondes in some sense to simple groups. The current research on fusion systems concernes mainly the classification of saturated fusion systems on “small" p-groups. The goal of this Thesis is to contribute to this classification, by classifying all reduced fusion systems on a Sylow 3-subgroup S of the sporadic McLaughlin group. Moreover, we prove that there exist some exotic fusion systems on S (where a saturated fusion system on S is exotic if its morphisms can not be “induced" by conjugations with elements of a finite group containing S). Also, we show that the fusion systems which are not exotic are “induced" by certain finite almost simple groups containing S as Sylow 3-subgroup, and we determine such groups.

L'attività di ricerca ha riguardato lo studio dell'esistenza e della molteplicità di soluzioni periodiche per un sistema differenziale ordinario del secondo ordine, con una struttura variazionale. La particolarità più significativa è che la parte principale del funzionale associato è il funzionale della lunghezza o il funzionale della variazione totale. Questo genere di problema è stato oggetto di particolare attenzione negli ultimi anni (si veda, ad esempio F. Obersnel and P. Omari, The periodic problem for curvature-like equations with asymmetric perturbations, J. Differential Equations 251 (2011), 1923—1971). Per applicare i metodi variazionali diretti, si deve considerare un funzionale rilassato, definito sullo spazio delle funzioni a variazione limitata. Mentre questo funzionale rilassato è adatto, sotto certe ipotesi, per la minimizzazione, la teoria dei punti critici non può essere applicata in maniera classica. Una prima difficoltà sta nel fatto che il funzionale non risulta differenziabile con continuità, ma il problema maggiore riguarda la verifica della condizione di Palais-Smale. Più precisamente, sotto le ipotesi naturali, non è chiaro come definire uno spazio funzionale in cui sia possibile dimostrare sia la semicontinuità inferiore del funzionale, sia la condizione di Palais-Smale. L’obiettivo principale della tesi è di proporre un approccio non standard che produca una risposta positiva a queste due necessità basilari. L’esistenza di una soluzione periodica è poi dimostrata quando il funzionale di ordine inferiore superlineare (mentre la parte principale del funzionale è asintoticamente lineare), in linea con un celebre teorema di P.H.Rabinowitz, e quando il funzionale di ordine inferiore è asintoticamente lineare, sotto una opportuna ipotesi di non risonanza. Inoltre, in linea con Z. Liu and Z.-Q. Wang, On Clark's theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), 1, The research activity has concerned the existence and multiplicity of periodic solutions for second order ordinary differential systems having a variational structure. The main feature is that the principal part of the associated functional is the length functional or the total variation functional. This kind of problem has received particular attention in the last years, see e.g. F. Obersnel and P. Omari, The periodic problem for curvature-like equations with asymmetric perturbations, J. Differential Equations 251 (2011), 1923—1971. In order to apply direct variational methods, one has to consider a relaxed functional defined on the space of functions with bounded variation. While the relaxed functional is suited, under appropriate assumptions, for minimization, critical point theory cannot be applied in a standard way. A first difficulty is that the functional is not continuously differentiable, but the major problem concerns the verification of the Palais-Smale condition. More precisely, under natural assumptions, it is not clear how to define a function space where one can prove both the lower semicontinuity of the functional and the Palais-Smale condition. The main purpose of the thesis is to propose a nonstandard functional approach which yields a positive answer to these two basic needs. Then the existence of a periodic solution is proved when the lower order functional is superlinear (while the principal part of the functional is asymptotically linear), in the line of a celebrated theorem of P.H. Rabinowitz, and when the lower order functional is asymptotically linear, under a suitable nonresonance condition. Moreover, in the line of Z. Liu and Z.-Q. Wang, On Clark's theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), 1015—1037, we prove the existence of infinitely many periodic solutions near the origin, under appropriate assumptions.

We consider Aharonov-Bohm operators with two poles and prove sharp asymptotics for simple eigenvalues as the poles collapse at an interior point out of nodal lines of the limit eigenfunction.

We study the behavior of certain eigenvalues for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We analyze the leading term in the Taylor expansion of the eigenvalue function as the pole moves in the interior of the domain, proving that it is a harmonic homogeneous polynomial and determining its exact coefficients.

We investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a planar domain. We provide sharp asymptotics for eigenvalues as the pole is moving in the interior of the domain, approaching a zero of an eigenfunction of the limiting problem along a nodal line. As a consequence, we verify theoretically some conjectures arising from numerical evidences in preexisting literature. The proof relies on an Almgren-type monotonicity argument for magnetic operators together with a sharp blow-up analysis.

We prove the sharp L1-L∞ time-decay estimate for the 2D -Schrödinger equation with a general family of scaling critical electromagnetic potentials.

In this thesis, we study Schrödinger equations with an external magnetic field. In the first part, we are interested in an eigenvalue problem. We work in an open, bounded and simply connected domain in dimension two. We consider a magnetic potential singular at one point in the domain, and related to the magnetic field being a multiple of a Dirac delta. Those two objects are related to the Bohm-Aharonov effect, in which a charged particle is influenced by the presence of the magnetic potential although it remains in a region where the magnetic field is zero. We consider the Schrödinger magnetic operator appearing in the Schrödinger equation in presence of an external magnetic field. We want to study the spectrum of this operator, and more particularly how it varies when the singular point moves in the domain. We prove some results of continuity and differentiability of the eigenvalues when the singular point moves in the domain or approaches its boundary. Finally, in case of half-integer circulation of the magnetic potential, we study some asymptotic behaviour of the eigenvalues close to their critical points. In the second part, we study nonlinear Schrödinger equations in a cylindrically setting. We are interested in the semiclassical limit of the equation. We prove the existence of a semiclassical solution concentrating on a circle. Moreover, the radius of that circle is determined by the electric potential, but also by the magnetic potential. This result is totally new with respect to the ones before, in which the concentration is driven only by the electric potential., Doctorat en Sciences, info:eu-repo/semantics/nonPublished

Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.

This paper is devoted to the study of essential self-adjointness of a relativistic Schrödinger operator with a singular homogeneous potential. From an explicit condition on the coefficient of the singular term, we provide a sufficient and necessary condition for essential self-adjointness.

In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a borderline Hardy potential, a proper variational setting has to be introduced in order to provide a weak formulation of the equation. An Almgren-type monotonicity formula is used to determine the exact asymptotic behavior of solutions.

Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations

In continuation with Part I, we investigate the asymptotic behavior of weighted eigenfunctions in two half-spaces connected by a thin tube. We provide several improvements about some convergences stated in Part I; most of all, we provide the exact asymptotic behavior of the implicit normalization for solutions given in Part I and thus describe the (N−1) -order singularity developed at a junction of the tube (where N is the space dimension)

Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.

We obtain a representation formula for solutions to Schrödinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp L1→L∞ time decay estimate for the 3D-inverse square and the 2D-Aharonov–Bohm potentials.

A monotonicity approach to the study of the asymptotic behaviour near corners of solutions to semilinear elliptic equations in domains with a conical boundary point is discussed. The presence of logarithms in the first term of the asymptotic expansion is excluded for boundary profiles sufficiently close to straight conical surfaces.

Consider two domains connected by a thin tube: it can be shown that, generically, the mass of a given eigenfunction of the Dirichlet Laplacian concentrates in only one of them. The restriction to the other domain, when suitably normalized, develops a singularity at the junction of the tube, as the channel section tends to zero. Our main result states that, under a nondegeneracy condition, the normalized limiting profile has a singularity of order N-1, where N is the space dimension. We give a precise description of the asymptotic behavior of eigenfunctions at the singular junction, which provides us with some important information about its sign near the tunnel entrance. More precisely, the solution is shown to be one-sign in a neighborhood of the singular junction. In other words, we prove that the nodal set does not enter inside the channel

This paper concerns the asymptotic behavior of solutions and their gradients to linear and nonlinear elliptic equations with singular coefficients of fuchsian type.

The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, pointwise upper estimate are derived.

Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.

A class of variational models describing ecological systems of k species competing for the same resources is investigated. The occurrence of coexistence in minimal energy solutions is discussed and positive results are proved for suitably differentiated internal dynamics.

Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using an Almgren type monotonicity formula, separation of variables, and an iterative Brezis–Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with homogeneity of order -1.

We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy.

We study positivity, localization of binding and essential self-adjointness properties of a class of Schrödinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials.

We will present the results obtained in collaboration with Alberto Ferrero and Susanna Terracini about the asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.

We will present the results obtained in collaboration with Alberto Ferrero and Susanna Terracini about the asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.

A class of nonlinear Schrödinger equations with critical power-nonlinearities and potentials exhibiting multiple anisotropic inverse square singularities is investigated. Conditions on strength, location, and orientation of singularities are given for the minimum of the associated Rayleigh quotient to be achieved, both in R^N and in bounded domains

We study the existence of different types of positive solutions to systems of elliptic equations con critical growth and singular potentials. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem.

Asymptotics of solutions to Schrodinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered.

Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established. (C) 2007 Elsevier Inc. All rights reserved.

This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show that existence of solutions heavily depends on the strength and the location of the singularities. We associate to the problem the corresponding Rayleigh quotient and give both sufficient and necessary conditions on masses and location of singularities for the minimum to be achieved. Both the cases of whole R N and bounded domains are taken into account.

The paper deals with the study of a quasilinear elliptic equation involving the $p$-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case related to Hardy inequality, providing a description of the behavior of radial solutions of the limiting problem and obtaining existence and multiplicity results for perturbed problems through variational and topological arguments.