William Borrelli

Verified

William verified their affiliation via an institutional email.

Learn more

Verified

William verified their affiliation via an institutional email.

Learn more

• PhD
• Associate Professor at Politecnico di Milano

Graphene is a monolayer graphitic film where electrons behave like two-dimensional Dirac fermions without mass. Its study has attracted a wide interest in the domain of condensed matter physics. In particular, it represents an ideal system to test the comprehension of 2D massless relativistic particles in a laboratory, the Fermi velocity being 300 t...

We correct a mistake in [J. Spectr. Theory 14 (2024), 1147–1193] in the computation of the square of a generic self-adjoint realization of the Dirac operator with an Aharonov–Bohm flux. We prove that only two self-adjoint realizations of the Dirac operator square to self-adjoint realizations of the Pauli operator with Aharonov–Bohm potential.

Graphene is a monolayer graphitic film where electrons behave like two-dimensional Dirac fermions without mass. Its study has attracted a wide interest in the domain of condensed matter physics. In particular, it represents an ideal system to test the comprehension of 2D massless relativistic particles in a laboratory, the Fermi velocity being $300...

We study a two-dimensional Pauli operator describing a charged quantum particle with spin 1/2 moving on a plane in presence of an orthogonal Aharonov–Bohm magnetic flux. We classify all the admissible self-adjoint realizations and give a complete picture of their spectral and scattering properties. Symmetries of the resulting Hamiltonians are also...

We study a two-dimensional Pauli operator describing a charged quantum particle with spin 1/2 moving on a plane in presence of an orthogonal Aharonov-Bohm magnetic flux. We classify all the admissible self-adjont realizations and give a complete picture of their spectral and scattering properties. Symmetries of the resulting Hamiltonians are also d...

Motivated by various geometric problems, we study the nodal set of solutions to Dirac equations on manifolds, of general form. We prove that such set has Hausdorff dimension less than or equal to n-2, n being the ambient dimension. We extend this result, previously known only in the smooth case or in specific cases, working with locally Lipschitz c...

We prove that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the p -Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions.

In this paper we study Dirac–Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence res...

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the p -Laplace operator and a general nonlinearity satisfying concavity-type assumptions. This provides an extension of results previously known in the literature only for the torsion and the first eigenvalue equation...

We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $\kappa$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schr\"odinger operator on $L^2(\mathbb R)$ \[ \ma...

We consider the two dimensional Schrödinger equation with a time dependent point interaction, which represents a model for the dynamics of a quantum particle subject to a point interaction whose strength varies in time. First, we prove global well-posedness of the associated Cauchy problem under general assumptions on the potential and on the initi...

In this paper we prove the convergence of solutions to discrete models for binary waveguide arrays toward those of their formal continuum limit, for which we also show the existence of localized standing waves. This work rigorously justifies formal arguments and numerical simulations present in the Physics literature.

We make a spectral analysis of the massive Dirac operator in a tubular neighbourhood of an unbounded planar curve, subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip...

In this paper we study Dirac-Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence res...

We prove that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the $p$-Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions.

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an extension of results previously known in the literature only for the torsion and the eigenfunction equations....

We consider the two dimensional Schr\"odinger equation with time dependent delta potential, which represents a model for the dynamics of a quantum particle subject to a point interaction whose strength varies in time. First, we prove global well-posedness of the associated Cauchy problem under general assumptions on the potential and on the initial...

It has been proven that the general relativistic Poynting-Robertson effect in the equatorial plane of the Kerr metric shows a chaotic behavior for a suitable range of parameters. As a further step, we calculate the timescale for the onset of chaos through the Lyapunov exponents, estimating how this trend impacts the observational dynamics. We concl...

It has been proved that the general relativistic Poynting-Robertson effect in the equatorial plane of Kerr metric shows a chaotic behavior for a suitable range of parameters. As a further step, we calculate the timescale for the onset of chaos through the Lyapunov exponents, estimating how this trend impacts on the observational dynamics. We conclu...

In this paper we prove the convergence of solutions to discrete models for binary waveguide arrays toward those of their formal continuum limit, for which we also show the existence of localized standing waves. This work rigorously justifies formal arguments and numerical simulations present in the Physics literature.

We prove a classification result for ground state solutions of the critical Dirac equation on \(\mathbb {R}^n\), \(n\geqslant 2\). By exploiting its conformal covariance, the equation can be posed on the round sphere \(\mathbb {S}^n\) and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreo...

The general relativistic Poynting-Robertson effect is a dissipative and nonlinear dynamical system obtained by perturbing through radiation processes the geodesic motion of test particles orbiting around a spinning compact object, described by the Kerr metric. Using the Melnikov method, we find that in a suitable range of parameters, chaotic behavi...

The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., |ψ|p−2ψ) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite N-star graphs, the existence of s...

The general relativistic Poynting-Robertson effect is a dissipative and non-linear dynamical system obtained by perturbing through radiation processes the geodesic motion of test particles orbiting around a spinning compact object, described by the Kerr metric. Using the Melnikov method we find that, in a suitable range of parameters, chaotic behav...

In this paper, we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we also prove smoothness and exponential...

We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip l...

In this note we present some properties of the Dirac operator on noncompact metric graphs with Kirchoff-type vertex conditions. In particular, we discuss the specific features of the spectrum of the operator and, finally, we give some further details on the associated quadratic form (and on the form domain).

In this paper we deal with two dimensional cubic Dirac equations appearing as effective model in gapped honeycomb structures. We give a formal derivation starting from cubic Schr\"odinger equations and prove the existence of standing waves bifurcating from one band-edge of the linear spectrum.

In this paper we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we prove also prove smoothness and exponen...

We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac–Einstein equations on \(\mathbb {R}^3\), which appear in the bubbling analysis of conformal Dirac–Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin–Talenti functions, while the...

We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $\mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spi...

We prove a classification result for ground state solutions of the critical Dirac equation on $\mathbb{R}^n$, $n\geq2$. By exploiting its conformal covariance, the equation can be posed on the round sphere $\mathbb{S}^n$ and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such grou...

The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $\psi^{p-2}\psi$) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite $N$-star graphs, the exi...

We present a brief overview of the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. First, we focus on the NLSE (both in the subcritical and the critical case) and, then, on the NLDE highlighting similarities and differences with the NL...

In this note we present some properties of the Dirac operator on noncompact metric graphs with Kirchoff-type vertex conditions. In particular, we discuss the specific features of the spectrum of the operator and, finally, we give some further details on the associated quadratic form (and on the form domain).

We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schr\"odinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the subcritical and the critical case, and then on the NLDE, highlighting similarities and differences with the NL...

In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L ² -subcritical...

Recently, new two-dimensional materials possessing unique properties have been discovered, the most famous being the graphene. In this materials, electrons at the Fermi level behave as massless particles and can be described by the massless Dirac equation. This phenomenon is quite general, and it is a common features of "honeycomb" periodic structu...

We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor c...

We prove sharp decay estimates for critical Dirac equations on $\mathbb{R}^{n}$, with $n\geq 2$. They appear, e.g., in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles (the so-called \emph{bubbles}) in the associated variational problem. We establish regularity and integrability properties of $L^{2^{\shar...

We prove sharp pointwise decay estimates for critical Dirac equations on R^n with n ≥ 2. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreo...

In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L 2-subcritical c...

We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor c...

This paper is devoted to the variational study of an effective model for the electron transport in a graphene sample. We prove the existence of infinitely many stationary solutions for a nonlin-ear Dirac equation which appears in the WKB limit for the Schr{\"o}dinger equation describing the semi-classical electron dynamics. The interaction term is...

This paper is devoted to the variational study of an effective model for the electron transport in a graphene sample. We prove the existence of infinitely many stationary solutions for a nonlin-ear Dirac equation which appears in the WKB limit for the Schr{\"o}dinger equation describing the semi-classical electron dynamics. The interaction term is...

In this paper we prove the existence of an exponentially localized stationary solution for a two-dimensional cubic Dirac equation. It appears as an effective equation in the description of nonlinear waves for some Condensed Matter (Bose-Einstein condensates) and Nonlinear Optics (optical fibers) systems. The nonlinearity is of Kerr-type, that is of...

In this paper we prove the existence of an exponentially localized stationary solution for a two-dimensional cubic Dirac equation. It appears as an effective equation in the description of nonlinear waves for some Condensed Matter (Bose-Einstein condensates) and Nonlinear Optics (optical fibers) systems. The nonlinearity is of Kerr-type, that is of...