Finite time extinction for nonlinear fractional evolution equations and related properties

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

Finite time extinction for nonlinear fractional evolution equations and related properties.

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

Toy model to describe the effect of positional blocklike disorder in metamaterials composites

We study theoretically the effect of a new type of blocklike positional disorder on the effective electromagnetic properties of one-dimensional chains of resonant, high-permittivity dielectric particles, where particles are arranged into perfectly well-ordered blocks whose relative position is a random variable. This creates a finite order correlation length that mimics the situation encountered in metamaterials fabricated through self-assembled techniques, whose structures often display short-range order between near neighbors but long-range disorder, due to stacking defects. Using a spectral theory approach combined with a principal component statistical analysis, we study, in the long-wavelength regime, the evolution of the electromagnetic response when the composite filling fraction and the block size are changed. Modifications in key features of the resonant response (amplitude, width, etc.) are investigated, showing a regime transition for a filling fraction around 50%.

Homogenization of the p-Laplacian with nonlinear boundary condition on critical size particles: Identifying the strange term for the some non smooth and multivalued operators

We extend previous papers in the literature concerning the homogenization of Robin type boundary conditions for quasilinear equations, in the case of microscopic obstacles of critical size: here we consider nonlinear boundary conditions involving some maximal monotone graphs which may correspond to discontinuous or non-Lipschitz functions arising in some catalysis problems.

A continuous model for quasinilpotent operators.

A classical result due to Foias and Pearcy establishes a discrete model for every quasinilpotent operator acting on a separable, infinite-dimensional complex Hilbert space HH . More precisely, given a quasinilpotent operator T on HH , there exists a compact quasinilpotent operator K in HH such that T is similar to a part of K⊕K⊕⋯⊕K⊕⋯K⊕K⊕⋯⊕K⊕⋯ acting on the direct sum of countably many copies of HH . We show that a continuous model for any quasinilpotent operator can be provided. The consequences of such a model will be discussed in the context of C0C0 -semigroups of quasinilpotent operators.

CB-norm estimates for maps between noncommutative Lp-spaces and quantum channel theory.

In the first part of this work, we show how certain techniques from quantum information theory can be used in order to obtain very sharp embeddings between noncommutative Lp-spaces. Then, we use these estimates to study the classical capacity with restricted assisted entanglement of the quantum erasure channel and the quantum depolarizing channel. In particular, we exactly compute the capacity of the first one and we show that certain nonmultiplicative results hold for the second one.

The spectra of linear fractional composition operators on weighted Dirichlet spaces

We completely determine the spectra of composition operators induced by linear fractional self-maps of the unit disc acting on weighted Dirichlet spaces; extending earlier results by Higdon [8] and answering the open questions in this context.

Rota’s universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In articular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Strictly singular operators in pairs of L (p) space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q aS, E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from L (p) to L (q) is found. There exists a strictly singular but not superstrictly singular operator on L (p) , provided that p not equal 2.

Rota's universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.