Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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A note on the spectrum of diagonal perturbation of weighted shift operator
Full text linkThis note provides a complete description of the spectrum of di- agonal perturbation of weighted shift operator acting on a separable Hilbert space
Computing the reciprocal distance signless Laplacian eigenvalues and energy of graphs
Full text linkIn this paper, we study the eigenvalues of the reciprocal distance signless Laplacian matrix of a connected graph and
obtain some bounds for the maximum
eigenvalue of this matrix. We also focus on bipartite graphs and find some bounds for the spectral radius of the reciprocal distance signless Laplacian matrix of this class of graphs. Moreover, we give bounds for the reciprocal distance signless Laplacian energy
Doubling inequality at the boundary for the Kirchhoff - Love plate\u27s equation with Dirichlet conditions
Full text linkThe main result of this paper is a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate\u27s equation satisfying homogeneous Dirichlet conditions. This result, like the three sphere inequality with optimal exponent at the boundary proved in Alessandrini, Rosset, Vessella, Arch. Ration. Mech. Anal. (2019), implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on a suitable Carleman estimate, and involves an ad hoc reflection of the solution. We also give a simple application of our main result, by weakening the standard hypotheses ensuring uniqueness for the Cauchy problem for the plate equation
An improved compact embedding theorem for degenerate Sobolev spaces
Full text linkThis short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain Ω with respect to the norm:where the weight v is comparable to a power of the pointwise operator norm of the matrix valued function Q=Q(x) in Ω. Following our main theorem, we give an explicit application where degeneracy is controlled through an ellipticity condition of the formfor a pair of p-admissible weights. We also give explicit examples demonstrating the sharpness of our hypotheses
On near-optimal time samplings for initial data best approximation
Full text linkLeveraging on the work of DeVore and Zuazua, we further explore their methodology and deal with two open questions presented in their paper. We show that for a class of linear evolutionary PDEs of order the admissible choice of the parameter which is used to construct the near-optimal sampling sequence is not influenced by the spectrum of of the operator controlling the spatial part of the PDE, but only by its order. Furthermore, we show that it is possible to extend their algorithm to a simple version of a non-autonomous heat equation in which the heat diffusivity coefficient depends explicitly on time
Three Weak Solutions to a degenerate quasilinear elliptic system
Full text linkSufficient conditions are established to guarantee the existence of at leastthree weak solutions to a degenerate quasilinear elliptic systemwith three parameters and Dirichlet boundary conditions. An application ofthe main theorem to a scalar elliptic problem is also presented.The proofs in the paper mainly make use of a variational argument andan abstract critical point theorem due to Ricceri
On trace theorems for Sobolev spaces
Full text linkWe survey a few trace theorems for Sobolev spaces on N-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov\u27s Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space H1(Ω) on a Lipschitz domain Ω, and which we have further developed for the trace spaces of Hk(Ω), k≥2, by using Fourier expansions associated with the eigenfunctions of new multi-parameter polyharmonic Steklov problems
Gamma-convergence for one-dimensional nonlocal phase transition energies
Full text linkWe study the asymptotic behavior as ε goes to 0 of an appropriate scaling of the following nonlocal Allen-Cahn energy,where I is an interval in R, and W is a double-well potential. We provide a Γ-convergence result for any s ∈ (0,1), by extending the case when s=1/2 studied by Alberti, Bouchittè and Seppecher in [2]. We also investigate the convergence as s↗1 of the related optimal profile problem to the local counterpart
Breaking through borders with σ-harmonic mappings
Full text linkWe consider mappings U=(u1,u2), whose components solve an arbitrary elliptic equation in divergence form in dimension two, and whose respective Dirichlet data φ1,φ2 constitute the parametrization of a simple closed curve γ. We prove that, if the interior of the curve γ is not convex, then we can find a parametrization Φ=(φ1,φ2) such that the mapping U is not invertible
A survey on the classical theory for Kolmogorov equation
Full text linkWe present a survey on the regularity theory for classic solutions to subelliptic degenerate Kolmogorov equations. In the last part of this note we present a detailed proof of a Harnack inequality and a strong maximum principle