arXiv subjects

T. Durante

Publications and source records attributed to T. Durante.

4 recordsLinked to original sources

arXiv2017

On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacia

In this paper we consider an optimal control problem for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued nonsmooth controls in its coefficients and a nonlinear equation of Hammerstein type. Using the direct method in calculus of variations, we prove the existence of an optimal control in considered problem and provide sensitivity analysis for a specific case of considered problem with respect to two-parameter regularization.

math.OC↗

arXiv2015

Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide $Π_{l}^{\varepsilon}$ obtained from a straight unit strip by a low box-shaped perturbation of size $2l\times\varepsilon,$ where $\varepsilon>0$ is a small parameter. We prove the existence of the length parameter $l_{k}^{\varepsilon}=πk+O\left( \varepsilon\right) $ with any $k=1,2,3,...$ such that the waveguide $Π_{l_{k}^{\varepsilon}}^{\varepsilon }$ supports a trapped mode with an eigenvalue $λ_{k}^{\varepsilon}% =π^{2}-4π^{4}l^{2}\varepsilon^{2}+O\left( \varepsilon^{3}\right) $ embedded into the continuous spectrum. This eigenvalue is unique in the segment $\left[ 0,π^{2}\right] $ and is absent in the case $l\neq l_{k}^{\varepsilon}.$ The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main technical difficulty is caused by corner points of the perturbed wall $\partialΠ_{l}^{\varepsilon}$ and we discuss available generalizations for other piecewise smooth boundaries.

math.SP↗

arXiv2009

The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends

A simple sufficient condition on curved end of a straight cylinder is found that provides a localization of the principal eigenfunction of the mixed boundary value for the Laplace operator with the Dirichlet conditions on the lateral side. Namely, the eigenfunction concentrates in the vicinity of the ends and decays exponentially in the interior. Similar effects are observed in the Dirichlet and Neumann problems, too.

math.SP↗

arXiv2009

Water-waves modes trapped in a canal by a body with the rough surface

The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter $ε>0$ while the distance of the body to the water surface is also of order $ε$. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely, it is proved that, for any given $d>0$ and integer $N>0$, there exists $ε(d,N)>0$ such that the problem has at least $N$ eigenvalues in the interval $(0,d)$ of the continuous spectrum in the case $ε\in(0,ε(d,N)) $. The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes.

math.SP↗