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We consider the Schrodinger type operator $$(1+|x|^\alpha )\Delta +c|x|^{\alpha -2}$$ , for $$\alpha > 2$$ , $$c2$$ . Heat kernel estimates of the associated semigroup are obtained using the equivalence between weighted Nash inequalities and “weighted” ultracontractivity of a symmetric Markov semigroup. Moreover we give estimates of the eigenfunctions of the operator for large values of |x|.

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide $\Pi_{l}^{\varepsilon}$ 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}=\pi k+O\left( \varepsilon\right) $ with any $k=1,2,3,...$ such that the waveguide $\Pi_{l_{k}^{\varepsilon}}^{\varepsilon }$ supports a trapped mode with an eigenvalue $\lambda_{k}^{\varepsilon}% =\pi^{2}-4\pi^{4}l^{2}\varepsilon^{2}+O\left( \varepsilon^{3}\right) $ embedded into the continuous spectrum. This eigenvalue is unique in the segment $\left[ 0,\pi^{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 $\partial\Pi_{l}^{\varepsilon}$ and we discuss available generalizations for other piecewise smooth boundaries., Comment: 36 pages, 6 figures

We consider an infinite strip perforated along a curve by small holes. In this perforated domain, we consider a scalar second-order elliptic differential operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic, we describe possible homogenized problems and prove the norm-resolvent convergence of the perturbed operator to a homogenized one. We also provide estimates for the rate of the convergence.

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued \(L^\infty (\varOmega ,\mathbb {R}^{N\times N})\)-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.

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide obtained from a straight unit strip by a low box-shaped perturbation. We prove that the waveguide supports a trapped mode with an eigenvalue embedded into the continuous spectrum. The main technical difficulty is caused by corner points of the perturbed wall.We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide obtained from a straight unit strip by a low box-shaped perturbation. We prove that the waveguide supports a trapped mode with an eigenvalue embedded into the continuous spectrum. The main technical difficulty is caused by corner points of the perturbed wall.

We consider the spectral Dirichlet problem for the Laplace operator in the plane $\Omega^{\circ}$ with double-periodic perforation but also in the domain $\Omega^{\bullet}$ with a semi-infinite foreign inclusion so that the Floquet-Bloch technique and the Gelfand transform do not apply directly. We describe waves which are localized near the inclusion and propagate along it. We give a formulation of the problem with radiation conditions that provides a Fredholm operator of index zero. The main conclusion concerns the spectra $\sigma^{\circ}$ and $\sigma^{\bullet}$ of the problems in $\Omega^{\circ}$ and $\Omega^{\bullet},$ namely we present a concrete geometry which supports the relation $\sigma^{\circ}\varsubsetneqq\sigma^{\bullet}$ due to a new non-empty spectral band caused by the semi-infinite inclusion called an open waveguide in the double-periodic medium., Comment: 25 pages, 5 figures

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 epsilon > 0 while the distance of the body to the water surface is also of order epsilon. 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 a number epsilon(d, N) > 0 such that the problem has at least N eigenvalues in the interval (0, d) of the continuous spectrum in the case epsilon is an element of (0, epsilon (d, N)). The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We study the asymptotic behavior (as e→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number 2N of thin rods with variable thickness of order \(\varepsilon =\mathcal{O}(N^{-1}).\) The thin rods are divided into two levels depending on the geometrical characteristics and on the controls given on their bases. In addition, the thin rods from each level are e-periodically alternated and the inhomogeneous perturbed Fourier boundary conditions are given on the lateral sides of the rods. Using the direct method of the calculus of variations and the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis of this problem for different kinds of controls is made as e→0.

The problem about a body with a rough surface in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established.

We consider a mixed boundary value problem for the Laplace operator in a straight cylinder with curved ends with the Dirichlet conditions on the lateral side. We prove that when the ratio between the diameter and the length of the cylinder tends to zero, the principal eigenfunction concentrates in the vicinity of the ends and decays exponentially in the interior.

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

In this note we give sufficient conditions for the essential self-adjointness of some Kolmogorov operators perturbed by singular potentials. As an application we show that the space of test functions $C_c^∞(R^N \backslash \{0\})$ is a core for the operator $Au= Δu-Bx∇u+\frac{c}{|x|^2} u=:Lu+\frac{c}{|x|^2} u, u ∈ C_c^∞(R^N \backslash \{0\}),$ in $L^2(R^N,\mu)$ provided that $c\le \frac{(N-2)^2}{4}-1$. Here $B$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$.

We consider quasilinear optimal control problems involving a thick two-level junction Ωe which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order O(e 2 ). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary conditions depend on parameters e, α, β and the thin cylinders from each level are e-periodically alternated. Using the Buttazzo-Dal Maso abstract scheme for varia- tional convergence of constrained minimization problems, the asymptotic analysis (as e → 0) of these problems are made for different values of α and β and different kinds of controls. We have showed that there are three qualitatively different cases. Application for an optimal control problem involving a thick one-level junction with cascade controls is presented as well.

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., Comment: 25 pages, 10 figures

On the basis of an asymptotic analysis of elliptic problems on thin domains and their junctions, a model of a mixed boundary value problem for a second-order scalar differential equation on the union of 3D thin beams and a plate is constructed. One end of each beam is attached to the plate, and on the other end, the Dirichlet conditions are imposed; on the remaining part of the joint boundary, the Neumann boundary conditions are set. An asymptotic expansion of the solution to such a problem has certain distinguishing features; namely, the expansion coefficients turn out to be rational functions of the large parameter |lnh| (where h ∈ (0, 1] is a small geometric parameter), and the solution to the limit problem in the longitudinal section of the plate has logarithmic singularities at the junction points with the beams. Thus, the classical settings of boundary value problems are inadequate to describe the asymptotics, and the technique of self-adjoint extensions and function spaces with separated asymptotics must be used.

In this paper the asymptotic behaviour of a second-order linear evolution problem is studied in a domain, a part of wich has an oscillating boundary. An homogeneous Neumann condition is given on the whole boundary of the domain. Moreover the behaviour of associated optimal control problem is analyzed.

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The uniform Dirichlet conditions and the nonuniform Neumann conditions are given respectively on the sides of the thin rods from the first level and the second level. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct the asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H¹ (Ωε) as ε → 0 (N → +∞). Розглядається мiшана крайова задача для рiвняння Пуассона у плоскому дворiвневому з’єднаннi Ωε, яке є об’єднанням деякої областi Ω₀ та великої кiлькостi 3N тонких стержнiв з товщиною порядку ε = O(N⁻¹). Тонкi стержнi роздiлено на два рiвнi в залежностi вiд їх довжини, i стержнi з кожного рiвня ε-перiодично чергуються. На сторонах тонких стержнiв з першого рiвня задано однорiднi крайовi умови Дiрiхле, а на сторонах стержнiв другого рiвня — неоднорiднi крайовi умови Неймана. З допомогою методу узгодження асимптотичних розвинень та спецiальних розв’язкiв типу примежового шару в зонi з’єднання побудовано асимптотичне наближення для розв’язку даної задачi та доведено вiдповiднi асимптотичнi оцiнки у просторi Соболєва H¹ (Ωε) при ε → 0 (N → +∞).