Asymptotic Analysis - Volume 66, issue 3-4

Authors: Ruzhansky, Michael | Sugimoto, Mitsuru

Article Type: Research Article

Abstract: A necessary and sufficient condition for the resolution of the weak extension problem is given. This criterion is applied to also give a criterion for the solvability of the classical Bochner's extension problem in the Lp -category. The solution of the Lp -extension problem by Bochner [J. Math. Pures Appl. 35 (1956), 193–202] giving the relation between the order of the operator, the dimension, and index p, for which the Lp -extension property holds, can be viewed as a subcritical case of the general Lp -extension problem. In general, this property fails in some critical and in all supercritical cases. …In this paper, the Lp -extension problem is investigated for operators of all orders and for all 1≤p≤∞. Necessary and sufficient conditions on the subset of Lp are given for which the Lp -extension property still holds, in the critical and supercritical cases. Show more

Keywords: extension problem, removable singularities

DOI: 10.3233/ASY-2009-0962

Citation: Asymptotic Analysis, vol. 66, no. 3-4, pp. 125-138, 2010

Authors: Micu, Sorin | de Teresa, Luz

Article Type: Research Article

Abstract: The paper studies the controllability properties of the linear 2-D wave equation in the rectangle Ω=(0, a)×(0, b). We consider two types of action, on an edge or on two adjacent edges of the boundary. Our analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency. In both analyzed cases we give a Fourier characterization of the controllable spaces of initial data and we construct particular controls for them.

Keywords: wave equation, control, Fourier expansion, biorthogonal

DOI: 10.3233/ASY-2009-0963

Citation: Asymptotic Analysis, vol. 66, no. 3-4, pp. 139-160, 2010

Authors: Tabet Tchamba, Thierry

Article Type: Research Article

Abstract: We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton–Jacobi equation ut −Δu+|Du|m =f in Ω×(0, +∞) with inhomogeneous Dirichlet boundary conditions, where Ω is a bounded domain of RN . We mainly focus on the superquadratic case (m>2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f, the initial and boundary data, we connect the problem studied to its associated stationary generalized Dirichlet problem on one hand and to a stationary problem with a state constraint boundary condition on the other hand.

DOI: 10.3233/ASY-2009-0965

Citation: Asymptotic Analysis, vol. 66, no. 3-4, pp. 161-186, 2010

Authors: Korotyaev, Evgeny L. | Kutsenko, Anton

Article Type: Research Article

Abstract: We consider the Schrödinger operator on nanoribbons (tight-binding models) in an external electric potential V on the plane. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V=0, then the spectrum of the Schrödinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on a weak electric potential V→0, then there are two cases: (1) this eigenvalue splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed eigenvalue remains the flat band. We describe all …potentials when the unperturbed eigenvalue remains the flat band and when one splits into the small band of the continuous spectrum. Show more

Keywords: nanoribbon, spectral band, Schrödinger operator, external electric field

DOI: 10.3233/ASY-2009-0966

Citation: Asymptotic Analysis, vol. 66, no. 3-4, pp. 187-206, 2010

Authors: Pastukhova, Svetlana

Article Type: Research Article

Abstract: We study parabolic equations with highly inhomogeneous locally periodic coefficients. The small parameter ε defines the degree of inhomogenity. The difference between the operator exponentials corresponding to the initial and the homogenized equations is of the main interest. We prove the estimate for this difference in operator L2 -norm of order ε1/2 .

Keywords: homogenization, operator estimates, locally periodic functions, parabolic equations

DOI: 10.3233/ASY-2009-0967

Citation: Asymptotic Analysis, vol. 66, no. 3-4, pp. 207-228, 2010

Authors: Morgulis, A.

Article Type: Research Article

Abstract: The paper addresses the dynamics of inviscid incompressible fluid confined within bounded domain with the inflow and outflow of fluid through certain parts of the boundary. This system is non-conservative essentially since the fluxes of energy and vorticity through the flow boundary are not equal to zero. Therefore, the dynamics of such flows should demonstrate the generic non-conservative phenomena such as the asymptotic stability of the equilibria, the onset of instability or the excitation of the self-oscillations, etc. These phenomena are studied extensively for the flows of the viscous fluids but not for the inviscid ones. In this paper, we …prove a sufficient condition for the non-linear asymptotic stability of the inviscid steady flows. Show more

Keywords: asymptotic stability, inviscid fluid, Euler equations

DOI: 10.3233/ASY-2009-0968

Citation: Asymptotic Analysis, vol. 66, no. 3-4, pp. 229-247, 2010

Article Type: Other

Citation: Asymptotic Analysis, vol. 66, no. 3-4, pp. 249-250, 2010