Article
Mathematics, Applied
Gokhan Mutlu, Ekin Ugurlu
Summary: We examine the spectral properties of two different boundary value problems on a compact star graph with vertex conditions dependent on the spectral parameter. Treating these problems as eigenvalue problems in extended Hilbert spaces associated with vector-valued operators, we prove that the corresponding operators are self-adjoint. We construct the characteristic functions and show that the operators have discrete spectra. Additionally, we provide examples where fundamental solutions are constructed and resolvent operators are derived.
ANALYSIS AND MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Paul Deuring
Summary: This article discusses the homogeneous Stokes resolvent system in a 3D exterior domain under inhomogeneous Dirichlet boundary conditions, estimating solutions in L-p norms with bounds depending on the absolute value of the resolvent parameter lambda. Two types of boundary data are considered, and it is shown that the L-p norm of the velocity outside the boundary, subtracting a certain harmonic function gradient, is bounded by a constant times |lambda|(-1) ||b||(p). This estimate carries over to the Oseen resolvent system, with applications in the theory of spatial asymptotics of solutions to the 3D time-dependent Navier-Stokes system with Oseen term.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Physics, Mathematical
Gino Biondini, Sitai Li, Dionyssios Mantzavinos
Summary: This study investigates the long-time behavior of solutions to the focusing nonlinear Schrodinger equation with symmetric, nonzero boundary conditions, exploring the nonlinear interactions between solitons and coherent oscillating structures. Combining inverse scattering transform and nonlinear steepest descent method, it reveals that the presence of a conjugate pair of discrete eigenvalues can lead to various outcomes such as soliton transmission, soliton trapping, or a mixed regime. The results are supported by accurate numerical simulations validating the soliton-induced position and phase shifts.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Renjun Duan, Shuangqian Liu
Summary: The study examines the uniform shear flow of rarefied gas and identifies the reason for temperature increase over time, as well as the characteristics of self-similar profiles. The non-negativity of these profiles is justified through large time asymptotic stability.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
Article
Mathematics, Applied
Muhammad Mustafa
Summary: This paper investigates a laminated beam model, where two identical uniform layers are bonded together with a small thickness adhesive, resulting in interfacial slip. By considering nonlinear boundary controls and using the multiplier method, a uniform stability result is established without imposing any conditions on the coefficients of the system, providing an explicit formula for the energy decay rates.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mechanics
V. Monchiet, G. Bonnet
Summary: In this paper, the FFT method is extended to handle the homogenization problem of composite conductors with uniform boundary conditions. The method applies a transformation to build a periodic problem from the solution with uniform boundary conditions. The extended domain obtained by mirror symmetry of the unit cell is used to solve the conductivity equation under an applied periodic polarization field. The effectiveness of the proposed method is validated by comparing the effective conductivity obtained with FFT to finite element solutions. The method can be applied to various microstructure geometries, including cells obtained through imaging devices.
EUROPEAN JOURNAL OF MECHANICS A-SOLIDS
(2024)
Article
Mathematics, Applied
Mifodijus Sapagovas, Jurij Novickij
Summary: This paper considers the stability of the alternating direction method for wave equation with integral boundary conditions in an energy norm. The proof of stability is based on the properties of eigenvalues and eigenvectors of the corresponding difference operators. The main properties of the alternating direction method are theoretically and numerically proven.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics
Veronica Felli, Benedetta Noris, Roberto Ognibene
Summary: This article deals with eigenvalue problems for the Laplacian with varying mixed boundary conditions, where homogeneous Neumann conditions are imposed on a vanishing portion of the boundary and Dirichlet conditions are imposed on the complement. Through the study of an Almgren-type frequency function, upper and lower bounds of the eigenvalue variation are derived, along with sharp estimates in the case of a strictly star-shaped Neumann region.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Automation & Control Systems
Michael Hinz, Anna Rozanova-Pierrat, Alexander Teplyaev
Summary: In this paper, we introduce new parametrized classes of shape admissible domains in R-n and prove their compactness in various senses. These domains are bounded (epsilon, infinity)-domains with possibly fractal boundaries, and we demonstrate the existence of optimal shapes for maximum energy dissipation in this framework. Additionally, we establish stability and convergence results for certain classes of domains and energy functionals.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Fanghua Lin, Zhongwei Shen
Summary: Sharp convergence rates for solutions of wave equations in a bounded domain with rapidly oscillating periodic coefficients are obtained using Dirichlet correctors and are used to prove exact boundary controllability. The results ensure uniformity in the projection of solutions to the subspace generated by eigenfunctions with eigenvalues less than C epsilon(-2/3).
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2022)
Article
Chemistry, Physical
Jicheng Wang, Yong Yu, Jiaqing He, Jitong Wang, Baopeng Ma, Xiaolian Chao, Zupei Yang, Di Wu
Summary: The study improved the performance of SnTe thermoelectric material through alloying and doping, achieving a significant reduction in lattice thermal conductivity while simultaneously achieving high ZT values.
ACS APPLIED ENERGY MATERIALS
(2021)
Article
Mathematics, Applied
Lorenzo Freddi, Peter Hornung, Maria Giovanna Mora, Roberto Paroni
Summary: This paper proves that when prescribed with affine boundary conditions on the short sides of the strip, the (extended) Sadowsky functional can still be deduced as the Gamma-limit of the Kirchhoff energy on a rectangular strip, providing a rigorous theoretical basis for Sadowsky's original argument about the equilibrium shape of a Mobius strip.
JOURNAL OF NONLINEAR SCIENCE
(2022)
Article
Mathematics, Applied
Sunil Kumar, Sumit, Higinio Ramos
Summary: The paper presents a parameter-uniform numerical method for solving singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions, using a modified Euler scheme in time, a central difference scheme in space, and a special finite difference scheme for the boundary conditions. The method is proven to converge of order two in space and order one in time, with numerical experiments supporting the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Satpal Singh, Devendra Kumar, Komal Deswal
Summary: In this paper, a non-polynomial-based trigonometric cubic B-spline collocation method is developed to solve reaction-diffusion singularly perturbed problems with Robin boundary conditions. The proposed scheme utilizes piecewise uniform mesh and modification of the mesh to enhance the accuracy of the numerical results. Numerical experiments validate the performance and theoretical findings of the method.
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS
(2022)
Article
Engineering, Multidisciplinary
Hannes Grimm-Strele, Matthias Kabel
Summary: The FFT-based homogenization method is established as a fast, accurate, and robust tool for periodic homogenization in solid mechanics. The implementation of nonperiodic boundary conditions efficiently reduces runtime and memory requirements compared to the domain mirroring approach. The use of periodic boundary conditions for nonperiodic geometries yields vastly different results than with PMUBC, and the influence of the discretization method on the results is examined.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2021)
Article
Physics, Mathematical
Denis Borisov, Matthias Taeufer, Ivan Veselic
Summary: This study focuses on multi-dimensional Schrödinger operators with weak random perturbations distributed in periodic lattices and random Hamiltonians defined on multi-dimensional layers. The location of the almost sure spectrum and its dependence on the global coupling constant are determined, particularly in cases where the spectrum expands when the perturbation is switched on. The study also involves deriving estimates and analyzing localization based on abstract perturbations, with implications for various known and new examples.
JOURNAL OF STATISTICAL PHYSICS
(2021)
Article
Mathematics, Applied
D. Borisov, G. Cardone, G. A. Chechkin, Yu O. Koroleva
Summary: The study focuses on a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition, involving singular perturbation and Dirichlet condition. It demonstrates the norm convergence of an operator related to the unperturbed problem, establishes a sharp estimate for the convergence rate, and shows the convergence of spectra and spectral projectors. Furthermore, the research examines perturbed eigenvalues converging to simple discrete limiting ones, constructing two-terms asymptotic expansions for these eigenvalues and associated eigenfunctions.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
D. Borisov, A. M. Golovina
Summary: This paper studies a general elliptic operator in an infinite multi-dimensional cylinder with several distant perturbations, showing that under certain conditions, the resonances of this type of operator are finite in number and the leading terms in the asymptotic expansion of these resonances are exponentially small. The authors conjecture that this scenario is unique when distant perturbations produce only a finite number of resonances near a real number lambda(0).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Nanoscience & Nanotechnology
Giuseppe Cardone, Tiziana Durante
Summary: This article studies a second order elliptic operator in a planar waveguide, where small holes are distributed along a curve and subject to classical boundary conditions on the holes. Under the weak assumptions on the perforation, all possible homogenized problems are described.
NANOSYSTEMS-PHYSICS CHEMISTRY MATHEMATICS
(2022)
Article
Physics, Multidisciplinary
Miloslav Znojil, Denis I. Borisov
Summary: This paper demonstrates the different applications of Arnold's one-dimensional polynomial potentials in classical catastrophe theory and quantum mechanics, particularly within specific dynamical regimes. By relaxing constraints and utilizing perturbative methods, the characteristics of these potentials are investigated.
Article
Mathematics, Applied
Srinivasan Aiyappan, Giuseppe Cardone, Carmen Perugia, Ravi Prakash
Summary: This paper studies the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary, while the Dirichlet condition is considered on the smooth separate part. By using the unfolding method and making natural hypotheses on the regularity of the domain, we prove the weak L-p-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2022)
Article
Mathematics
D. I. Borisov, P. Exner
Summary: In this paper, a new type of approximation for a second-order elliptic operator with a point interaction in a planar domain is presented. The approximation is of a geometric nature and consists of operators with the same symbol and regular coefficients on the domain with a small hole. The boundary condition is imposed at the boundary of the hole with a coefficient depending on the linear size of the hole. It is shown that as the hole shrinks to a point and the parameter in the boundary condition is scaled appropriately, the approximating family converges in the norm-resolvent sense to the operator with the point interaction. The convergence is established with respect to several operator norms and the convergence rates are estimated.
BULLETIN OF MATHEMATICAL SCIENCES
(2023)
Article
Mathematics, Applied
D. I. Borisov, A. L. Piatnitski, E. A. Zhizhina
Summary: This paper focuses on the spectral properties of a bounded self-adjoint operator in L-2(R-d) that is the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. The essential and discrete spectra of this operator are studied. The essential spectrum of the sum is shown to be the union of the essential spectrum of the convolution operator and the image of the potential. Sufficient conditions for the existence of discrete spectrum are provided, and lower and upper bounds for the number of discrete eigenvalues are obtained. The spectral properties of the operators considered in this work are compared with those of classical Schrodinger operators.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
G. Cardone, A. Fouetio, S. Talla Lando, J. L. Woukeng
Summary: This work investigates the global dynamics of 2D stochastic tidal equations in a highly heterogeneous environment. Using the stochastic version of the sigma-convergence method and the Prokhorov and Skorokhod compactness theorems, the paper proves that the dynamics at the macroscopic level are of the same type as those at the microscopic level, but with non-oscillating parameters. A corrector-type result is also established.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics, Applied
D. I. Borisov, J. Kriz
Summary: In this paper, we consider a second order linear elliptic equation in a finely perforated domain. The shapes and distribution of cavities in the domain are arbitrary and non-periodic, and the boundary conditions can be either Dirichlet or nonlinear Robin conditions. We show that under certain conditions, the solution to our problem tends to zero as the perforation becomes finer.
ANALYSIS AND MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
D. I. Borisov
Summary: In this paper, a boundary value problem for a general second-order linear equation in a perforated domain is considered. The perforation is made by small cavities with a minimal distance between them also being small. Minimal natural geometric conditions are imposed on the shapes of the cavities, while no conditions are imposed on their distribution in the domain. A nonlinear Robin condition is imposed on the boundaries of the cavities. The main results of the paper demonstrate the convergence of the solution of the perturbed problem to that of the homogenized one, providing estimates for the convergence rates in W-2(1)- and L-2-norms uniformly in the L-2-norm of the right-hand side in the equation.
APPLICABLE ANALYSIS
(2023)
Article
Mathematics
Denis Ivanovich Borisov
Summary: We introduce the concept of point interaction for general non-self-adjoint elliptic operators in planar domains. By cutting out a small cavity around the point, we show that these operators can be geometrically approximated. A special Robin-type boundary condition with a nonlocal term is imposed on the boundary of the cavity. As the cavity shrinks to a point, the perturbed operator converges in the norm resolvent sense to a limiting operator with a point interaction containing an arbitrary complex-valued coupling constant. We establish convergence rates for several operator norms. The convergence of the spectrum is proven as a corollary of the norm resolvent convergence.
Article
Mathematics, Applied
S. Aiyappan, Giuseppe Cardone, Carmen Perugia
Summary: In this study, we examine the asymptotic behavior of a linear optimal control problem posed on a locally periodic rapidly oscillating domain. The problem involves an L2-cost functional constrained by a Poisson problem with a mixed boundary condition: a homogeneous Neumann condition on the oscillating part of the boundary and a homogeneous Dirichlet condition on the remaining part.
APPLICABLE ANALYSIS
(2023)
Article
Mathematics
D. I. Borisov
Summary: We study the existence of a limiting operator and the convergence of resolvent norm for a general second order matrix operator subject to a classical boundary condition in a multi-dimensional domain, perturbed by a first order differential operator depending on a small multi-dimensional parameter. Our main results show that the convergence of resolvent norm is equivalent to the convergence of coefficients in the perturbing operator in certain spaces of multipliers. Furthermore, we find that the convergence in these spaces is equivalent to the convergence of local mean values over small pieces of the domain. Our results are supported by examples of non-periodically oscillating perturbations.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
D. I. Borisov, A. L. Piatnitski, E. A. Zhizhina
Summary: This article considers a multiplication operator in L2(R) multiplied by a complex-valued potential, to which a convolution operator multiplied by a small parameter is added. The essential spectrum of such an operator is found in an explicit form, and it is shown that the entire spectrum is located in a thin neighborhood of the spectrum of the multiplication operator.
