Article
Mathematics, Applied
Alberto Ferrero, Pier Domenico Lamberti
Summary: This paper investigates the stability properties of the spectrum of the classical Steklov problem under domain perturbation, finding conditions for spectral stability and showing their optimality. The convergence of eigenfunctions can be expressed in terms of the H-1 strong convergence.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics
Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, David A. Sher
Summary: This paper investigates the inverse Steklov spectral problem for curvilinear polygons and shows that under specific conditions, the asymptotics of Steklov eigenvalues can determine the number of vertices, the properly ordered sequence of side lengths, and the angles. Counterexamples are presented for when the generic assumptions fail. Additionally, the paper demonstrates the existence of non-isometric triangles with closely related Steklov spectra and utilizes the Hadamard-Weierstrass factorization theorem and other techniques to reconstruct trigonometric functions from their roots' asymptotics.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Applied
Shixian Ren, Yu Zhang, Ziqiang Wang
Summary: An efficient spectral method is proposed for a new Steklov eigenvalue problem in inverse scattering. The weak form and the associated discrete scheme are established by introducing an appropriate Sobolev space and a corresponding approximation space. The error estimates of approximated eigenvalues and eigenfunctions are proved using the spectral approximation results and the approximation properties of orthogonal projection operators. The algorithm is extended to the circular domain and validated through numerical experiments.
Article
Mathematics
A. G. Chechkina
Summary: The spectral problem of the Steklov type for the Laplacian in an unbounded domain with a smooth boundary is considered, where the Steklov condition rapidly alternates with the homogeneous Dirichlet condition on a part of the boundary. Operator estimates are obtained to study the asymptotic behavior of the eigenelements of the original problem as the small parameter approaches zero. The small parameter characterizes the size of the boundary parts with the Dirichlet condition, with distances between them in order of the logarithm of the small parameter to a negative power.
DOKLADY MATHEMATICS
(2021)
Article
Mathematics, Applied
Nunzia Gavitone, Rossano Sannipoli
Summary: In this paper, we investigate the Steklov-Robin eigenvalue problem for the Laplacian in annular domains. We consider a specific type of annular domain and impose Steklov and Robin conditions on the boundaries. We study the first eigenvalue and its properties, including the behavior when varying the norm of beta and the radius of the inner ball. We also analyze the asymptotic behavior of the corresponding eigenfunctions as beta approaches infinity.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Daniel Fortunato, Nicholas Hale, Alex Townsend
Summary: The novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincare-Steklov scheme is competitive and efficient for solving second-order linear PDEs on polygonal domains with unstructured quadrilateral or triangular meshes. The method achieves almost banded linear systems with high polynomial order, enabling fast elliptic solves and hp-adaptivity. The open-source software system, ultraSEM, provides flexible and user-friendly spectral element computations in MATLAB.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics
Leonard Tschanz
Summary: We introduce a graph gamma that is approximately isometric to the hyperbolic plane and investigate the Steklov eigenvalues of a subgraph with boundary omega of gamma. For a sequence of subgraphs (omega(l))(l >= 1) of gamma such that the size of omega(l) approaches infinity, we prove that the k(th) eigenvalue tends to 0 in proportion to 1/|B-l| for each k is an element of N. The proof idea involves finding a bounded domain N in the hyperbolic plane that is approximately isometric to omega, establishing an upper bound for the Steklov eigenvalues of N, and transferring this bound to omega through a process called discretization.
JOURNAL OF GEOMETRIC ANALYSIS
(2023)
Article
Mathematics
Isaac Harris
Summary: This paper discusses the numerical approximation of the Steklov eigenvalue problem in inverse acoustic scattering, focusing on the Galerkin method using Neumann eigenfunctions of the Laplacian as basis functions. Error estimates are proven and the method is tested against separation of variables for validation. The inverse spectral problem of estimating/refractive index from Steklov eigenvalues is also considered, with numerical examples provided.
RESEARCH IN THE MATHEMATICAL SCIENCES
(2021)
Article
Engineering, Multidisciplinary
Onder Turk
Summary: This study proposes a novel approach based on the dual reciprocity boundary element method to approximate solutions for various Steklov eigenvalue problems. The method weights the governing differential equation with fundamental solutions of the Laplace equation, does not require interior nodes, and relates eigenfunctions to boundary flux in a novel way. It efficiently solves moderate-sized generalized eigenvalue problems at a smaller cost compared to full domain discretization alternatives.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2021)
Article
Mathematics
Michel Alexis, Alexander Aptekarev, Sergey Denisov
Summary: This article considers the continuous dependence of weighted operators on L-p(R-d) space, and uses this result to estimate the norm of orthogonal polynomials on the unit circle. Additionally, the asymptotics of polynomial entropy is obtained as an application.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics
A. G. Chechkina
Summary: This problem involves a rapidly changing Steklov problem in an n-dimensional domain. The condition alternates between Steklov condition and homogeneous Dirichlet condition. The coefficient in the Steklov condition is a rapidly oscillating function and has different orders inside and outside the inclusions. The convergence rate of the solution to the original problem is estimated when the small parameter tends to zero in the case of weak singularity.
DOKLADY MATHEMATICS
(2022)
Article
Mathematics, Applied
Hicham Maadan, Nour Eddine Askour, Jamal Messaho
Summary: This study focuses on the limit behavior of weak solutions of an elliptic problem with variable exponent in a specific structure, involving an oscillating nanolayer with thickness and periodicity parameters depending on epsilon. A generalized Sobolev space is constructed, and the epiconvergence method is used to determine the limit problem with interface conditions.
JOURNAL OF FUNCTION SPACES
(2021)
Article
Mathematics, Applied
A. Nachaoui, M. Nachaoui, A. Chakib, M. A. Hilal
Summary: This paper addresses an elliptic inverse Cauchy problem by establishing a relationship between the Cauchy problem and an interface problem, developing regularizing and stable algorithms using classical methods of non-overlapping domain decomposition, and reformulating the inverse problem into a fixed point one. The existence result is shown based on the topological degree of Leray-Schauder, and the efficiency and accuracy of the developed algorithms are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Physics, Mathematical
Yunyun Ma, Jiguang Sun
Summary: A numerical method for solving a non-selfadjoint Steklov eigenvalue problem of the Helmholtz equation is proposed, using boundary integrals and the Nystro??m method for discretization, and the spectral indicator method for eigenvalue calculation. Convergence is proven using spectral approximation theory for compact operators, and the method's effectiveness is validated through numerical examples.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Felipe Lepe, David Mora, Gonzalo Rivera, Ivan Velasquez
Summary: This paper analyzes the influence of small edges in the computation of the spectrum of the Steklov eigenvalue problem using a lowest order virtual element method. The scheme is shown to provide a correct approximation of the spectrum under weaker assumptions on the polygonal meshes. Optimal error estimates for eigenfunctions and double order for eigenvalues are proven, with numerical tests supporting the theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Andrea Braides, Andrey Piatnitski
Summary: In this article, we prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The limit of the corresponding energy is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem that applies to a wide class of domains containing compact periodic perforations.
ADVANCES IN CALCULUS OF VARIATIONS
(2022)
Article
Mathematics
Andrea Braides, Andrey Piatnitski
Summary: The study establishes a homogenization theorem for a class of quadratic convolution energies with random coefficients, showing that the Gamma-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional. The proof relies on results on the asymptotic behavior of subadditive processes and uses a blow-up technique common for local energies.
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
(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
Andrea Braides, Andrey Piatnitski
Summary: By scaling Poisson random sets in the plane, we can obtain an isotropic perimeter energy with an asymptotic formula, which can be achieved by neglecting cells with very long or very short edges in the Voronoi model. Using tools from Geometry Measure Theory and limit theorems, we can define compact convergence and characterize the metric properties of clusters of Voronoi cells.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Mathematics, Applied
Roberto Alicandro, Andrea Braides, Marco Cicalese, Lucia De Luca, Andrey Piatnitski
Summary: This study examines the emergence of topological singularities in periodic media within the Ginzburg-Landau model and the core-radius approach. Through Gamma-convergence analysis, it is found that there are finite number of vortex-like point singularities of integer degree at specific scales. Additionally, a separation-of-scale effect is demonstrated, where concentration processes occur around vortices at certain scales before subsequent optimization, followed by homogenization at larger scales.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Mathematics
B. Amaziane, L. Pankratov, A. Piatnitski
Summary: The paper discusses the stochastic homogenization of a system modeling immiscible compressible two-phase flow in random porous media, and successfully proves the convergence of solutions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Gregory A. Chechkin
Summary: This paper considers an elliptic problem in a domain perforated along the boundary. By imposing certain boundary conditions, the higher integrability of the gradient of the solution is proven.
Article
Mathematics
M. A. Kisatov, V. N. Samokhin, G. A. Chechkin
Summary: We generalize the existence and uniqueness theorem for a classical solution to the system of equations describing thermal boundary layers in viscous media with the Ladyzhenskaya rheological law.
DOKLADY MATHEMATICS
(2022)
Article
Mathematics, Applied
Yurij A. Alkhutov, Gregory A. Chechkin, Vladimir G. Maz'ya
Summary: This study considers the variational solution to the Zaremba problem for a divergent linear second-order elliptic equation with measurable coefficients. The problem is set in a local Lipschitz graph domain. An estimate in L2+delta, delta > 0, for the gradient of a solution is proved, and an example of the problem with specific Dirichlet data supported by a fractal set of zero (n-1)-dimensional measure and non-zero p-capacity, p > 1, is constructed.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
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.
Correction
Mathematics
M. A. Kisatov, V. N. Samokhin, G. A. Chechkin
DOKLADY MATHEMATICS
(2022)
Article
Mathematics
K. A. Bekmaganbetov, V. V. Chepyzhov, G. A. Chechkin
Summary: The system of reaction-diffusion equations in a perforated domain with rapidly oscillating terms was studied, showing weak convergence of trajectory attractors to a homogeneous system with a strange term in the corresponding topology.
DOKLADY MATHEMATICS
(2021)
Article
Mathematics
Yu. A. Alkhutov, G. A. Chechkin
Summary: The study provides an estimate for the increased integrability of the gradient of the solution to the Zaremba problem in a bounded plane domain with a Lipschitz boundary and rapidly alternating Dirichlet and Neumann boundary conditions, with the increased integrability exponent being independent of the frequency of the boundary condition change.
DOKLADY MATHEMATICS
(2021)
Article
Mechanics
Yurij A. Alkhutov, Gregory A. Chechkin
Summary: This paper provides an estimate for the increased integrability of the gradient of the solution to the Zaremba problem for a divergent elliptic operator in a bounded domain with nontrivial capacity of the Dirichlet boundary conditions.
COMPTES RENDUS MECANIQUE
(2021)
Article
Mechanics
Gregory A. Chechkin, Tudor S. Ratiu, Maxim S. Romanov
Summary: This paper introduces the three-dimensional Eringen system of equations for nematodynamics of liquid crystals, proving the short-time existence and uniqueness of strong solutions for the one-dimensional problem in the periodic case, and demonstrating the continuous dependence of the solution on the initial data.
COMPTES RENDUS MECANIQUE
(2021)
Article
Mathematics, Applied
Geunsu Choi, Mingu Jung, Sun Kwang Kim, Miguel Martin
Summary: This paper studies weak-star quasi norm attaining operators and proves that the set of such operators is dense in the space of bounded linear operators regardless of the choice of Banach spaces. It is also shown that weak-star quasi norm attaining operators have distinct properties from other types of norm attaining operators, although they may share some equivalent properties under certain conditions.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Maria Lorente, Francisco J. Martin-Reyes, Israel P. Rivera-Rios
Summary: In this paper, we provide quantitative one-sided estimates that recover the dependences in the classical setting. We estimate the one-sided maximal function in Lorentz spaces and demonstrate the applicability of the conjugation method for commutators in this context.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Fernando Cobos, Luz M. Fernandez-Cabrera
Summary: We provide a necessary and sufficient condition for the weak compactness of bilinear operators interpolated using the real method. However, this characterization does not hold for interpolated operators using the complex method.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Ovgue Gurel Yilmaz, Sofiya Ostrovska, Mehmet Turan
Summary: The Lupas q-analogue Rn,q, the first q-version of the Bernstein polynomials, was originally proposed by A. Lupas in 1987 but gained popularity 20 years later when q-analogues of classical operators in approximation theory became a focus of intensive research. This work investigates the continuity of operators Rn,q with respect to the parameter q in both the strong operator topology and the uniform operator topology, considering both fixed and infinite n.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
M. Agranovsky, A. Koldobsky, D. Ryabogin, V. Yaskin
Summary: This article modifies the concept of polynomial integrability for even dimensions and proves that ellipsoids are the only convex infinitely smooth bodies satisfying this property.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Abel Komalovics, Lajos Molnar
Summary: In this paper, a parametric family of two-variable maps on positive cones of C*-algebras is defined and studied from various perspectives. The square roots of the values of these maps under a faithful tracial positive linear functional are considered as a family of potential distance measures. The study explores the problem of well-definedness and whether these distance measures are true metrics, and also provides some related trace characterizations. Several difficult open questions are formulated.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Frederic Bayart
Summary: The passage describes the construction of an operator on a separable Hilbert space that is 5-hypercyclic for all δ in the range (ε, 1) and is not 5-hypercyclic for all δ in the range (0, ε).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Helene Frankowska, Nikolai P. Osmolovskii
Summary: This paper investigates second-order optimality conditions for the minimization problem of a C2 function f on a general set K in a Banach space X. Both necessary and sufficient conditions are discussed, with the sufficiency condition requiring additional assumptions. The paper demonstrates the validity of these assumptions for the case when the set K is an intersection of sets described by smooth inequalities and equalities, such as in mathematical programming problems. The novelty of the approach lies in the arbitrary nature of the set K and the straightforward proofs.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Ole Fredrik Brevig, Kristian Seip
Summary: This paper studies the Hankel operator on the Hardy space and discusses its minimal and maximal norms, as well as the relationship between the maximal norm and the properties of the function.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Alexander Meskhi
Summary: Rubio de Francia's extrapolation theorem is established for new weighted grand Morrey spaces Mp),lambda,theta w (X) with weights w beyond the Muckenhoupt Ap classes. This result implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. The necessary conditions for the boundedness of the Hardy-Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Maud Szusterman
Summary: In this work, the necessary conditions on the structure of the boundary of a convex body K to satisfy all inequalities are investigated. A new solution for the 3-dimensional case is obtained in particular.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Rami Ayoush, Michal Wojciechowski
Summary: In this article, lower bounds for the lower Hausdorff dimension of finite measures are provided under certain restrictions on their quaternionic spherical harmonics expansions. This estimate is analogous to a result previously obtained by the authors for complex spheres.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
F. G. Abdullayev, V. V. Savchuk
Summary: This paper investigates the convergence and theorem proof of the Takenaka-Malmquist system and Fejer-type operator on the unit circle, and provides relevant results on the class of holomorphic functions representable by Cauchy-type integrals with bounded densities.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Sofiya Ostrovska, Mikhail I. Ostrovskii
Summary: This work aims to establish new results on the structure of transportation cost spaces. The main outcome of this paper states that if a metric space X contains an isometric copy of L1 in its transportation cost space, then it also contains a 1-complemented isometric copy of $1.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Pilar Rueda, Enrique A. Sanchez Perez
Summary: We prove a factorization theorem for Lipschitz operators acting on certain subsets of metric spaces of measurable functions and with values on general metric spaces. Our results show how a Lipschitz operator can be extended to a subset of other metric space of measurable functions that satisfies the following optimality condition: it is the biggest metric space, formed by measurable functions, to which the operator can be extended preserving the Lipschitz constant. Also, we demonstrate the coarsest metric that can be given for a metric space in which an order bounded lattice-valued-Lipschitz map is defined, and provide concrete examples involving the relevant space L0(mu).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
