Article
Mathematics, Applied
Xiaomeng Zhao, Ganghua Yuan
Summary: In this paper, we study the partial data inverse boundary value problem for the Schrodinger operator at a high frequency in a bounded domain with smooth boundary. We obtained logarithmic stability and Holder-type stability by combining the CGO solution, Runge approximation, and Carleman estimate.
Article
Mathematics, Applied
Jean Carlos Nakasato, Marcone Correa Pereira
Summary: In this work, the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating (N+1)-dimensional thin domains is analyzed. Monotone nonlinear boundary conditions on the rough border are also permitted, with their magnitude depending on the squeezing of the domain. Different regimes establishing effective homogenized limits in N-dimensional open bounded sets are obtained based on the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition. Monotone operator analysis techniques and the unfolding method are combined to tackle asymptotic analysis and homogenization problems.
ADVANCED NONLINEAR STUDIES
(2023)
Article
Engineering, Multidisciplinary
Parikshit Boregowda, Gui-Rong Liu
Summary: This work presents a novel particle consistent gradient formulation for imposing Dirichlet boundary condition on a Laplacian with boundary integrals. Boundary integral formulation allows accurate imposition of boundary conditions without creating virtual particles to extend boundaries. The convergence characteristics of different Laplacian formulations with boundary integrals are discussed and numerical experiments are conducted to demonstrate the robustness of various Laplacian operators for practical use.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Mathematics, Applied
Steve Hofmann, Guoming Zhang
Summary: In this paper, commutator estimates for the Dirichlet-to-Neumann Map associated with a divergence form elliptic operator in the upper half-space R-+(n+1) := {(x, t) ∈ R-n × (0, infinity)} with uniformly complex elliptic, L-infinity, t-independent coefficients are established. These results extend corresponding results for the Laplacian in a Lipschitz domain by Kenig, Lin, and Shen, with applications to the theory of homogenization.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Engineering, Multidisciplinary
Oliver Boolakee, Martin Geier, Laura De Lorenzis
Summary: We propose novel, second-order accurate boundary formulations for Dirichlet and Neumann boundary conditions on arbitrary curved boundaries. The proposed methodology is based on the asymptotic expansion technique and is expected to have general applicability beyond the scope of this paper. We develop a modified version of the bounce-back method for Dirichlet boundary conditions, and a novel generalized ansatz for Neumann boundary conditions that requires information from one additional neighbor node.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Santiago Cano-Casanova
Summary: This article investigates the existence of positive weak solutions for a non-uniform elliptic boundary value problem of logistic type in a general annulus. The novelty of this research lies in considering non-classical mixed glued boundary conditions, specifically Dirichlet conditions on one part of the boundary and glued Dirichlet-Neumann conditions on the other part. The paper provides a comprehensive analysis of the existence of positive weak solutions, presenting necessary and sufficient conditions depending on the lambda parameter, spatial dimension N >= 2, and exponent q > 1 of the reaction term. The main mathematical techniques employed in this study are variational and monotonicity techniques. The results obtained in this paper are pioneering in this field as, to the author's knowledge, this is the first analysis of such logistic problems.
Article
Statistics & Probability
Sigurd Assing, John Herman
Summary: Recent research shows that complete Bernstein functions of the Laplace operator can map the Dirichlet boundary condition to the Neumann boundary condition for related elliptic PDEs. This mapping is important as it allows problems involving non-local operators to be converted into problems involving only differential operators. This result is generalized to diffusion operators associated with stochastic differential equations using a method based on stochastic analysis.
ELECTRONIC JOURNAL OF PROBABILITY
(2021)
Article
Mathematics
S. Mayboroda, B. Poggi
Summary: This study establishes an analog perturbation result for degenerate elliptic operators, developed by David, Feneuil, and Mayboroda, similar to the result by Fefferman, Kenig, and Pipher for divergence form elliptic operators. It shows the stability of the solvability of the Dirichlet problem under certain boundary conditions and perturbations.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics, Applied
Bastien Chaudet-Dumas, Martin J. Gander
Summary: This paper studies the convergence of the Dirichlet-Neumann method at the continuous level in the presence of cross-points for the first time. The study shows that the iterates of the method can be decomposed into two parts: an even symmetric part that converges geometrically when there are no cross-points, and an odd symmetric part that generates singularity at the cross-point and is not convergent.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi
Summary: This paper considers the homogenization problem for a nonlocal equation involving different smooth kernels, and shows the existence of a homogenized limit system with three kernels and a limit function under specific conditions. Both Neumann and Dirichlet boundary conditions are addressed, and a probabilistic interpretation of the results is also provided.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Wolfgang Arendt, James B. Kennedy
Summary: This paper proves that if an operator intertwining two appropriate realizations of the Laplacian on a pair of domains preserves disjoint supports, and satisfies additional assumptions, then the domains are congruent, even without the strong requirement of unitarity.
JOURNAL OF SPECTRAL THEORY
(2021)
Article
Mathematics
Hans-Christoph Grunau, Giulio Romani, Guido Sweers
Summary: This study focuses on fundamental solutions of elliptic operators with constant coefficients in large dimensions, exploring how the sign of the fundamental solution can change near singularities. An inductive argument by space dimension reveals that sign change in some dimension implies sign change in larger dimensions for certain operators. Analysis of symbol expressions indicates that the sign of the fundamental solution depends on the dimension for such operators.
MATHEMATISCHE ANNALEN
(2021)
Article
Mathematics
S. Bortz, T. Toro, Z. Zhao
Summary: The article focuses on the quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator. It proves that the elliptic measure of an operator with coefficients satisfying a specific condition is an asymptotically optimal weight. The article introduces a new framework through quantitative estimates on a quantity that controls the A(infinity) constant, which may be applied to similar problems.
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics, Applied
Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi
Summary: This paper discusses the homogenization of an evolution problem involving a jump process with three different smooth kernels. By dividing the spatial domain into two subdomains and assuming weak convergence of certain functions, the existence of a homogenized limit system is proven. The probabilistic interpretation of the evolution equation is also presented, along with the convergence of the underlying process to a limit process. The analysis focuses on Neumann type boundary conditions, with a brief mention of how to handle Dirichlet boundary conditions.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Ali BenAmor
Summary: In this work, necessary and sufficient conditions for defining a Dirichlet-to-Neumann operator via Dirichlet principle in the framework of Hilbert spaces are given. The analysis of singular Dirichlet-to-Neumann operators includes establishing Laurent expansion near singularities and Mittag-Leffler expansion for related quadratic forms. The results obtained are applied to definitively solve the positivity problem of the related semigroup in the framework of Lebesgue spaces.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics
Jose M. Arrieta, Rosa Pardo, Anibal Rodriguez-Bernal
JOURNAL OF DIFFERENTIAL EQUATIONS
(2015)
Article
Mathematics, Applied
Jose M. Arrieta, Francesco Ferraresso, Pier Domenico Lamberti
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2018)
Article
Mathematics, Applied
Jose M. Arrieta, Manuel Villanueva-Pesqueira
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2016)
Article
Mathematics
Jose M. Arrieta, Francesco Ferraresso, Pier Domenico Lamberti
INTEGRAL EQUATIONS AND OPERATOR THEORY
(2017)
Article
Mathematics
Jose M. Arrieta, Esperanza Santamaria
JOURNAL OF DIFFERENTIAL EQUATIONS
(2017)
Article
Mathematics, Applied
Jose M. Arrieta, Manuel Villanueva-Pesqueira
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2017)
Article
Mathematics, Applied
Jose M. Arrieta, Esperanza Santamaria
COLLECTANEA MATHEMATICA
(2018)
Article
Mathematics, Applied
Jose M. Arrieta, Gerassimos Barbatis
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2014)
Article
Mathematics, Applied
Jose M. Arrieta, Manuel Villanueva-Pesqueira
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2014)
Article
Mathematics, Applied
Jose M. Arrieta, Ariadne Nogueira, Marcone C. Pereira
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2019)
Article
Mathematics, Applied
Jose M. Arrieta, Ariadne Nogueira, Marcone C. Pereira
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2019)
Proceedings Paper
Mathematics, Interdisciplinary Applications
Jose M. Arrieta, Rosa Pardo, Anibal Rodriguez-Bernal
ADVANCES IN DIFFERENTIAL EQUATIONS AND APPLICATIONS
(2014)
Proceedings Paper
Mathematics, Interdisciplinary Applications
Jose M. Arrieta, Manuel Villanueva-Pesqueira
ADVANCES IN DIFFERENTIAL EQUATIONS AND APPLICATIONS
(2014)
Article
Mathematics, Applied
Jose M. Arrieta, Rosa Pardo, Anibal Rodriguez-Bernal
ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS
(2014)
Article
Mathematics
Daniele Cassani, Zhisu Liu, Giulio Romani
Summary: This article investigates the strongly coupled nonlinear Schrodinger equation and Poisson equation in two dimensions. The existence of solutions is proved using a variational approximating procedure, and qualitative properties of the solutions are established through the moving planes technique.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Giovanni Alessandrini, Romina Gaburro, Eva Sincich
Summary: This paper considers the inverse problem of determining the conductivity of a possibly anisotropic body Ω, subset of R-n, by means of the local Neumann-to-Dirichlet map on a curved portion Σ of its boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities, the paper provides a Hölder stability estimate on Σ when the conductivity is a priori known to be a constant matrix near Σ.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nuno Costa Dias, Cristina Jorge, Joao Nuno Prata
Summary: This article studies the time dependent Euler-Bernoulli beam equation with discontinuous and singular coefficients, and obtains an explicit formulation of the differential problem using an extension of the Hormander product of distributions. The dynamics of the Euler-Bernoulli beam model with discontinuous flexural stiffness and structural cracks are further explored, and the relationship between the characteristic frequencies of the beam and the singularities in the flexural stiffness is investigated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Baoquan Zhou, Hao Wang, Tianxu Wang, Daqing Jiang
Summary: This paper is Part I of a two-part series that presents a mathematical framework for approximating the invariant probability measures and density functions of stochastic generalized Kolmogorov systems with small diffusion. It introduces two new approximation methods and demonstrates their utility in various applications.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Yun Li, Danhua Jiang, Zhi-Cheng Wang
Summary: In this study, a nonlocal reaction-diffusion equation is used to model the growth of phytoplankton species in a vertical water column with changing-sign advection. The species relies solely on light for metabolism. The paper primarily focuses on the concentration phenomenon of phytoplankton under conditions of large advection amplitude and small diffusion rate. The findings show that the phytoplankton tends to concentrate at certain critical points or the surface of the water column under these conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Monica Conti, Stefania Gatti, Alain Miranville
Summary: The aim of this paper is to study a perturbation of the Cahn-Hilliard equation with nonlinear terms of logarithmic type. By proving the existence, regularity and uniqueness of solutions, as well as the (strong) separation properties of the solutions from the pure states, we finally demonstrate the convergence to the Cahn-Hilliard equation on finite time intervals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Xiaolong He
Summary: This article employs the CWB method to construct quasi-periodic solutions for nonlinear delayed perturbation equations, and combines the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, thus extending the applicability of the CWB method. As an application, it studies the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nicolas Camps, Louise Gassot, Slim Ibrahim
Summary: In this paper, we consider the probabilistic local well-posedness problem for the Schrodinger half-wave equation with a cubic nonlinearity in quasilinear regimes. Due to the lack of probabilistic smoothing in the Picard's iterations caused by high-low-low nonlinear interactions, we need to use a refined ansatz. The proof is an adaptation of Bringmann's method on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, ill-posedness results for this equation are discussed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Elie Abdo, Mihaela Ignatova
Summary: In this study, we investigate the Nernst-Planck-Navier-Stokes system with periodic boundary conditions and prove the exponential nonlinear stability of constant steady states without constraints on the spatial dimension. We also demonstrate the exponential stability from arbitrary large data in the case of two spatial dimensions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Peter De Maesschalck, Joan Torregrosa
Summary: This paper provides the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. The key idea is the perturbation of a vector field with many cusp equilibria, which is constructed using elements of catastrophe theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Leyi Jiang, Taishan Yi, Xiao-Qiang Zhao
Summary: This paper studies the propagation dynamics of a class of integro-difference equations with a shifting habitat. By transforming the equation using moving coordinates and establishing the spreading properties of solutions and the existence of nontrivial forced waves, the paper contributes to the understanding of the propagation properties of the original equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Mckenzie Black, Changhui Tan
Summary: This article investigates a family of nonlinear velocity alignments in the compressible Euler system and shows the asymptotic emergent phenomena of alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are studied, resulting in a variety of different asymptotic behaviors.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Lorenzo Cavallina
Summary: In this paper, the concept of variational free boundary problem is introduced, and a unified functional-analytical framework is provided for constructing families of solutions. The notion of nondegeneracy of a critical point is extended to this setting.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Ying-Chieh Lin, Kuan-Hsiang Wang, Tsung-Fang Wu
Summary: In this study, we investigate a linearly coupled Schrodinger system and establish the existence of positive ground states under suitable assumptions and by using variational methods. We also relax some of the conditions and provide some results on the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
