We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity.
对于h，我们考虑E >0，半经典Schrödinger算子- h2 Δ + V−e的解析估计。接近无穷时，势的形式为V = V L + V S，其中V L是一个长范围势，它是关于径向变量的Lipschitz势，而V S = O (| x |−1 (log | x |)−ρ)对于某些ρ >1. 在原点附近，如果0≤β <，则| V |可能表现为| x |−β;2(3−1)。我们发现，对于任意ρ≈>1、有C, h 0 >对于所有h∈(0,h 0)，我们有一个形式为exp (C h−2 (log (h−1))1 + ρ≈的可解界。如果V S以更快的速度向无穷远处衰减，边界的h依赖性就会提高。
{"title":"Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin","authors":"Jacob Shapiro","doi":"10.3233/asy-231872","DOIUrl":"https://doi.org/10.3233/asy-231872","url":null,"abstract":"We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135413834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions.
{"title":"Higher order evolution inequalities involving Leray–Hardy potential singular on the boundary","authors":"Mohamed Jleli, Bessem Samet, Calogero Vetro","doi":"10.3233/asy-231873","DOIUrl":"https://doi.org/10.3233/asy-231873","url":null,"abstract":"We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136142340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a class of ( p , q )-Laplacian equations with critical growth is taken into consideration: − Δ p u − Δ q u + ( | u | p − 2 + | u | q − 2 ) u + λ ϕ | u | q − 2 u = μ g ( u ) + | u | q ∗ − 2 u , x ∈ R 3 , − Δ ϕ = | u | q , x ∈ R 3 , where Δ ξ u = div ( | ∇ u | ξ − 2 ∇ u ) is the ξ-Laplacian operator ( ξ = p , q ), 3 2 < p < q < 3, λ and μ are positive parameters, q ∗ = 3 q / ( 3 − q ) is the Sobolev critical exponent. We use a primary technique of constrained minimization to determine the existence, energy estimate and convergence property of nodal (that is, sign-changing) solutions under appropriate conditions on g, and thus generalize the existing results.
{"title":"Nodal solutions to ( p , q )-Laplacian equations with critical growth","authors":"Hongling Pu, Sihua Liang, Shuguan Ji","doi":"10.3233/asy-231871","DOIUrl":"https://doi.org/10.3233/asy-231871","url":null,"abstract":"In this paper, a class of ( p , q )-Laplacian equations with critical growth is taken into consideration: − Δ p u − Δ q u + ( | u | p − 2 + | u | q − 2 ) u + λ ϕ | u | q − 2 u = μ g ( u ) + | u | q ∗ − 2 u , x ∈ R 3 , − Δ ϕ = | u | q , x ∈ R 3 , where Δ ξ u = div ( | ∇ u | ξ − 2 ∇ u ) is the ξ-Laplacian operator ( ξ = p , q ), 3 2 < p < q < 3, λ and μ are positive parameters, q ∗ = 3 q / ( 3 − q ) is the Sobolev critical exponent. We use a primary technique of constrained minimization to determine the existence, energy estimate and convergence property of nodal (that is, sign-changing) solutions under appropriate conditions on g, and thus generalize the existing results.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136142604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we construct a Jacobi spectral collocation scheme to approximate the Caputo fractional derivative based on Jacobi–Gauss quadrature. The convergence analysis is provided in anisotropic Jacobi-weighted Sobolev spaces. Furthermore, the convergence rate is presented for solving Caputo fractional derivative with noisy data by invoking the mollification regularization method. Lastly, numerical examples illustrate the effectiveness and stability of the proposed method.
{"title":"A Jacobi spectral method for calculating fractional derivative based on mollification regularization","authors":"Wen Zhang, Changxing Wu, Zhousheng Ruan, Shufang Qiu","doi":"10.3233/asy-231869","DOIUrl":"https://doi.org/10.3233/asy-231869","url":null,"abstract":"In this article, we construct a Jacobi spectral collocation scheme to approximate the Caputo fractional derivative based on Jacobi–Gauss quadrature. The convergence analysis is provided in anisotropic Jacobi-weighted Sobolev spaces. Furthermore, the convergence rate is presented for solving Caputo fractional derivative with noisy data by invoking the mollification regularization method. Lastly, numerical examples illustrate the effectiveness and stability of the proposed method.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135857355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order O ( k − 5 / 6 ) where k is the wave number in Helmholtz.
{"title":"Error estimates for Gaussian beams at a fold caustic","authors":"Olivier Lafitte, Olof Runborg","doi":"10.3233/asy-231852","DOIUrl":"https://doi.org/10.3233/asy-231852","url":null,"abstract":"In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order O ( k − 5 / 6 ) where k is the wave number in Helmholtz.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present article deals with the homogenization of a distributive optimal control problem (OCP) subjected to the more generalized stationary Stokes equation involving unidirectional oscillating coefficients posed in a two-dimensional oscillating domain. The cost functional considered is of the Dirichlet type involving a unidirectional oscillating coefficient matrix. We characterize the optimal control and study the homogenization of this OCP with the aid of the unfolding operator. Due to the presence of oscillating matrices both in the governing Stokes equations and the cost functional, one obtains the limit OCP involving a perturbed tensor in the convergence analysis.
{"title":"Homogenization of distributive optimal control problem governed by Stokes system in an oscillating domain","authors":"Swati Garg, Bidhan Chandra Sardar","doi":"10.3233/asy-231867","DOIUrl":"https://doi.org/10.3233/asy-231867","url":null,"abstract":"The present article deals with the homogenization of a distributive optimal control problem (OCP) subjected to the more generalized stationary Stokes equation involving unidirectional oscillating coefficients posed in a two-dimensional oscillating domain. The cost functional considered is of the Dirichlet type involving a unidirectional oscillating coefficient matrix. We characterize the optimal control and study the homogenization of this OCP with the aid of the unfolding operator. Due to the presence of oscillating matrices both in the governing Stokes equations and the cost functional, one obtains the limit OCP involving a perturbed tensor in the convergence analysis.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135352423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system.
{"title":"Global attractors for a partially damped Timoshenko–Ehrenfest system without the hypothesis of equal wave speeds","authors":"M.M. Freitas, D.S. Almeida Júnior, L.G.R. Miranda, A.J.A. Ramos, R.Q. Caljaro","doi":"10.3233/asy-231843","DOIUrl":"https://doi.org/10.3233/asy-231843","url":null,"abstract":"This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.
{"title":"On the distribution of Born transmission eigenvalues in the complex plane","authors":"Narek Hovsepyan","doi":"10.3233/asy-231868","DOIUrl":"https://doi.org/10.3233/asy-231868","url":null,"abstract":"We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135245912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the d-dimensional ( d ⩾ 2) Oldroyd-B model with only dissipation in the equation of stress tensor, and establish a small data global well-posedness result in critical L p framework. Particularly, we give a positive answer to the problem proposed recently by Wu-Zhao (J. Differ. Equ. 316 (2022)) involving the upper bound for the time integral of the lower frequency piece of the stress tensor, and show that it is indeed independent of the time. Moreover, we improve the results in (J. Math. Fluid Mech. 24 (2022)) by relaxing the space dimension d = 2 , 3 to any d ⩾ 2.
{"title":"Global regularity for Oldroyd-B model with only stress tensor dissipation","authors":"Weixun Feng, Zhi Chen, Dongdong Qin, Xianhua Tang","doi":"10.3233/asy-231861","DOIUrl":"https://doi.org/10.3233/asy-231861","url":null,"abstract":"In this paper, we consider the d-dimensional ( d ⩾ 2) Oldroyd-B model with only dissipation in the equation of stress tensor, and establish a small data global well-posedness result in critical L p framework. Particularly, we give a positive answer to the problem proposed recently by Wu-Zhao (J. Differ. Equ. 316 (2022)) involving the upper bound for the time integral of the lower frequency piece of the stress tensor, and show that it is indeed independent of the time. Moreover, we improve the results in (J. Math. Fluid Mech. 24 (2022)) by relaxing the space dimension d = 2 , 3 to any d ⩾ 2.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47764526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider quasilinear elliptic equations Δ p u + f ( u ) = 0 in the quarter-plane Ω, with zero Dirichlet data. For some general nonlinearities f, we prove the existence of a positive solution with a prescribed limiting profile. The question is motivated by the result in (Adv. Nonlinear Stud. 13(1) (2013) 115–136), where the authors establish that for solutions u ( x 1 , x 2 ) of the preceding Dirichlet problem, lim x 1 → ∞ u ( x 1 , x 2 ) = V ( x 2 ), where V is a solution of the corresponding one-dimensional problem with V ( + ∞ ) = z and z is a root of f. Starting with such a profile V and a carefully selected z, the authors of this paper apply Perron’s method in order to prove the existence of a solution u with limiting profile V. The work in this paper is similar in spirit to that in (Math. Methods Appl. Sci. 39(14) (2016) 4129–4138), where the authors compare the sub and the super solutions by using arguments based on the strong maximum principle for semilinear equations. However, for the quasilinear case, such a maximum principle is lacking. This difficulty is overcome by employing a less classical weak sweeping principle that requires a careful boundary analysis.
{"title":"Existence of quasilinear elliptic equations with prescribed limiting behavior","authors":"H. Ibrahim, R. Younes","doi":"10.3233/asy-231846","DOIUrl":"https://doi.org/10.3233/asy-231846","url":null,"abstract":"We consider quasilinear elliptic equations Δ p u + f ( u ) = 0 in the quarter-plane Ω, with zero Dirichlet data. For some general nonlinearities f, we prove the existence of a positive solution with a prescribed limiting profile. The question is motivated by the result in (Adv. Nonlinear Stud. 13(1) (2013) 115–136), where the authors establish that for solutions u ( x 1 , x 2 ) of the preceding Dirichlet problem, lim x 1 → ∞ u ( x 1 , x 2 ) = V ( x 2 ), where V is a solution of the corresponding one-dimensional problem with V ( + ∞ ) = z and z is a root of f. Starting with such a profile V and a carefully selected z, the authors of this paper apply Perron’s method in order to prove the existence of a solution u with limiting profile V. The work in this paper is similar in spirit to that in (Math. Methods Appl. Sci. 39(14) (2016) 4129–4138), where the authors compare the sub and the super solutions by using arguments based on the strong maximum principle for semilinear equations. However, for the quasilinear case, such a maximum principle is lacking. This difficulty is overcome by employing a less classical weak sweeping principle that requires a careful boundary analysis.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42658549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：