This note addresses the question of convergence of critical points of the Ambrosio–Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford–Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio–Tortorelli functional satisfying the energy convergence assumption as in (Babadjian, Millot and Rodiac (2022)) are proved to exist.
{"title":"A note on the one-dimensional critical points of the Ambrosio–Tortorelli functional","authors":"Jean-François Babadjian, V. Millot, Rémy Rodiac","doi":"10.3233/asy-231857","DOIUrl":"https://doi.org/10.3233/asy-231857","url":null,"abstract":"This note addresses the question of convergence of critical points of the Ambrosio–Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford–Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio–Tortorelli functional satisfying the energy convergence assumption as in (Babadjian, Millot and Rodiac (2022)) are proved to exist.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48831720","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}
Railane Antonia, Giovanni Molica Bisci, Henrique F. de Lima, Márcio S. Santos
We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.
{"title":"Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs","authors":"Railane Antonia, Giovanni Molica Bisci, Henrique F. de Lima, Márcio S. Santos","doi":"10.3233/asy-231858","DOIUrl":"https://doi.org/10.3233/asy-231858","url":null,"abstract":"We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46371180","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 deals with global stability dynamics for the Klein–Gordon–Zakharov system in R 2 . We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space H s + 1 ( R 2 ) × H s + 1 ( R 2 ) × H s + 1 ( R 2 ) for any s ⩾ 1. Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n-dimension dissipative Klein–Gordon–Zakharov system for any n ⩾ 1.
本文研究r2中Klein-Gordon-Zakharov系统的全局稳定性动力学问题。首先证明了该系统存在一类线性模态不稳定的显式拟周期波解。接下来，我们证明Kelvin-Voigt阻尼可以帮助稳定Sobolev空间H s + 1 (r2) × H s + 1 (r2) × H s + 1 (r2)中的Klein-Gordon-Zakharov系统的那些线性模式不稳定的显式准周期波解对于任何s大于或等于1。此外，Kelvin-Voigt阻尼Klein-Gordon-Zakharov系统允许一个唯一的Sobolev正则解指数收敛于它的一些特解(包括拟周期波解)。我们的结果可以扩展到任何n小于1的n维耗散Klein-Gordon-Zakharov系统。
{"title":"Stabilization for the Klein–Gordon–Zakharov system","authors":"Weijia Li, Yuqi Shangguan, Weiping Yan","doi":"10.3233/asy-231856","DOIUrl":"https://doi.org/10.3233/asy-231856","url":null,"abstract":"This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in R 2 . We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space H s + 1 ( R 2 ) × H s + 1 ( R 2 ) × H s + 1 ( R 2 ) for any s ⩾ 1. Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n-dimension dissipative Klein–Gordon–Zakharov system for any n ⩾ 1.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43480472","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 hyperbolic inequalities with Hardy potential u t t − Δ u + λ | x | 2 u ⩾ | x | − a | u | p in ( 0 , ∞ ) × B 1 ∖ { 0 } , u ( t , x ) ⩾ f ( x ) on ( 0 , ∞ ) × ∂ B 1 , where B 1 is the unit ball in R N , N ⩾ 3, λ > − ( N − 2 2 ) 2 , a ⩾ 0, p > 1 and f is a nontrivial L 1 -function. We study separately the cases: λ = 0, − ( N − 2 2 ) 2 < λ < 0 and λ > 0. For each case, we obtain an optimal criterium for the nonexistence of weak solutions. Our study yields naturally optimal nonexistence results for the corresponding stationary problem. The novelty of this work lies in two facts: (i) To the best of our knowledge, in all previous works dealing with nonexistence results for evolution equations with Hardy potential in a bounded domain, only the parabolic case has been investigated, making use of some comparison principles. (ii) To the best of our knowledge, in all previous works, the issue of nonexistence has been studied only in the case of positive solutions. In this paper, there is no restriction on the sign of solutions.
{"title":"Semilinear hyperbolic inequalities with Hardy potential in a bounded domain","authors":"M. Jleli, B. Samet","doi":"10.3233/asy-231854","DOIUrl":"https://doi.org/10.3233/asy-231854","url":null,"abstract":"We consider hyperbolic inequalities with Hardy potential u t t − Δ u + λ | x | 2 u ⩾ | x | − a | u | p in ( 0 , ∞ ) × B 1 ∖ { 0 } , u ( t , x ) ⩾ f ( x ) on ( 0 , ∞ ) × ∂ B 1 , where B 1 is the unit ball in R N , N ⩾ 3, λ > − ( N − 2 2 ) 2 , a ⩾ 0, p > 1 and f is a nontrivial L 1 -function. We study separately the cases: λ = 0, − ( N − 2 2 ) 2 < λ < 0 and λ > 0. For each case, we obtain an optimal criterium for the nonexistence of weak solutions. Our study yields naturally optimal nonexistence results for the corresponding stationary problem. The novelty of this work lies in two facts: (i) To the best of our knowledge, in all previous works dealing with nonexistence results for evolution equations with Hardy potential in a bounded domain, only the parabolic case has been investigated, making use of some comparison principles. (ii) To the best of our knowledge, in all previous works, the issue of nonexistence has been studied only in the case of positive solutions. In this paper, there is no restriction on the sign of solutions.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49345373","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 work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology optimization problems in fluid mechanics.
{"title":"Topological sensitivity analysis for the 3D nonlinear Navier–Stokes equations","authors":"M. Hassine, M. Ouni","doi":"10.3233/asy-231855","DOIUrl":"https://doi.org/10.3233/asy-231855","url":null,"abstract":"This work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology optimization problems in fluid mechanics.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46908542","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}
N. Almousa, Claudia Bucur, Roberta Cornale, M. Squassina
In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion.
{"title":"Concavity principles for nonautonomous elliptic equations and applications","authors":"N. Almousa, Claudia Bucur, Roberta Cornale, M. Squassina","doi":"10.3233/asy-231863","DOIUrl":"https://doi.org/10.3233/asy-231863","url":null,"abstract":"In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45446365","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 aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with a strong uniform electric field and Dirichlet boundary conditions on a smooth bounded domain of R N , N ⩾ 2. This work aims at generalizing the recent results of Cornean, Krejčiřik, Pedersen, Raymond, and Stockmeyer in dimension 2. More precisely, in dimension N, in the strong electric field limit, we derive, under certain local convexity conditions, a full asymptotic expansion of the low-lying eigenvalues. To establish our main result, we perform the construction of quasi-modes. The “optimality” of our constructions is then established thanks to a reduction to model operators and localization estimates.
本文的目的是在R N, N大于或等于2的光滑有界域上用强均匀电场和狄利克雷边界条件建立斯塔克哈密顿量的特征值的渐近扩展。这项工作旨在推广Cornean, Krejčiřik, Pedersen, Raymond和Stockmeyer在2维的最新结果。更确切地说，在N维强电场极限下，在一定的局部凸性条件下，我们导出了低洼特征值的完全渐近展开式。为了建立我们的主要结果，我们进行了准模的构造。然后，通过减少模型算子和定位估计，我们的结构的“最优性”得以建立。
{"title":"On the semi-classical analysis of Schrödinger operators with linear electric potentials on a bounded domain","authors":"Rayan Fahs","doi":"10.3233/asy-231848","DOIUrl":"https://doi.org/10.3233/asy-231848","url":null,"abstract":"The aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with a strong uniform electric field and Dirichlet boundary conditions on a smooth bounded domain of R N , N ⩾ 2. This work aims at generalizing the recent results of Cornean, Krejčiřik, Pedersen, Raymond, and Stockmeyer in dimension 2. More precisely, in dimension N, in the strong electric field limit, we derive, under certain local convexity conditions, a full asymptotic expansion of the low-lying eigenvalues. To establish our main result, we perform the construction of quasi-modes. The “optimality” of our constructions is then established thanks to a reduction to model operators and localization estimates.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45871466","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}
G.E. Bittencourt Moraes, S.J. de Camargo, M.A. Jorge Silva
This is the second paper of a trilogy intended by the authors in what concerns a unified approach to the stability of thermoelastic arched beams of Bresse type under Fourier’s law. Differently of the first one, where the thermal couplings are regarded on the axial and bending displacements, here the thermal couplings are taken over the shear and bending forces. Such thermal effects still result in a new prototype of partially damped Bresse system whose stability results demand a proper approach. Combining a novel path of local estimates by means of the resolvent equation along with a control-observability analysis developed for elastic non-homogeneous systems of Bresse type proposed in trilogy’s first paper, we are able to provide a unified methodology of the asymptotic stability results, by proving the pattern of them with respect to boundary conditions and the action of temperature couplings, which is in compliance with our previous and present goal.
{"title":"Arched beams of Bresse type: New thermal couplings and pattern of stability","authors":"G.E. Bittencourt Moraes, S.J. de Camargo, M.A. Jorge Silva","doi":"10.3233/asy-231850","DOIUrl":"https://doi.org/10.3233/asy-231850","url":null,"abstract":"This is the second paper of a trilogy intended by the authors in what concerns a unified approach to the stability of thermoelastic arched beams of Bresse type under Fourier’s law. Differently of the first one, where the thermal couplings are regarded on the axial and bending displacements, here the thermal couplings are taken over the shear and bending forces. Such thermal effects still result in a new prototype of partially damped Bresse system whose stability results demand a proper approach. Combining a novel path of local estimates by means of the resolvent equation along with a control-observability analysis developed for elastic non-homogeneous systems of Bresse type proposed in trilogy’s first paper, we are able to provide a unified methodology of the asymptotic stability results, by proving the pattern of them with respect to boundary conditions and the action of temperature couplings, which is in compliance with our previous and present goal.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48031232","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 the present paper, we study the influence of oscillations of the time-dependent damping term b ( t ) u t on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ-evolution equation u t t + ( − Δ ) σ u + b ( t ) u t = 0 , ( t , x ) ∈ [ 0 , ∞ ) × R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , where σ > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b ( t ) = β ( 1 + t ) − 1 , with β > 0, and we discuss the influence of oscillations of b on the energy estimates according to the size of β.
在本文中,我们研究的影响时间的振荡阻尼项b (t) u t解的渐近性态的能量σ进化论方程的柯西问题u t t +(−Δ)σu + b (t) t = 0时,(t, x)∈(0,∞)×R n, u (0, x) = 0 (x), u t (0, x) = 1 (x), x∈R n,其中σ> 0和b是一个持续的和积极的作用。我们主要考虑的阻尼项是尺度不变情况b (t) = β (1 + t)−1，β > 0的扰动，并根据β的大小讨论了b的振荡对能量估计的影响。
{"title":"On the asymptotic behavior of the energy for evolution models with oscillating time-dependent damping","authors":"Halit Sevki Aslan, Marcelo Rempel Ebert","doi":"10.3233/asy-231851","DOIUrl":"https://doi.org/10.3233/asy-231851","url":null,"abstract":"In the present paper, we study the influence of oscillations of the time-dependent damping term b ( t ) u t on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ-evolution equation u t t + ( − Δ ) σ u + b ( t ) u t = 0 , ( t , x ) ∈ [ 0 , ∞ ) × R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , where σ > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b ( t ) = β ( 1 + t ) − 1 , with β > 0, and we discuss the influence of oscillations of b on the energy estimates according to the size of β.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46884889","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 investigate the Laplace’s equation for the electrical potential of charge drops on exterior domain, and overdetermined boundary conditions are prescribed. We determine the local bifurcation structure with respect to the surface tension coefficient as bifurcation parameter. Furthermore, we establish the stability and the instability near the bifurcation point.
{"title":"Bifurcation and stability for charged drops","authors":"Guowei Dai, Ben Duan, Fang Liu","doi":"10.3233/asy-231853","DOIUrl":"https://doi.org/10.3233/asy-231853","url":null,"abstract":"In this paper, we investigate the Laplace’s equation for the electrical potential of charge drops on exterior domain, and overdetermined boundary conditions are prescribed. We determine the local bifurcation structure with respect to the surface tension coefficient as bifurcation parameter. Furthermore, we establish the stability and the instability near the bifurcation point.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42607144","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}
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