Let $Delta_{Omega_varepsilon}$ be the Dirichlet Laplacian in the domain $Omega_varepsilon:=Omegasetminusleft(cup_i D_{i varepsilon}right)$. Here $Omegasubsetmathbb{R}^n$ and ${D_{i varepsilon}}_{i}$ is a family of tiny identical holes ("ice pieces") distributed periodically in $mathbb{R}^n$ with period $varepsilon$. We denote by $mathrm{cap}(D_{i varepsilon})$ the capacity of a single hole. It was known for a long time that $-Delta_{Omega_varepsilon}$ converges to the operator $-Delta_{Omega}+q$ in strong resolvent sense provided the limit $q:=lim_{varepsilonto 0} mathrm{cap}(D_{ivarepsilon}) varepsilon^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded $Omega$) an estimate for the difference of the $k$-th eigenvalue of $-Delta_{Omega_varepsilon}$ and $-Delta_{Omega_varepsilon}+q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.
{"title":"Operator estimates for the crushed ice problem","authors":"A. Khrabustovskyi, O. Post","doi":"10.3233/ASY-181480","DOIUrl":"https://doi.org/10.3233/ASY-181480","url":null,"abstract":"Let $Delta_{Omega_varepsilon}$ be the Dirichlet Laplacian in the domain $Omega_varepsilon:=Omegasetminusleft(cup_i D_{i varepsilon}right)$. Here $Omegasubsetmathbb{R}^n$ and ${D_{i varepsilon}}_{i}$ is a family of tiny identical holes (\"ice pieces\") distributed periodically in $mathbb{R}^n$ with period $varepsilon$. We denote by $mathrm{cap}(D_{i varepsilon})$ the capacity of a single hole. It was known for a long time that $-Delta_{Omega_varepsilon}$ converges to the operator $-Delta_{Omega}+q$ in strong resolvent sense provided the limit $q:=lim_{varepsilonto 0} mathrm{cap}(D_{ivarepsilon}) varepsilon^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded $Omega$) an estimate for the difference of the $k$-th eigenvalue of $-Delta_{Omega_varepsilon}$ and $-Delta_{Omega_varepsilon}+q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"SE-13 1","pages":"137-161"},"PeriodicalIF":0.0,"publicationDate":"2017-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84641284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects. The pore-scale model consists of a strongly coupled system of time-dependent Stokes-Cahn-Hilliard equations. In the considered model the fluids are separated by an evolving diffuse interface of a finite width, which is assumed to be independent of the scale parameter ε. We obtain upscaled equations for the considered model by a rigorous two-scale convergence approach.
{"title":"Homogenization of evolutionary Stokes-Cahn-Hilliard equations for two-phase porous media flow","authors":"L. Baňas, H. Mahato","doi":"10.3233/ASY-171436","DOIUrl":"https://doi.org/10.3233/ASY-171436","url":null,"abstract":"We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects. The pore-scale model consists of a strongly coupled system of time-dependent Stokes-Cahn-Hilliard equations. In the considered model the fluids are separated by an evolving diffuse interface of a finite width, which is assumed to be independent of the scale parameter ε. We obtain upscaled equations for the considered model by a rigorous two-scale convergence approach.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"6 1","pages":"77-95"},"PeriodicalIF":0.0,"publicationDate":"2017-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81169719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to some Lipschitz estimates between sub-and super-solutions of Fully Nonlinear equations on the model of the anisotropic ~ p-Laplacian. In particular we derive from the results enclosed that the continuous viscosity solutions for the equation ∑N 1 ∂i(∂iu| i∂iu) = f are Lipschitz continuous when supi pi < infi pi + 1, where ~ p = ∑ i piei.
研究了各向异性~ p- laplace模型上的完全非线性方程的子解和超解之间的Lipschitz估计。特别地，我们从所附的结果中推导出方程∑N 1∂i(∂iu| i∂iu) = f的连续粘度解在supi pi < infi pi + 1时是Lipschitz连续的，其中~ p =∑i pii。
{"title":"Regularity properties of viscosity solutions for fully nonlinear equations on the model of the anisotropic p →-Laplacian","authors":"F. Demengel","doi":"10.3233/ASY-171433","DOIUrl":"https://doi.org/10.3233/ASY-171433","url":null,"abstract":"This paper is devoted to some Lipschitz estimates between sub-and super-solutions of Fully Nonlinear equations on the model of the anisotropic ~ p-Laplacian. In particular we derive from the results enclosed that the continuous viscosity solutions for the equation ∑N 1 ∂i(∂iu| i∂iu) = f are Lipschitz continuous when supi pi < infi pi + 1, where ~ p = ∑ i piei.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"17 1","pages":"27-43"},"PeriodicalIF":0.0,"publicationDate":"2017-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74346488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type $u_t+H(x,Du)=l(x),$ set in the whole space $R^Ntimes [0,infty).$ We assume that $l$ is bounded from below but may have arbitrary growth and therefore the solutions may also have arbitrary growth. A complete study of the structure of solutions of the ergodic problem $H(x,Dv)=l(x)+c$ is provided : contrarily to the periodic setting, the ergodic constant is not anymore unique, leading to different large time behavior for the solutions. We establish the ergodic behavior of the solutions of the Cauchy problem (i) when starting with a bounded from below initial condition and (ii) for some particular unbounded from below initial condition, two cases for which we have different ergodic constants which play a role. When the solution is not bounded from below, an example showing that the convergence may fail in general is provided.
{"title":"Large time behavior of unbounded solutions of first-order Hamilton-Jacobi equations in R N","authors":"G. Barles, Olivier Ley, Thi-Tuyen Nguyen, T. Phan","doi":"10.3233/ASY-181488","DOIUrl":"https://doi.org/10.3233/ASY-181488","url":null,"abstract":"We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type $u_t+H(x,Du)=l(x),$ set in the whole space $R^Ntimes [0,infty).$ We assume that $l$ is bounded from below but may have arbitrary growth and therefore the solutions may also have arbitrary growth. A complete study of the structure of solutions of the ergodic problem $H(x,Dv)=l(x)+c$ is provided : contrarily to the periodic setting, the ergodic constant is not anymore unique, leading to different large time behavior for the solutions. We establish the ergodic behavior of the solutions of the Cauchy problem (i) when starting with a bounded from below initial condition and (ii) for some particular unbounded from below initial condition, two cases for which we have different ergodic constants which play a role. When the solution is not bounded from below, an example showing that the convergence may fail in general is provided.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"1 1","pages":"1-22"},"PeriodicalIF":0.0,"publicationDate":"2017-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90248151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.
{"title":"Initial-boundary value problem for distributed order time-fractional diffusion equations","authors":"Zhi-yuan Li, Yavar Kian, É. Soccorsi","doi":"10.3233/asy-191532","DOIUrl":"https://doi.org/10.3233/asy-191532","url":null,"abstract":"We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"36 1","pages":"95-126"},"PeriodicalIF":0.0,"publicationDate":"2017-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86109677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter $epsilon > 0$) whose limit (as $epsilon$ tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the $epsilon$ tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that e is small enough. Numerical experiments illustrate the theoretical results.
{"title":"Trapped modes in thin and infinite ladder like domains. Part 1: Existence results","authors":"B. Delourme, S. Fliss, P. Joly, E. Vasilevskaya","doi":"10.3233/ASY-171422","DOIUrl":"https://doi.org/10.3233/ASY-171422","url":null,"abstract":"The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter $epsilon > 0$) whose limit (as $epsilon$ tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the $epsilon$ tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that e is small enough. Numerical experiments illustrate the theoretical results.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"36 1","pages":"103-134"},"PeriodicalIF":0.0,"publicationDate":"2017-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73212962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Hilbert space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time $t^*$ of the limit. For our class of functionals, $t^*$ coincides with the blow-up time of the solutions of the linearized system around the equilibrium, and is in particular strictly greater than the time $t_c$ where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of $t^*$, rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini, where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times.
{"title":"Delayed loss of stability in singularly perturbed finite-dimensional gradient flows","authors":"G. Scilla, Francesco Solombrino","doi":"10.3233/ASY-181475","DOIUrl":"https://doi.org/10.3233/ASY-181475","url":null,"abstract":"In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Hilbert space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time $t^*$ of the limit. For our class of functionals, $t^*$ coincides with the blow-up time of the solutions of the linearized system around the equilibrium, and is in particular strictly greater than the time $t_c$ where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of $t^*$, rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini, where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"31 1","pages":"1-19"},"PeriodicalIF":0.0,"publicationDate":"2017-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80961353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schr"odinger equation in presence of a smooth external potential.
本文研究了光滑外势存在下对数Schr odinger方程的高斯(孤子)动力学的有效性。
{"title":"Gausson dynamics for logarithmic Schrödinger equations","authors":"Alex H. Ardila, M. Squassina","doi":"10.3233/ASY-171458","DOIUrl":"https://doi.org/10.3233/ASY-171458","url":null,"abstract":"In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schr\"odinger equation in presence of a smooth external potential.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"36 1","pages":"203-226"},"PeriodicalIF":0.0,"publicationDate":"2017-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85499143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the resonances of $2times 2$ systems of one dimensional Schrodinger operators which are related to the mathematical theory of molecular predissociation. We determine the precise positions of the resonances with real parts below the energy where bonding and anti-bonding potentials intersect transversally. In particular, we find that imaginary parts (widths) of the resonances are exponentially small and that the indices are determined by Agmon distances for the minimum of two potentials.
{"title":"Molecular predissociation resonances below an energy level crossing","authors":"Sohei Ashida","doi":"10.3233/ASY-171453","DOIUrl":"https://doi.org/10.3233/ASY-171453","url":null,"abstract":"We study the resonances of $2times 2$ systems of one dimensional Schrodinger operators which are related to the mathematical theory of molecular predissociation. We determine the precise positions of the resonances with real parts below the energy where bonding and anti-bonding potentials intersect transversally. In particular, we find that imaginary parts (widths) of the resonances are exponentially small and that the indices are determined by Agmon distances for the minimum of two potentials.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"138 1","pages":"135-167"},"PeriodicalIF":0.0,"publicationDate":"2017-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76548199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L2 on the real axis with the weight |t|2α e –t2 and |t|2β e –t2, respectively. We determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity.
本文研究多项式的高阶导数的范数与多项式本身的范数之间的马尔可夫型不等式中的最佳常数。多项式的范数及其导数在实轴的L2上分别取权值为|t|2α e -t2和|t|2β e -t2。当多项式的次数趋于无穷时，我们确定了常数渐近的前项。
{"title":"Asymptotically sharp inequalities for polynomials involving mixed Gegenbauer norms","authors":"Holger Langenau","doi":"10.3233/ASY-171425","DOIUrl":"https://doi.org/10.3233/ASY-171425","url":null,"abstract":"The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L2 on the real axis with the weight |t|2α e –t2 and |t|2β e –t2, respectively. We determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"67 1","pages":"221-233"},"PeriodicalIF":0.0,"publicationDate":"2017-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84039971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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