In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in $L^2(mathbb{R}^2)$. The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of Schroedinger operators with point interactions.
{"title":"Spectral analysis of photonic crystals made of thin rods","authors":"M. Holzmann, V. Lotoreichik","doi":"10.3233/ASY-181478","DOIUrl":"https://doi.org/10.3233/ASY-181478","url":null,"abstract":"In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in $L^2(mathbb{R}^2)$. The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of Schroedinger operators with point interactions.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"260 1","pages":"83-112"},"PeriodicalIF":0.0,"publicationDate":"2017-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77143412","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 energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal characteristic function defined by means of a simplified two dimensional Ginzburg-Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog-Helffer and Fournais-Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor's surface.
{"title":"On the Ginzburg-Landau energy with a magnetic field vanishing along a curve","authors":"Ayman Kachmar, M. Nasrallah","doi":"10.3233/ASY-171424","DOIUrl":"https://doi.org/10.3233/ASY-171424","url":null,"abstract":"The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal characteristic function defined by means of a simplified two dimensional Ginzburg-Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog-Helffer and Fournais-Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor's surface.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"84 1","pages":"135-163"},"PeriodicalIF":0.0,"publicationDate":"2017-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80877855","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 work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Γ-convergence theory.
{"title":"Stochastic homogenization of rate-dependent models of monotone type in plasticity","authors":"M. Heida, Sergiy Nesenenko","doi":"10.3233/ASY-181502","DOIUrl":"https://doi.org/10.3233/ASY-181502","url":null,"abstract":"In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Γ-convergence theory.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"28 1","pages":"185-212"},"PeriodicalIF":0.0,"publicationDate":"2017-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84865221","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}
{"title":"Multiple solutions for a class of quasilinear problems in Orlicz-Sobolev spaces","authors":"K. Ait-Mahiout, C. O. Alves","doi":"10.3233/ASY-171428","DOIUrl":"https://doi.org/10.3233/ASY-171428","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"37 1","pages":"49-66"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74924653","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}
C. G. Lopes, R. B. Santos, A. Novotny, J. Sokołowski
Contact problems with given friction are considered for plane elasticity in the framework of shape-topological optimization. The asymptotic analysis of the second kind variational inequalities in plane elasticity is performed for the purposes of shapetopological optimization. To this end, the saddle point formulation for the associated Lagrangian is introduced for the variational inequality. The non-smooth term in the energy functional is replaced by pointwise constraints for the multipliers. The one term expansion of the strain energy with respect to the small parameter which governs the size of the singular perturbation of geometrical domain is obtained. The topological derivatives of energy functional are derived in closed form adapted to the numerical methods of shape-topological optimization. In general, the topological derivative (TD) of the elastic energy is defined through a limit passage when the small parameter governing the size of the topological perturbation goes to zero. TD can be used as a steepestdescent direction in an optimization process like in any method based on the gradient of the cost functional. In this paper, we deal with the topological asymptotic analysis in the context of contact problems with given friction. Since the problem is nonlinear, the domain decomposition technique combined with the Steklov-Poincaré pseudodifferential boundary operator is used for asymptotic analysis purposes with respect to the small parameter associated with the size of the topological perturbation. As a fundamental result, the expansion of the strain energy coincides with the expansion of the SteklovPoincaré operator on the boundary of the truncated domain, leading to the expression for TD. Finally, the obtained TD is applied in the context of topology optimization of mechanical structures under contact condition with given friction.
{"title":"Asymptotic analysis of variational inequalities with applications to optimum design in elasticity","authors":"C. G. Lopes, R. B. Santos, A. Novotny, J. Sokołowski","doi":"10.3233/ASY-171416","DOIUrl":"https://doi.org/10.3233/ASY-171416","url":null,"abstract":"Contact problems with given friction are considered for plane elasticity in the framework of shape-topological optimization. The asymptotic analysis of the second kind variational inequalities in plane elasticity is performed for the purposes of shapetopological optimization. To this end, the saddle point formulation for the associated Lagrangian is introduced for the variational inequality. The non-smooth term in the energy functional is replaced by pointwise constraints for the multipliers. The one term expansion of the strain energy with respect to the small parameter which governs the size of the singular perturbation of geometrical domain is obtained. The topological derivatives of energy functional are derived in closed form adapted to the numerical methods of shape-topological optimization. In general, the topological derivative (TD) of the elastic energy is defined through a limit passage when the small parameter governing the size of the topological perturbation goes to zero. TD can be used as a steepestdescent direction in an optimization process like in any method based on the gradient of the cost functional. In this paper, we deal with the topological asymptotic analysis in the context of contact problems with given friction. Since the problem is nonlinear, the domain decomposition technique combined with the Steklov-Poincaré pseudodifferential boundary operator is used for asymptotic analysis purposes with respect to the small parameter associated with the size of the topological perturbation. As a fundamental result, the expansion of the strain energy coincides with the expansion of the SteklovPoincaré operator on the boundary of the truncated domain, leading to the expression for TD. Finally, the obtained TD is applied in the context of topology optimization of mechanical structures under contact condition with given friction.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"115 1","pages":"227-242"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78115948","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}
{"title":"On the constants in Markov inequalities for the Laplace operator on polynomials with the Laguerre norm","authors":"A. Böttcher, Christian Rebs","doi":"10.3233/ASY-161400","DOIUrl":"https://doi.org/10.3233/ASY-161400","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"8 1","pages":"227-239"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88950969","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}
{"title":"A turning point asymptotic expansion for a rigid-dumbbell polymer fluid probability configurational equation for fast shear flows","authors":"I. Ciuperca, L. Palade","doi":"10.3233/ASY-171435","DOIUrl":"https://doi.org/10.3233/ASY-171435","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"4 1","pages":"45-76"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74509634","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}
Population growth is governed by many external and internal factors. In order to study their effects on population dynamics, we develop an age-structured time-dependent population model with logist ...
{"title":"Large time behavior of the logistic age-structured population model in a changing environment","authors":"V. Kozlov, S. Radosavljevic, U. Wennergren","doi":"10.3233/ASY-171409","DOIUrl":"https://doi.org/10.3233/ASY-171409","url":null,"abstract":"Population growth is governed by many external and internal factors. In order to study their effects on population dynamics, we develop an age-structured time-dependent population model with logist ...","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"102 1","pages":"21-54"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87198896","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 the Cauchy problem for the one-dimensional Bresse system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the solution using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. In fact we prove that the dissipation induced by the heat conduction is very weak and produces very slow decay rates. In addition in some cases, those decay rates are of regularity-loss type. Also, we prove that there is a number (depending on the parameters of the system) that controls the decay rate of the solution and the regularity assumptions on the initial data. In addition, we show that in the absence of the frictional damping, the memory damping term is not strong enough to produce a decay rate for the solution. In fact, we show in this case, despite the fact that the energy is still dissipative, the solution does not decay at all. This result improves and extends several results, such as those in Appl. Math. Optim. (2016), to appear, Communications in Contemporary Mathematics 18(4) (2016), 1550045, Math. Methods Appl. Sci. 38(17) (2015), 3642–3652 and others.
{"title":"On the decay rate of solutions of the Bresse system with Gurtin-Pipkin thermal law","authors":"Maisa Khader, B. Houari","doi":"10.3233/ASY-171417","DOIUrl":"https://doi.org/10.3233/ASY-171417","url":null,"abstract":"We consider the Cauchy problem for the one-dimensional Bresse system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the solution using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. In fact we prove that the dissipation induced by the heat conduction is very weak and produces very slow decay rates. In addition in some cases, those decay rates are of regularity-loss type. Also, we prove that there is a number (depending on the parameters of the system) that controls the decay rate of the solution and the regularity assumptions on the initial data. In addition, we show that in the absence of the frictional damping, the memory damping term is not strong enough to produce a decay rate for the solution. In fact, we show in this case, despite the fact that the energy is still dissipative, the solution does not decay at all. This result improves and extends several results, such as those in Appl. Math. Optim. (2016), to appear, Communications in Contemporary Mathematics 18(4) (2016), 1550045, Math. Methods Appl. Sci. 38(17) (2015), 3642–3652 and others.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"16 1","pages":"1-32"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82100709","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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