Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.30
L. Barbu, G. Morosanu
In this paper we consider in a bounded domain Q C RN a Steklov- like eigenvalue problem involving the (p, q)-Laplacian plus some potentials. Under suitable assumptions, using the Nehari manifold method and a variational approach, we are able to determine the full eigenvalue set of this problem as being an open interval (A*, +^) with A* > 0.
{"title":"FULL DESCRIPTION OF THE SPECTRUM OF A STEKLOV-LIKE EIGENVALUE PROBLEM INVOLVING THE (p, q)-LAPLACIAN","authors":"L. Barbu, G. Morosanu","doi":"10.56082/annalsarscimath.2023.1-2.30","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.30","url":null,"abstract":"In this paper we consider in a bounded domain Q C RN a Steklov- like eigenvalue problem involving the (p, q)-Laplacian plus some potentials. Under suitable assumptions, using the Nehari manifold method and a variational approach, we are able to determine the full eigenvalue set of this problem as being an open interval (A*, +^) with A* > 0.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009552","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.286
F. Troltzsch
Optimal control problems for the linear heat equation with final observation and pointwise constraints on the control are considered, where the control depends only on the time. It is shown that to each finite number of given switching points, there is a final target such that the optimal objective value is positive, the optimal control is bang bang, and has the desired switching structure. The theory is completed by numerical examples.
{"title":"ON THE BANG-BANG PRINCIPLE FOR PARABOLIC OPTIMAL CONTROL PROBLEMS","authors":"F. Troltzsch","doi":"10.56082/annalsarscimath.2023.1-2.286","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.286","url":null,"abstract":"Optimal control problems for the linear heat equation with final observation and pointwise constraints on the control are considered, where the control depends only on the time. It is shown that to each finite number of given switching points, there is a final target such that the optimal objective value is positive, the optimal control is bang bang, and has the desired switching structure. The theory is completed by numerical examples.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009897","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.352
C. M. Murea
We study numerically the dynamic impact/contact of an elastic body on a moving foundation using the mid-point algorithm. Stability results are presented when foundation is decreasing. Numerical simulations on two-dimensional problems are included and we show that the energy is absorbed in the case of decreasing foundation compared to the fixed one.
{"title":"IMPACT/CONTACT OF ELASTIC BODY ON A MOVING FOUNDATION","authors":"C. M. Murea","doi":"10.56082/annalsarscimath.2023.1-2.352","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.352","url":null,"abstract":"We study numerically the dynamic impact/contact of an elastic body on a moving foundation using the mid-point algorithm. Stability results are presented when foundation is decreasing. Numerical simulations on two-dimensional problems are included and we show that the energy is absorbed in the case of decreasing foundation compared to the fixed one.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010119","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.175
P. Colli, G. Gilardi, A. Signori, J. Sprekels
In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails further mathematical difficulties because the mass conservation of the order parameter is no longer satisfied. In this paper, we study the case that the double-well potential driving the evolution of the phase transition is given by the nondifferentiable double obstacle potential, thereby complementing recent results obtained for the differentiable cases of regular and logarithmic potentials. Besides existence results, we derive first-order necessary optimality conditions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondifferentiable double obstacle potential is approximated by a family of potentials of logarithmic structure for which meaningful first-order necessary optimality conditions in terms of suitable adjoint systems and variational inequalities are available. Since the results for the logarithmic potentials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situation, our whole analysis is restricted to the two-dimensional case.
{"title":"OPTIMAL TEMPERATURE DISTRIBUTION FOR A NONISOTHERMAL CAHN-HILLIARD SYSTEM IN TWO DIMENSIONS WITH SOURCE TERM AND DOUBLE OBSTACLE POTENTIAL","authors":"P. Colli, G. Gilardi, A. Signori, J. Sprekels","doi":"10.56082/annalsarscimath.2023.1-2.175","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.175","url":null,"abstract":"In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails further mathematical difficulties because the mass conservation of the order parameter is no longer satisfied. In this paper, we study the case that the double-well potential driving the evolution of the phase transition is given by the nondifferentiable double obstacle potential, thereby complementing recent results obtained for the differentiable cases of regular and logarithmic potentials. Besides existence results, we derive first-order necessary optimality conditions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondifferentiable double obstacle potential is approximated by a family of potentials of logarithmic structure for which meaningful first-order necessary optimality conditions in terms of suitable adjoint systems and variational inequalities are available. Since the results for the logarithmic potentials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situation, our whole analysis is restricted to the two-dimensional case.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009888","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.5
Frederic Bonnans, Vasile Dragan
This issue is dedicated to the 70-th anniversary of Dan Tiba, a Romanian mathematician appreciated at the international level for his scientific contributions in the fields of optimization, optimal control for PDEs, shape and topology optimization, nonlinear analysis. His activity was marked by the passion for mathematics and his results were published by many prestigious journals and publishing houses. The colleagues and the collaborators wish him, at this anniversary moment, a long and active life in good health and more achievements in the mathematical research
{"title":"DAN TIBA AT HIS 70th ANNIVERSARY","authors":"Frederic Bonnans, Vasile Dragan","doi":"10.56082/annalsarscimath.2023.1-2.5","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.5","url":null,"abstract":"This issue is dedicated to the 70-th anniversary of Dan Tiba, a Romanian mathematician appreciated at the international level for his scientific contributions in the fields of optimization, optimal control for PDEs, shape and topology optimization, nonlinear analysis. His activity was marked by the passion for mathematics and his results were published by many prestigious journals and publishing houses. The colleagues and the collaborators wish him, at this anniversary moment, a long and active life in good health and more achievements in the mathematical research","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009890","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.408
A. Matei
We consider an obstacle model mathematically described by means of a boundary value problem governed by PDE. Three possible variational formulations are highlighted. The first one is a variational inequality of the first kind and the other two are mixed variational formulations with Lagrange multipliers in dual spaces. After we discuss the solvability of the three variational formulations under consideration we focus on the relationship between them. Subsequently, we address the recovery of the formulation in terms of PDE starting from the mixed variational formulations.
{"title":"THREE WEAK FORMULATIONS FOR AN OBSTACLE MODEL AND THEIR RELATIONSHIP","authors":"A. Matei","doi":"10.56082/annalsarscimath.2023.1-2.408","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.408","url":null,"abstract":"We consider an obstacle model mathematically described by means of a boundary value problem governed by PDE. Three possible variational formulations are highlighted. The first one is a variational inequality of the first kind and the other two are mixed variational formulations with Lagrange multipliers in dual spaces. After we discuss the solvability of the three variational formulations under consideration we focus on the relationship between them. Subsequently, we address the recovery of the formulation in terms of PDE starting from the mixed variational formulations.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010110","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.366
M. Buliga
"We explain a dissipative version of hamiltonian mechanics, based on the information content of the deviation from hamiltonian dynam¬ics. From this formulation we deduce minimal dissipation principles, dynamical inclusions, or constrained evolution with hamiltonian drift reformulations. Among applications we recover a dynamics generaliza¬tion of Mielke et al quasistatic rate-independent processes. This article gives a clear and unitary presentation of the theory of hamiltonian inclusions with convex dissipation or symplectic Brezis- Ekeland-Nayroles principle, presented under various conventions first in [3] arXiv:0810.1419, then in [4] arXiv:1408.3102 and, for the ap¬pearance of bipotentials in relation to the symplectic duality, in [2] arXiv:1902.04598v1."
{"title":"DISSIPATION AND THE INFORMATION CONTENT OF THE DEVIATION FROM HAMILTONIAN DYNAMICS","authors":"M. Buliga","doi":"10.56082/annalsarscimath.2023.1-2.366","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.366","url":null,"abstract":"\"We explain a dissipative version of hamiltonian mechanics, based on the information content of the deviation from hamiltonian dynam¬ics. From this formulation we deduce minimal dissipation principles, dynamical inclusions, or constrained evolution with hamiltonian drift reformulations. Among applications we recover a dynamics generaliza¬tion of Mielke et al quasistatic rate-independent processes. This article gives a clear and unitary presentation of the theory of hamiltonian inclusions with convex dissipation or symplectic Brezis- Ekeland-Nayroles principle, presented under various conventions first in [3] arXiv:0810.1419, then in [4] arXiv:1408.3102 and, for the ap¬pearance of bipotentials in relation to the symplectic duality, in [2] arXiv:1902.04598v1.\"","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009900","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.308
M. Sofonea, D.A. Tarzia
We consider an abstract problem P in a metric space X which has a unique solution u G X. Our aim in this current paper is two folds: first, to provide a convergence criterion to the solution of Problem P , that is, to give necessary and sufficient conditions on a sequence {un} C X which guarantee the convergence un ^ u in the space X; second, to find a Tyknonov triple T such that a sequence {un} C X is a T -approximating sequence if and only if it converges to u. The two problems stated above, associated to the original Problem P , are closely related. We illustrate how they can be solved in three particular cases of Problem P: a variational inequality in a Hilbert space, a fixed point problem in a metric space and a minimization problem in a reflexive Banach space. For each of these problems we state and prove a convergence criterion that we use to define a convenient Tykhonov triple T which requires the condition stated above. We also show how the convergence criterion and the corresponding T -well posedness concept can be used to deduce convergence and classical well-posedness results, respectively.
考虑度量空间X中具有唯一解u gx的抽象问题P，本文的目的有两个方面:第一，给出问题P解的收敛判据，即给出序列{un} C X在空间X中收敛un ^ u的充分必要条件;第二，找到一个Tyknonov三重T，使得序列{un} C X是一个T逼近序列当且仅当它收敛于u。上面所述的两个问题与原问题P相关，是密切相关的。在Hilbert空间中的变分不等式问题、度量空间中的不动点问题和自反Banach空间中的最小化问题这三种特殊情况下，我们说明了如何解决它们。对于这些问题中的每一个，我们陈述并证明了一个收敛准则，我们用它来定义一个方便的Tykhonov三重T，它需要上述条件。我们还展示了如何使用收敛准则和相应的T -适定性概念分别推导收敛和经典适定性结果。
{"title":"CONVERGENCE CRITERIA, WELL-POSEDNESS CONCEPTS AND APPLICATIONS","authors":"M. Sofonea, D.A. Tarzia","doi":"10.56082/annalsarscimath.2023.1-2.308","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.308","url":null,"abstract":"We consider an abstract problem P in a metric space X which has a unique solution u G X. Our aim in this current paper is two folds: first, to provide a convergence criterion to the solution of Problem P , that is, to give necessary and sufficient conditions on a sequence {un} C X which guarantee the convergence un ^ u in the space X; second, to find a Tyknonov triple T such that a sequence {un} C X is a T -approximating sequence if and only if it converges to u. The two problems stated above, associated to the original Problem P , are closely related. We illustrate how they can be solved in three particular cases of Problem P: a variational inequality in a Hilbert space, a fixed point problem in a metric space and a minimization problem in a reflexive Banach space. For each of these problems we state and prove a convergence criterion that we use to define a convenient Tykhonov triple T which requires the condition stated above. We also show how the convergence criterion and the corresponding T -well posedness concept can be used to deduce convergence and classical well-posedness results, respectively.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009904","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.520
O. Kiși, C. Choudhury
In the present article, we set forth with the new notion of rough A—statistical convergence in the gradual normed linear spaces. We produce significant results that present several fundamental properties of this notion. We also introduce the notion of Arst(G)—limit set and prove that it is convex, gradually closed, and plays an important role for the gradually A—statistical boundedness of a sequence.
{"title":"CERTAIN ASPECTS OF Xrst(G)-CONVERGENCE OF SEQUENCES IN GRADUAL NORMED LINEAR SPACES","authors":"O. Kiși, C. Choudhury","doi":"10.56082/annalsarscimath.2023.1-2.520","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.520","url":null,"abstract":"In the present article, we set forth with the new notion of rough A—statistical convergence in the gradual normed linear spaces. We produce significant results that present several fundamental properties of this notion. We also introduce the notion of Arst(G)—limit set and prove that it is convex, gradually closed, and plays an important role for the gradually A—statistical boundedness of a sequence.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009911","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}
Pub Date : 2023-01-01DOI: 10.56082/annalsarscimath.2023.1-2.535
H. A. Abass
In this paper, we study and introduce a self adaptive method together with a Halpern iterative algorithm for approximating solutions of multiple-sets split monotone variational inclusion problem which includes the multiple-sets split feasibility problem, split feasibility problem, split monotone variational inclusion problem and split variational inclusion problem, to mention a few. Using our iterative algorithm, we prove a strong convergence result for approximating the solution of the aforementioned problems. Numerical examples on finite-dimensional and infinite-dimensional spaces are displayed to illustrate the performance of our iterative method. The result discussed in this article extends and complements many related results in literature.
{"title":"SOLVING MULTIPLE-SETS SPLIT MONOTONE VARIATIONAL INCLUSION PROBLEM IN REAL HILBERT SPACES.","authors":"H. A. Abass","doi":"10.56082/annalsarscimath.2023.1-2.535","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2023.1-2.535","url":null,"abstract":"In this paper, we study and introduce a self adaptive method together with a Halpern iterative algorithm for approximating solutions of multiple-sets split monotone variational inclusion problem which includes the multiple-sets split feasibility problem, split feasibility problem, split monotone variational inclusion problem and split variational inclusion problem, to mention a few. Using our iterative algorithm, we prove a strong convergence result for approximating the solution of the aforementioned problems. Numerical examples on finite-dimensional and infinite-dimensional spaces are displayed to illustrate the performance of our iterative method. The result discussed in this article extends and complements many related results in literature.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010115","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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