Pub Date : 2021-12-10DOI: 10.1142/s1664360722300043
Y. Giga, N. Požár
We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.
我们考虑一类称为结晶曲率流的各向异性曲率流。我们对这类流进行了研究，特别强调了其初值问题的适定性。
{"title":"Motion by Crystalline-Like Mean Curvature: A Survey","authors":"Y. Giga, N. Požár","doi":"10.1142/s1664360722300043","DOIUrl":"https://doi.org/10.1142/s1664360722300043","url":null,"abstract":"We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78662966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-30DOI: 10.1142/s1664360721500119
F. A. da Silva, M. Federson, E. Toon
In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron [Formula: see text]-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron [Formula: see text]-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.
{"title":"Existence, uniqueness, variation-of-constant formula and controllability for linear dynamic equations with Perron Δ-integrals","authors":"F. A. da Silva, M. Federson, E. Toon","doi":"10.1142/s1664360721500119","DOIUrl":"https://doi.org/10.1142/s1664360721500119","url":null,"abstract":"In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron [Formula: see text]-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron [Formula: see text]-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75146635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-06DOI: 10.1142/s1664360721500120
M. Jleli, M. Kirane, B. Samet
In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.
{"title":"A general blow-up result for a degenerate hyperbolic inequality in an exterior domain","authors":"M. Jleli, M. Kirane, B. Samet","doi":"10.1142/s1664360721500120","DOIUrl":"https://doi.org/10.1142/s1664360721500120","url":null,"abstract":"In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79320555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-03DOI: 10.1142/s1664360722300031
A. Cialdea, V. Maz'ya
Various notions of dissipativity for partial differential operators and their applications are surveyed. We deal with functional dissipativity and its particular case [Formula: see text]-dissipativity. Most of the results are due to the authors.
{"title":"A survey of functional and Lp-dissipativity theory","authors":"A. Cialdea, V. Maz'ya","doi":"10.1142/s1664360722300031","DOIUrl":"https://doi.org/10.1142/s1664360722300031","url":null,"abstract":"Various notions of dissipativity for partial differential operators and their applications are surveyed. We deal with functional dissipativity and its particular case [Formula: see text]-dissipativity. Most of the results are due to the authors.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91389631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-20DOI: 10.1142/s1664360721500090
Jinwan Park
In this paper, we study the tangential touch and [Formula: see text] regularity of the free boundary near the fixed boundary of the double obstacle problem for Laplacian and fully nonlinear operator. The main idea to have the properties is regarding the upper obstacle as a solution of the single obstacle problem. Then, in the classification of global solutions of the double problem, it is enough to consider only two cases for the upper obstacle, [Formula: see text] The second one is a new type of upper obstacle, which does not exist in the study of local regularity of the free boundary of the double problem. Thus, in this paper, a new type of difficulties that come from the second type upper obstacle is mainly studied.
{"title":"Properties of the free boundary near the fixed boundary of the double obstacle problems","authors":"Jinwan Park","doi":"10.1142/s1664360721500090","DOIUrl":"https://doi.org/10.1142/s1664360721500090","url":null,"abstract":"In this paper, we study the tangential touch and [Formula: see text] regularity of the free boundary near the fixed boundary of the double obstacle problem for Laplacian and fully nonlinear operator. The main idea to have the properties is regarding the upper obstacle as a solution of the single obstacle problem. Then, in the classification of global solutions of the double problem, it is enough to consider only two cases for the upper obstacle, [Formula: see text] The second one is a new type of upper obstacle, which does not exist in the study of local regularity of the free boundary of the double problem. Thus, in this paper, a new type of difficulties that come from the second type upper obstacle is mainly studied.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87764495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-12DOI: 10.1142/s1664360721500107
V. Ambrosio
{"title":"A Note on the Boundedness of Solutions for Fractional Relativistic Schrodinger Equations","authors":"V. Ambrosio","doi":"10.1142/s1664360721500107","DOIUrl":"https://doi.org/10.1142/s1664360721500107","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79162039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-05DOI: 10.1142/S1664360721500089
K. Kefi, N. Irzi, M. M. Al-Shomrani, Dušan D. Repovš
We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].
{"title":"On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions","authors":"K. Kefi, N. Irzi, M. M. Al-Shomrani, Dušan D. Repovš","doi":"10.1142/S1664360721500089","DOIUrl":"https://doi.org/10.1142/S1664360721500089","url":null,"abstract":"We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83217918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-21DOI: 10.1142/S166436072230002X
Gui-Qiang G. Chen, M. Feldman
We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.
{"title":"Multidimensional transonic shock waves and free boundary problems","authors":"Gui-Qiang G. Chen, M. Feldman","doi":"10.1142/S166436072230002X","DOIUrl":"https://doi.org/10.1142/S166436072230002X","url":null,"abstract":"We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82691101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-02DOI: 10.1142/s1664360722500072
Wei Dai, J. Fu
In this paper, we investigate various properties (e.g., nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear triharmonic equations in R (n ≥ 2) and bi-harmonic equations in R with negative exponents. Such kind of equations arise from conformal geometry.
{"title":"On properties of positive solutions to nonlinear tri-harmonic and bi-harmonic equations with negative exponents","authors":"Wei Dai, J. Fu","doi":"10.1142/s1664360722500072","DOIUrl":"https://doi.org/10.1142/s1664360722500072","url":null,"abstract":"In this paper, we investigate various properties (e.g., nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear triharmonic equations in R (n ≥ 2) and bi-harmonic equations in R with negative exponents. Such kind of equations arise from conformal geometry.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90468439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-01DOI: 10.1142/S166436071950019X
A. D. Gonzalez, G. Lagomasino, H. P. Cabrera
We consider extremal polynomials with respect to a Sobolev-type [Formula: see text]-norm, with [Formula: see text] and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the [Formula: see text]th root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.
{"title":"Asymptotic zero distribution for a class of extremal polynomials","authors":"A. D. Gonzalez, G. Lagomasino, H. P. Cabrera","doi":"10.1142/S166436071950019X","DOIUrl":"https://doi.org/10.1142/S166436071950019X","url":null,"abstract":"We consider extremal polynomials with respect to a Sobolev-type [Formula: see text]-norm, with [Formula: see text] and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the [Formula: see text]th root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90070934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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