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":"50 1","pages":""},"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":"3 1","pages":""},"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":"102 1","pages":""},"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":"25 1","pages":""},"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":"101 1","pages":""},"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":"79 1","pages":""},"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":"241 1","pages":""},"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}
Pub Date : 2021-03-13DOI: 10.1142/s1664360722500138
J. Giacomoni, Lais Moreira dos Santos, C. Santos
A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $lambda-$direction and bifurcates from infinity at $lambda=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.
{"title":"Multiplicity for a strongly singular quasilinear problem via bifurcation theory","authors":"J. Giacomoni, Lais Moreira dos Santos, C. Santos","doi":"10.1142/s1664360722500138","DOIUrl":"https://doi.org/10.1142/s1664360722500138","url":null,"abstract":"A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $lambda-$direction and bifurcates from infinity at $lambda=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89790121","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 : 2021-01-05DOI: 10.1142/S1664360721500053
Simion Breaz, Tomasz Brzezi'nski
It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.
{"title":"The Baer–Kaplansky Theorem for All Abelian Groups and Modules","authors":"Simion Breaz, Tomasz Brzezi'nski","doi":"10.1142/S1664360721500053","DOIUrl":"https://doi.org/10.1142/S1664360721500053","url":null,"abstract":"It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"337 1","pages":"1-12"},"PeriodicalIF":1.2,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75347203","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 : 2020-08-02DOI: 10.1142/s1664360722500035
D. Borisov, P. Exner
We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.
{"title":"Approximation of point interactions by geometric perturbations in two-dimensional domains","authors":"D. Borisov, P. Exner","doi":"10.1142/s1664360722500035","DOIUrl":"https://doi.org/10.1142/s1664360722500035","url":null,"abstract":"We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"18 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80036499","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：