Advanced Nonlinear Studies最新文献

{"title":"Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture","authors":"Yahong Guo, Lingwei Ma, Zhenqiu Zhang","doi":"10.1515/ans-2023-0114","DOIUrl":"https://doi.org/10.1515/ans-2023-0114","url":null,"abstract":"In this paper, we consider the general dual fractional parabolic problem <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msubsup> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi mathvariant=\"script\">L</m:mi> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mspace width=\"0.3333em\" /> <m:mspace width=\"0.3333em\" /> <m:mtext>in</m:mtext> <m:mspace width=\"0.3333em\" /> <m:mspace width=\"0.3333em\" /> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:mi mathvariant=\"double-struck\">R</m:mi> <m:mo>.</m:mo> </m:math> <jats:tex-math>${partial }_{t}^{alpha }uleft(x,tright)+mathcal{L}uleft(x,tright)=fleft(t,uleft(x,tright)right) text{in} {mathbb{R}}^{n}{times}mathbb{R}.$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0114_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula> We show that the bounded entire solution <jats:italic>u</jats:italic> satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on <jats:italic>f</jats:italic>. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi mathvariant=\"script\">L</m:mi> </m:math> <jats:tex-math>$mathcal{L}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0114_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula> and the Marchaud time derivative <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msubsup> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> </m:math> <jats:tex-math>${partial }_{t}^{alpha }$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0114_ineq_003.png","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"2 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140108299","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}

{"title":"Infinite energy harmonic maps from quasi-compact Kähler surfaces","authors":"Georgios Daskalopoulos, Chikako Mese","doi":"10.1515/ans-2023-0122","DOIUrl":"https://doi.org/10.1515/ans-2023-0122","url":null,"abstract":"We construct infinite energy harmonic maps from a quasi-compact Kähler surface with a Poincaré-type metric into an NPC space. This is the first step in the construction of pluriharmonic maps from quasiprojective varieties into symmetric spaces of non-compact type, Euclidean and hyperbolic buildings and Teichmüller space.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"42 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140108251","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}

{"title":"Nonlinear problems inspired by the Born–Infeld theory of electrodynamics","authors":"Yisong Yang","doi":"10.1515/ans-2023-0123","DOIUrl":"https://doi.org/10.1515/ans-2023-0123","url":null,"abstract":"It is shown that nonlinear electrodynamics of the Born–Infeld theory type may be exploited to shed insight into a few fundamental problems in theoretical physics, including rendering electromagnetic asymmetry to energetically exclude magnetic monopoles, achieving finite electromagnetic energy to relegate curvature singularities of charged black holes, and providing theoretical interpretation of equations of state of cosmic fluids via k-essence cosmology. Also discussed are some nonlinear differential equation problems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140074032","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}

{"title":"Liouville type theorems involving fractional order systems","authors":"Qiuping Liao, Zhao Liu, Xinyue Wang","doi":"10.1515/ans-2023-0108","DOIUrl":"https://doi.org/10.1515/ans-2023-0108","url":null,"abstract":"In this paper, let <jats:italic>α</jats:italic> be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mfenced close=\"\" open=\"{\"> <m:mrow> <m:mtable> <m:mtr> <m:mtd columnalign=\"left\"> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width=\"0.3333em\" /> <m:mspace width=\"0.3333em\" /> <m:mspace width=\"0.3333em\" /> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>v</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>g</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width=\"0.3333em\" /> <m:mspace width=\"0.3333em\" /> <m:mspace width=\"0.3333em\" /> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>.</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> <jats:tex-math> $begin{cases}{left(-{Delta}right)}^{alpha /2}uleft(xright)=fleft(uleft(xright),vleft(xright)right), xin {mathbb{R}}^{n},quad hfill {left(-{Delta}right)}^{alpha /2}vleft(xright)=gleft(uleft(xright),vleft(xright)right), xin {mathbb{R}}^{n}.quad hfill end{cases}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://ww","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"5 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140074036","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}

{"title":"Annuloids and Δ-wings","authors":"David Hoffman, Francisco Martín, Brian White","doi":"10.1515/ans-2023-0111","DOIUrl":"https://doi.org/10.1515/ans-2023-0111","url":null,"abstract":"We describe new annular examples of complete translating solitons for the mean curvature flow and how they are related to a family of translating graphs, the Δ-wings. In addition, we will prove several related results that answer questions that arise naturally in this investigation. These results apply to translators in general, not just to graphs or annuli.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"44 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140073958","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}

{"title":"Eigenvalue lower bounds and splitting for modified Ricci flow","authors":"Tobias Holck Colding, William P. Minicozzi II","doi":"10.1515/ans-2022-0083","DOIUrl":"https://doi.org/10.1515/ans-2022-0083","url":null,"abstract":"We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"32 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140074029","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}

{"title":"Stability and critical dimension for Kirchhoff systems in closed manifolds","authors":"Emmanuel Hebey","doi":"10.1515/ans-2022-0066","DOIUrl":"https://doi.org/10.1515/ans-2022-0066","url":null,"abstract":"The Kirchhoff equation was proposed in 1883 by Kirchhoff [<jats:italic>Vorlesungen über Mechanik</jats:italic>, Leipzig, Teubner, 1883] as an extension of the classical D’Alembert’s wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [“On some questions in boundary value problems of mathematical physics,” in <jats:italic>Contemporary Developments in Continuum Mechanics and PDE’s</jats:italic>, G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msup> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mfenced close=\")\" open=\"(\"> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:msub> <m:mo>∫</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:msub> <m:mo stretchy=\"false\">|</m:mo> <m:mi>∇</m:mi> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant=\"normal\">d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> <jats:tex-math> $frac{{partial }^{2}u}{partial {t}^{2}}+left(a+b{int }_{{Omega}}vert nabla u{vert }^{2}mathrm{d}xright){Delta}u=fleft(x,uright),$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2022-0066_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula> where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mo form=\"prefix\" movablelimits=\"false\">∑</m:mo> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msubsup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:mfrac> </m:math> <jats:tex-math> ${Delta}=-sum frac{{partial }^{2}}{partial {x}_{i}^{2}}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2022-0066_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula> is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when <jats:italic>u</jats:italic","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"6 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019654","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}

{"title":"A Liouville theorem for superlinear parabolic equations on the Heisenberg group","authors":"Juncheng Wei, Ke Wu","doi":"10.1515/ans-2023-0119","DOIUrl":"https://doi.org/10.1515/ans-2023-0119","url":null,"abstract":"We consider a parabolic nonlinear equation on the Heisenberg group. Applying the Gidas–Spruck type estimates, we prove that under suitable conditions, the equation does not have positive solutions. As an application of the nonexistence result, we provide optimal universal estimates for positive solutions.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"51 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019658","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}

{"title":"On subsolutions and concavity for fully nonlinear elliptic equations","authors":"Bo Guan","doi":"10.1515/ans-2023-0116","DOIUrl":"https://doi.org/10.1515/ans-2023-0116","url":null,"abstract":"Subsolutions and concavity play critical roles in classical solvability, especially <jats:italic>a priori</jats:italic> estimates, of fully nonlinear elliptic equations. Our first primary goal in this paper is to explore the possibility to weaken the concavity condition. The second is to clarify relations between weak notions of subsolution introduced by Székelyhidi and the author, respectively, in attempt to treat equations on closed manifolds. More precisely, we show that these weak notions of subsolutions are equivalent for equations defined on convex cones of type 1 in the sense defined by Caffarelli, Nirenberg and Spruck.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"69 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019536","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}

{"title":"Perturbation compactness and uniqueness for a class of conformally compact Einstein manifolds","authors":"Sun-Yung Alice Chang, Yuxin Ge, Xiaoshang Jin, Jie Qing","doi":"10.1515/ans-2023-0124","DOIUrl":"https://doi.org/10.1515/ans-2023-0124","url":null,"abstract":"In this paper, we establish compactness results for some classes of conformally compact Einstein metrics defined on manifolds of dimension d ≥ 4. In the special case when the manifold is the Euclidean ball with the unit sphere as the conformal infinity, the existence of such class of metrics has been established in the earlier work of C. R. Graham and J. Lee (“Einstein metrics with prescribed conformal infinity on the ball,” <jats:italic>Adv. Math.</jats:italic>, vol. 87, no. 2, pp. 186–225, 1991). As an application of our compactness result, we derive the uniqueness of the Graham–Lee metrics. As a second application, we also derive some gap theorem, or equivalently, some results of non-existence CCE fill-ins.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019650","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}