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
本文考虑 R n × R 中的一般二元分式抛物线问题 ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) 。 ${partial }_{t}^{alpha }uleft(x,tright)+mathcal{L}uleft(x,tright)=fleft(t,uleft(x,tright)right) text{in}.{$ 我们证明，满足某些单向渐近假设的有界全解 u 必须是单调递增的，并且在 f 的适当递减条件下沿该方向是一维对称的。我们这里的结果实际上解决了一个著名的问题，即对偶分式抛物方程中的吉本斯猜想。为了克服非局部发散型算子 L $mathcal{L}$ 和 Marchaud 时间导数 ∂ t α ${partial }_{t}^{alpha }$ 带来的困难，我们引入了几个新思路。首先，我们推导出与非局部算子 L $mathcal{L}$ 相对应的一般加权平均不等式，它在证明无界域中的最大原则时起到了基本的桥梁作用。然后，我们将这两个基本要素结合起来，用滑动方法建立了吉本斯猜想。值得注意的是，即使对于 L $mathcal{L}$的一个特例，即分数拉普拉斯（-Δ）s，我们的结果也是新颖的，本文所发展的方法将适用于涉及更一般的马尔查时间导数和更一般的非局部椭圆算子的广泛的非局部抛物方程。
{"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}
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.
{"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}
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.
{"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}
在本文中，设 α 为 0 至 2 之间的任意实数，我们研究以下涉及分数拉普拉卡的半线性椭圆系统： ( - Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x∈ R n , ( - Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x∈ R n , ( - Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x∈ R n . $begin{cases}{left(-{Delta}right)}^{alpha /2}uleft(xright)=fleft(uleft(xright),vleft(xright)right), xin {mathbb{R}}^{n}、vleft(xright)=gleft(uleft(xright),vleft(xright)/right), xin {mathbb{R}}^{n}.quad hfill end{cases}$ 在 f 和 g 的性质结构条件下，我们使用由 W. Chen、Y. Li 和 R. Zhang 引入的移动球直接法（"A direct method of moving spheres on fractional order equations," J. Funct. Analations, vol. 272, No.Anal.》，第 272 卷，第 4131-4157 页，2017 年）。在半空间中，我们通过结合移动平面和移动球的直接方法，在没有任何可积分性假设的情况下建立了一个 Liouville 型定理，这改进了 W. Dai、Z. Liu 和 G. Lu 所证明的结果（"Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space，" Potential Anal.，第 46 卷，第 569-588 页，2017 年）。
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.
{"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}
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.
{"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}
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
基尔霍夫方程由基尔霍夫于 1883 年提出[Vorlesungen über Mechanik, Leipzig, Teubner, 1883]，作为经典的达朗贝尔波方程对弹性弦振动的扩展。将近一个世纪后，雅克-路易斯-里昂（Jacques Louis Lions）["论数学物理边界值问题中的一些问题"，载于《连续介质力学和 PDE 的当代发展》（Contemporary Developments in Continuum Mechanics and PDE's），G. M. de la Penha, and L. A. Medeiros, Eds、阿姆斯特丹，North-Holland，1978 年]回到了方程，并提出了一个任意维度的带外力项的一般基尔霍夫方程，其写法为 ∂ 2 u∂ t 2 + a + b ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) 、 $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)、其中 Δ = - ∑ ∂ 2 ∂ x i 2 ${Delta}=-sum frac{{partial }^{2}}{partial {x}_{i}^{2}}$ 是拉普拉斯-贝尔特拉米欧几里得拉普拉斯。本文将研究在封闭流形情况下，当 u 为矢量值且 f 为纯临界幂非线性时，该方程的一个密切相关的静态版本。我们所考虑的是方程的稳定性，这个问题在现代非线性椭圆 PDE 理论中源于 Gidas 和 Spruck 的开创性工作。
{"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}
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.
{"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}
Subsolutions and concavity play critical roles in classical solvability, especially a priori 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.
{"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}
Sun-Yung Alice Chang, Yuxin Ge, Xiaoshang Jin, Jie Qing
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,” Adv. Math., 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.
在本文中，我们建立了定义在维数 d ≥ 4 的流形上的几类保角紧凑爱因斯坦度量的紧凑性结果。在流形是以单位球为保角无穷的欧几里得球的特殊情况下，这类度量的存在已在 C. R. Graham 和 J. Lee 的早期研究中得到证实（"球上具有规定保角无穷的爱因斯坦度量"，《数学研究》，第 87 卷第 2 期，第 186-225 页，1991 年）。作为紧凑性结果的一个应用，我们推导出了格雷厄姆-李度量的唯一性。作为第二个应用，我们还推导出了一些间隙定理，或者等同于一些不存在 CCE 填充的结果。
{"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}
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