Giuseppe Buttazzo, Juan Casado-Díaz, Faustino Maestre
In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BVmathrm{BV} one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BVmathrm{BV} regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases.
{"title":"On the regularity of optimal potentials in control problems governed by elliptic equations","authors":"Giuseppe Buttazzo, Juan Casado-Díaz, Faustino Maestre","doi":"10.1515/acv-2023-0010","DOIUrl":"https://doi.org/10.1515/acv-2023-0010","url":null,"abstract":"In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>BV</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0010_eq_0228.png\"/> <jats:tex-math>mathrm{BV}</jats:tex-math> </jats:alternatives> </jats:inline-formula> one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>BV</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0010_eq_0228.png\"/> <jats:tex-math>mathrm{BV}</jats:tex-math> </jats:alternatives> </jats:inline-formula> regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Giacomo Canevari, Federico Luigi Dipasquale, Giandomenico Orlandi
We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n≥3{ngeq 3}. Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2.
我们考虑在维数为 n ≥ 3 {ngeq 3} 的封闭黎曼流形上的赫米线束上的轨距不变金兹堡-朗道函数（也称为阿贝尔杨-米尔斯-希格斯模型）。假定耦合参数有对数能量约束，我们研究伦敦极限临界点的渐近行为。在方便地选择了量规之后，我们证明了有限能量临界点在索波列夫规范中的紧凑性。此外，得益于一个合适的单调性公式，我们证明了临界点的能量密度经耦合参数的对数重估后，会收敛到一个静止的、可整流的、标度为 2 的变折点的权重度量。
{"title":"The Yang–Mills–Higgs functional on complex line bundles: Asymptotics for critical points","authors":"Giacomo Canevari, Federico Luigi Dipasquale, Giandomenico Orlandi","doi":"10.1515/acv-2023-0064","DOIUrl":"https://doi.org/10.1515/acv-2023-0064","url":null,"abstract":"We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0064_eq_1143.png\"/> <jats:tex-math>{ngeq 3}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"79 3 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian operator of a level set function for the open set bounded by the hypersurface. As application, we give a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff’s soap bubble theorem in space forms, Serrin’s overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff–Fenchel inequalities.
{"title":"Stability from rigidity via umbilicity","authors":"Julian Scheuer","doi":"10.1515/acv-2023-0119","DOIUrl":"https://doi.org/10.1515/acv-2023-0119","url":null,"abstract":"We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian operator of a level set function for the open set bounded by the hypersurface. As application, we give a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff’s soap bubble theorem in space forms, Serrin’s overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff–Fenchel inequalities.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"22 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140811699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d>n-22{d>frac{n-2}{2}}. Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.
我们研究了具有非紧凑边界的非紧凑黎曼流形上共形度量的存在性，这些度量作为度量空间是完整的，在内部具有负常标量曲率，在边界上具有负常平均曲率。这些度量是在光滑流形上构造的，光滑流形是通过从某些以广义实心圆锥为局部模型的 n 维紧凑空间中移除 d 维子流形而得到的。我们证明了当且仅当 d > n - 2 2 {d>frac{n-2}{2}} 时这种度量的存在。 .我们的主要定理受 Aviles-McOwen 和 Loewner-Nirenberg 的经典结果启发，在文献中被称为 "奇异 Yamabe 问题"。
{"title":"A singular Yamabe problem on manifolds with solid cones","authors":"Juan Alcon Apaza, Sérgio Almaraz","doi":"10.1515/acv-2022-0105","DOIUrl":"https://doi.org/10.1515/acv-2022-0105","url":null,"abstract":"We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing <jats:italic>d</jats:italic>-dimensional submanifolds from certain <jats:italic>n</jats:italic>-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>d</m:mi> <m:mo>></m:mo> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0105_eq_0720.png\"/> <jats:tex-math>{d>frac{n-2}{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"33 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai, Seis and Spadaro [L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 2016, 1, Article ID 18]. This method is based on the gradient flow approach due to Almgren, Taylor and Wang [F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 1993, 2, 387–438] and Luckhaus and Sturzenhecker [S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 1995, 2, 253–271] and my aim is to replace the volume penalization with the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on second variation argument.
在本文中，我将重温 Mugnai、Seis 和 Spadaro 通过离散最小化运动方案构建的体积保持平均曲率流的全局弱解[L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume preserving mean-curvature flow, Calc.Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc.Var.Partial Differential Equations 55 2016，1，Article ID 18]。该方法基于 Almgren、Taylor 和 Wang [F. Almgren, J. E. Wang] 提出的梯度流方法。Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim.31 1993, 2, 387-438] 和 Luckhaus and Sturzenhecker [S. Luckhaus and T. Sturzenhecker].Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc.Var.Partial Differential Equations 3 1995, 2, 253-271]，而我的目的是在离散方案中直接用体积约束取代体积惩罚，从实用角度来看，这也许更自然。技术上的新颖之处在于基于二次变分论证的密度估计证明。
{"title":"Flat flow solution to the mean curvature flow with volume constraint","authors":"Vesa Julin","doi":"10.1515/acv-2023-0047","DOIUrl":"https://doi.org/10.1515/acv-2023-0047","url":null,"abstract":"In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai, Seis and Spadaro [L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 2016, 1, Article ID 18]. This method is based on the gradient flow approach due to Almgren, Taylor and Wang [F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 1993, 2, 387–438] and Luckhaus and Sturzenhecker [S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 1995, 2, 253–271] and my aim is to replace the volume penalization with the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on second variation argument.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"27 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Esther Cabezas-Rivas, Salvador Moll, Marcos Solera
We obtain existence of minimizers for the p-capacity functional defined with respect to a centrally symmetric anisotropy for 1<p<∞{1<p<infty}, including the case of a crystalline norm in ℝN{mathbb{R}^{N}}. The result is obtained by a characterization of the corresponding subdifferential and it applies to unbounded domains of the form ℝN∖Ω¯{mathbb{R}^{N}setminusoverline{Omega}} under mild regularity assumptions (Lipschitz-continuous boundary) and no convexity requirements on the bounded domain Ω. If we further assume an interior ball condition (where the Wulff shape plays the role of a ball), then any minimizer is shown to be Lipschitz continuous.
我们得到了针对 1 < p < ∞ {1<p<infty} 的中心对称各向异性定义的 p 容量函数的最小值存在性，包括ℝ N {mathbb{R}^{N}} 中的结晶规范。 包括ℝ N {mathbb{R}^{N}} 中的晶体规范的情况。 .这个结果是通过相应子微分的特征得到的，它适用于形式为 ℝ N ∖ Ω ¯ {mathbb{R}^{N}setminusoverline{Omega}} 的无界域，前提是温和的正则性假设（Lipschitz-连续边界）以及对有界域 Ω 没有凸性要求。如果我们进一步假设一个内部球条件（Wulff 形状扮演球的角色），那么任何最小化都可以证明是 Lipschitz 连续的。
{"title":"Characterization of the subdifferential and minimizers for the anisotropic p-capacity","authors":"Esther Cabezas-Rivas, Salvador Moll, Marcos Solera","doi":"10.1515/acv-2023-0057","DOIUrl":"https://doi.org/10.1515/acv-2023-0057","url":null,"abstract":"We obtain existence of minimizers for the <jats:italic>p</jats:italic>-capacity functional defined with respect to a centrally symmetric anisotropy for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0057_eq_0885.png\" /> <jats:tex-math>{1<p<infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, including the case of a crystalline norm in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0057_eq_1021.png\" /> <jats:tex-math>{mathbb{R}^{N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The result is obtained by a characterization of the corresponding subdifferential and it applies to unbounded domains of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>∖</m:mo> <m:mover accent=\"true\"> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>¯</m:mo> </m:mover> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0057_eq_1019.png\" /> <jats:tex-math>{mathbb{R}^{N}setminusoverline{Omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> under mild regularity assumptions (Lipschitz-continuous boundary) and no convexity requirements on the bounded domain Ω. If we further assume an interior ball condition (where the Wulff shape plays the role of a ball), then any minimizer is shown to be Lipschitz continuous.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Simone Di Marino, Lorenzo Portinale, Emanuela Radici
We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N→∞{Ntoinfty} to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ>0{tau>0}. This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.
我们通过 N 个有序粒子锥体上合适的离散度量，研究了实线上具有非线性流动性的广义瓦瑟斯坦距离的离散化问题，这一问题自然出现在偏微分方程的确定性粒子逼近框架中。特别是，我们提供了相关离散度量在 N → ∞ {Ntoinfty} 到连续度量时的Γ-收敛结果，并讨论了通过所谓广义最小化运动逼近一维守恒定律（梯度流类型）的应用，证明了这些方案在任何给定离散时间步长 τ > 0 {tau>0} 时的收敛结果。这是系列研究的第一项成果，旨在揭示广义梯度流结构、守恒定律和具有非线性运动的瓦瑟斯坦距离之间的相互作用。
{"title":"Optimal transport with nonlinear mobilities: A deterministic particle approximation result","authors":"Simone Di Marino, Lorenzo Portinale, Emanuela Radici","doi":"10.1515/acv-2022-0076","DOIUrl":"https://doi.org/10.1515/acv-2022-0076","url":null,"abstract":"We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>N</m:mi> <m:mo>→</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0076_eq_0466.png\" /> <jats:tex-math>{Ntoinfty}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>τ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0076_eq_0751.png\" /> <jats:tex-math>{tau>0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"10 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the area-preserving Willmore evolution of surfaces ϕ that are close to a half-sphere with a small radius, sliding on the boundary S of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a “half-spherical” shape. Additionally, we investigate the asymptotic behavior of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of S, we then establish convergence of the flow.
我们考虑了接近小半径半球的表面 j 的面积保全 Willmore 演化，这些表面在域 Ω 的边界 S 上滑动，同时与域 S 正交。我们证明，流动在任何时候都存在，并保持 "半球 "形状。此外，我们还研究了流动的渐近行为，并证明在较大时间内，曲面的原点近似遵循一个显式常微分方程。对 S 的平均曲率施加附加条件后，我们确定了流的收敛性。
{"title":"On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary","authors":"Jan-Henrik Metsch","doi":"10.1515/acv-2023-0023","DOIUrl":"https://doi.org/10.1515/acv-2023-0023","url":null,"abstract":"We consider the area-preserving Willmore evolution of surfaces ϕ that are close to a half-sphere with a small radius, sliding on the boundary <jats:italic>S</jats:italic> of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a “half-spherical” shape. Additionally, we investigate the asymptotic behavior of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of <jats:italic>S</jats:italic>, we then establish convergence of the flow.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Roberto Alicandro, Lucia De Luca, Mariapia Palombaro, Marcello Ponsiglione
We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence.
{"title":"Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions","authors":"Roberto Alicandro, Lucia De Luca, Mariapia Palombaro, Marcello Ponsiglione","doi":"10.1515/acv-2023-0053","DOIUrl":"https://doi.org/10.1515/acv-2023-0053","url":null,"abstract":"We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the <jats:italic>dilute regime</jats:italic>, we analyze the asymptotic behavior of the nonlinear elastic energy, as the <jats:italic>core-radius</jats:italic> (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"14 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>×</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0092_eq_0192.png" /> <jats:tex-math>{mathbb{R}^{n}timesmathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0092_eq_0220.png" /> <jats:tex-math>{ngeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0092_eq_0150.png" /> <jats:tex-math>{(r,y(r))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0092_eq_0149.png" /> <jats:tex-math>{(r(y),y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0092_eq_0232.png" /> <jats:tex-math>{r=|x|}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0092_eq_0257.png" /> <jats:tex-math>{xinmathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, is the radially symmetric coordinate and <jats:inline-formula>
我们将给出一个新的证明：在 ℝ n × ℝ {mathbb{R}^{n}timesmathbb{R}} 中存在反均值曲率流的非紧凑同调孤子。 , n ≥ 2 {ngeq 2} , 形式为 ( r , y ( r ) ) {(r,y(r))} 或 ( r ( y ) , y ) {(r(y),y)} , 其中 r = | x | {r=|x|} x ∈ ℝ n {xinmathbb{R}^{n}} ，是径向对称坐标。 是径向对称坐标，y∈ ℝ {yinmathbb{R}} 。 .更确切地说，对于任意 1 n < λ < 1 n - 1 {frac{1}{n}<lambda<frac{1}{n-1}} 和 μ < 0 {mu<0} ，我们将给出新的证明。 我们将给出一个新的证明，证明存在一个唯一的解 r ( y ) ∈ C 2 ( μ , ∞ ) ∩ C ( [ μ , ∞ ) ) {r(y)in C^{2}(mu,infty)cap C([mu,infty))}的方程 r y y ( y ) 1 + r y ( y ) 2 = n - 1 r ( y ) - 1 + r y ( y ) 2 λ ( r ( y ) - y r y ( y ) ) , r ( y ) > 0 , frac{r_{yy}(y)}{1+r_{y}(y)^{2}}=frac{n-1}{r(y)}-frac{1+r_{y}(y)^{2}}{% lambda(r(y)-yr_{y}(y))},quad r(y)>；0, in ( μ , ∞ ) {(mu,infty)} which satisfies r ( μ ) = 0 {r(mu)=0} and r y ( μ ) = lim y ↘ μ r y ( y ) = + ∞ {r_{y}(mu)=lim_{ysearrowmu}r_{y}(y)=+infty} .我们证明存在常数 y 2 > y 1 > 0 y_{2}>y_{1}>0，使得 r y ( y ) > 0 r_{y}(y)>0 for any μ < y < y 1 mu<y<y_{1}。 , r y ( y 1 ) = 0 r_{y}(y_{1})=0 , r y ( y ) < 0 r_{y}(y)<0 for any y > y 1 y>y_{1} , r y y ( y ) < 0 r_{y}(y)<0 for any y > y 1 y>y_{1} , r y y ( y ) < 0 r_{yy}(y)<0 for any μ < y < y 2 mu<y<y_{2} , r y y ( y ) < 0 r_{yy}(y)<0 for any μ < y < y 2 对于任意 y > y 2 {y>y_{2}} ，r y y ( y 2 ) = 0 r_{yy}(y_{2})=0 且 r y y ( y ) > 0 {r_{yy}(y)>0} 。 .此外，lim y → + ∞ r ( y ) = 0 {lim_{yto+infty}r(y)=0} 和 lim y → + ∞ y r y ( y ) = 0 {lim_{yto+infty}yr_{y}(y)=0} .
{"title":"Another proof of the existence of homothetic solitons of the inverse mean curvature flow","authors":"Shu-Yu Hsu","doi":"10.1515/acv-2022-0092","DOIUrl":"https://doi.org/10.1515/acv-2022-0092","url":null,"abstract":"We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>×</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0192.png\" /> <jats:tex-math>{mathbb{R}^{n}timesmathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0220.png\" /> <jats:tex-math>{ngeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>r</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0150.png\" /> <jats:tex-math>{(r,y(r))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0149.png\" /> <jats:tex-math>{(r(y),y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0232.png\" /> <jats:tex-math>{r=|x|}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0257.png\" /> <jats:tex-math>{xinmathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, is the radially symmetric coordinate and <jats:inline-formula>","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"10 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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