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Regularity in the two-phase Bernoulli problem for the p-Laplace operator p 拉普拉斯算子两相伯努利问题中的正则性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-16 DOI: 10.1007/s00526-024-02789-3
Masoud Bayrami, Morteza Fotouhi
{"title":"Regularity in the two-phase Bernoulli problem for the p-Laplace operator","authors":"Masoud Bayrami, Morteza Fotouhi","doi":"10.1007/s00526-024-02789-3","DOIUrl":"https://doi.org/10.1007/s00526-024-02789-3","url":null,"abstract":"<p>We show that any minimizer of the well-known ACF functional (for the <i>p</i>-Laplacian) constitutes a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, which boils down to <span>(C^{1,eta })</span> regularity of the flat part of the free boundary. This result, in turn, is used to prove the Lipschitz regularity of minimizers by a dichotomy argument. It is noteworthy that the analysis of branch points is also included.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"37 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718856","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}
引用次数: 0
Extension operators and Korn inequality for variable coefficients in perforated domains with applications to homogenization of viscoelastic non-simple materials 穿孔域中可变系数的扩展算子和科恩不等式及其在粘弹性非简单材料均质化中的应用
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-16 DOI: 10.1007/s00526-024-02793-7
Markus Gahn
{"title":"Extension operators and Korn inequality for variable coefficients in perforated domains with applications to homogenization of viscoelastic non-simple materials","authors":"Markus Gahn","doi":"10.1007/s00526-024-02793-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02793-7","url":null,"abstract":"<p>In this paper we present the homogenization for nonlinear viscoelastic second-grade non-simple perforated materials at large strain in the quasistatic setting. The reference domain <span>(Omega _{varepsilon })</span> is periodically perforated and is depending on the scaling parameter <span>(varepsilon )</span> which describes the ratio between the size of the whole domain and the small periodic perforations. The mechanical energy depends on the gradient and also the second gradient of the deformation, and also respects positivity of the determinant of the deformation gradient. For the viscous stress we assume dynamic frame indifference and it is therefore depending on the rate of the Cauchy-stress tensor. For the derivation of the homogenized model for <span>(varepsilon rightarrow 0)</span> we use the method of two-scale convergence. For this uniform <i>a priori</i> estimates with respect to <span>(varepsilon )</span> are necessary. The most crucial part is to estimate the rate of the deformation gradient. Due to the time-dependent frame indifference of the viscous term, we only get coercivity with respect to the rate of the Cauchy-stress tensor. To overcome this problem we derive a Korn inequality for non-constant coefficients on the perforated domain. The crucial point is to verify that the constant in this inequality, which is usually depending on the domain, can be chosen independently of the parameter <span>(varepsilon )</span>. Further, we construct an extension operator for second order Sobolev spaces on perforated domains with operator norm independent of <span>(varepsilon )</span>.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718857","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}
引用次数: 0
Variational aspects of the generalized Seiberg–Witten functional 广义塞伯格-维滕函数的变量问题
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-16 DOI: 10.1007/s00526-024-02771-z
Wanjun Ai, Shuhan Jiang, Jürgen Jost
{"title":"Variational aspects of the generalized Seiberg–Witten functional","authors":"Wanjun Ai, Shuhan Jiang, Jürgen Jost","doi":"10.1007/s00526-024-02771-z","DOIUrl":"https://doi.org/10.1007/s00526-024-02771-z","url":null,"abstract":"<p>In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg–Witten functional that in particular includes the Kapustin–Witten functional as a special case. We first demonstrate the smoothness of weak solutions to this generalized functional. We then establish the existence of weak solutions under the assumption that the structure group of the bundle is abelian, by verifying the Palais–Smale compactness.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718854","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}
引用次数: 0
Second order necessary condition for a strong minimum in the classical problem of calculus of variations 变化微积分经典问题中强最小值的二阶必要条件
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-13 DOI: 10.1007/s00526-024-02795-5
A. D. Ioffe
{"title":"Second order necessary condition for a strong minimum in the classical problem of calculus of variations","authors":"A. D. Ioffe","doi":"10.1007/s00526-024-02795-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02795-5","url":null,"abstract":"<p>The paper offers a second order necessary condition for a strong minimum in the standard problem of calculus of variations. No idea of such a result seems to have appeared in the classical theory. But a simple example given in the paper shows that the condition can work when all known conditions fail. At the same time, the proof of the proposition is fairly simple. It is also explained in the paper that the condition effectively works only for problems with integrands not convex with respect to the last (derivative) argument.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"41 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612924","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}
引用次数: 0
Minkowski content estimates for generic area minimizing hypersurfaces 一般面积最小超曲面的闵科夫斯基内容估计值
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-13 DOI: 10.1007/s00526-024-02791-9
Xuanyu Li
{"title":"Minkowski content estimates for generic area minimizing hypersurfaces","authors":"Xuanyu Li","doi":"10.1007/s00526-024-02791-9","DOIUrl":"https://doi.org/10.1007/s00526-024-02791-9","url":null,"abstract":"<p>Let <span>(Gamma )</span> be a smooth, closed, oriented, <span>((n-1))</span>-dimensional submanifold of <span>(mathbb {R}^{n+1})</span>. It was shown by Chodosh–Mantoulidis–Schulze [6] that one can perturb <span>(Gamma )</span> to a nearby <span>(Gamma ')</span> such that all minimizing currents with boundary <span>(Gamma ')</span> are smooth away from a set with Hausdorff dimension less than <span>(n-9)</span>. We prove that the perturbation can be made such that the singular set of the minimizing current with boundary <span>(Gamma ')</span> has Minkowski dimension less than <span>(n-9)</span>.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"58 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612816","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}
引用次数: 0
A strong form of the quantitative Wulff inequality for crystalline norms 晶体规范的定量伍尔夫不等式的强形式
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-13 DOI: 10.1007/s00526-024-02796-4
Kenneth DeMason
{"title":"A strong form of the quantitative Wulff inequality for crystalline norms","authors":"Kenneth DeMason","doi":"10.1007/s00526-024-02796-4","DOIUrl":"https://doi.org/10.1007/s00526-024-02796-4","url":null,"abstract":"<p>Quantitative stability for crystalline anisotropic perimeters, with control on the oscillation of the boundary with respect to the corresponding Wulff shape, is proven for <span>(nge 3)</span>. This extends a result of Neumayer (SIAM J Math Anal 48:172–1772, 2016) in <span>(n=2)</span>.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"81 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612814","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}
引用次数: 0
On the Neumann (p, q)-eigenvalue problem in Hölder singular domains 论荷尔德奇异域中的诺依曼（p，q）特征值问题
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-10 DOI: 10.1007/s00526-024-02788-4
Prashanta Garain, Valerii Pchelintsev, Alexander Ukhlov
{"title":"On the Neumann (p, q)-eigenvalue problem in Hölder singular domains","authors":"Prashanta Garain, Valerii Pchelintsev, Alexander Ukhlov","doi":"10.1007/s00526-024-02788-4","DOIUrl":"https://doi.org/10.1007/s00526-024-02788-4","url":null,"abstract":"<p>In the article we study the Neumann (<i>p</i>, <i>q</i>)-eigenvalue problems in bounded Hölder <span>(gamma )</span>-singular domains <span>(Omega _{gamma }subset {mathbb {R}}^n)</span>. In the case <span>(1&lt;p&lt;infty )</span> and <span>(1&lt;q&lt;p^{*}_{gamma })</span> we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of (<i>p</i>, <i>q</i>)-eigenvalues.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"18 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141571344","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}
引用次数: 0
BV estimates on the transport density with Dirichlet region on the boundary 边界上有德里赫特区域的传输密度 BV 估计值
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-10 DOI: 10.1007/s00526-024-02746-0
Samer Dweik
{"title":"BV estimates on the transport density with Dirichlet region on the boundary","authors":"Samer Dweik","doi":"10.1007/s00526-024-02746-0","DOIUrl":"https://doi.org/10.1007/s00526-024-02746-0","url":null,"abstract":"<p>In this paper, we prove BV regularity on the transport density in the mass transport problem to the boundary in two dimensions under certain conditions on the domain, the boundary cost and the mass distribution. Moreover, we show by a counter-example that the smoothness of the mass distribution, the boundary and the boundary cost does not imply that the transport density is <span>(W^{1,p})</span>, for some <span>(p&gt;1)</span>.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141571347","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}
引用次数: 0
Phase transition of an anisotropic Ginzburg–Landau equation 各向异性金兹堡-朗道方程的相变
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-10 DOI: 10.1007/s00526-024-02779-5
Yuning Liu
{"title":"Phase transition of an anisotropic Ginzburg–Landau equation","authors":"Yuning Liu","doi":"10.1007/s00526-024-02779-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02779-5","url":null,"abstract":"<p>We study the effective geometric motions of an anisotropic Ginzburg–Landau equation with a small parameter <span>(varepsilon &gt;0)</span> which characterizes the width of the transition layer. For well-prepared initial datum, we show that as <span>(varepsilon )</span> tends to zero the solutions will develop a sharp interface limit which evolves under mean curvature flow. The bulk limits of the solutions correspond to a vector field <span>({textbf{u}}(x,t))</span> which is of unit length on one side of the interface, and is zero on the other side. The proof combines the modulated energy method and weak convergence methods. In particular, by a (boundary) blow-up argument we show that <span>({textbf{u}})</span> must be tangent to the sharp interface. Moreover, it solves a geometric evolution equation for the Oseen–Frank model in liquid crystals.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141571343","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}
引用次数: 0
Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities 具有导数非线性的非线性薛定谔方程系统的变量问题
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-10 DOI: 10.1007/s00526-024-02782-w
Hiroyuki Hirayama, Masahiro Ikeda
{"title":"Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities","authors":"Hiroyuki Hirayama, Masahiro Ikeda","doi":"10.1007/s00526-024-02782-w","DOIUrl":"https://doi.org/10.1007/s00526-024-02782-w","url":null,"abstract":"<p>We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"18 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141571345","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}
引用次数: 0
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