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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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
Higher Robin eigenvalues for the p-Laplacian operator as p approaches 1 当 p 接近 1 时 p 拉普拉斯算子的高罗宾特征值
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-09 DOI: 10.1007/s00526-024-02769-7
José C. Sabina de Lis, Sergio Segura de León
{"title":"Higher Robin eigenvalues for the p-Laplacian operator as p approaches 1","authors":"José C. Sabina de Lis, Sergio Segura de León","doi":"10.1007/s00526-024-02769-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02769-7","url":null,"abstract":"<p>This work addresses several aspects of the dependence on <i>p</i> of the higher eigenvalues <span>(lambda _n)</span> to the Robin problem,\u0000</p><span>$$begin{aligned} {left{ begin{array}{ll} -Delta _p u = lambda |u|^{p-2}u &amp;{} qquad xin Omega , |nabla u|^{p-2}dfrac{partial u}{partial nu }+ b |u|^{p-2}u= 0&amp;{}qquad xin partial Omega . end{array}right. } end{aligned}$$</span><p>Here, <span>(Omega subset {{mathbb {R}}}^N)</span> is a <span>(C^1)</span> bounded domain, <span>(nu )</span> is the outer unit normal, <span>(Delta _p u = text {div} (|nabla u|^{p-2}nabla u))</span> stands for the <i>p</i>-Laplacian operator and <span>(bin L^infty (partial Omega ))</span>. Main results concern: (a) the existence of the limits of <span>(lambda _n)</span> as <span>(prightarrow 1)</span>, (b) the ‘limit problems’ satisfied by the ‘limit eigenpairs’, (c) the continuous dependence of <span>(lambda _n)</span> on <i>p</i> when <span>(1&lt; p &lt;infty )</span> and (d) the limit profile of the eigenfunctions as <span>(prightarrow 1)</span>. The latter study is performed in the one dimensional and radially symmetric cases. Corresponding properties on the Dirichlet and Neumann eigenvalues are also studied in these two special scenarios.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141571346","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
An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces 局部尤多维奇空间中欧拉流的存在性和唯一性的基本证明
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-05 DOI: 10.1007/s00526-024-02750-4
Gianluca Crippa, Giorgio Stefani
{"title":"An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces","authors":"Gianluca Crippa, Giorgio Stefani","doi":"10.1007/s00526-024-02750-4","DOIUrl":"https://doi.org/10.1007/s00526-024-02750-4","url":null,"abstract":"<p>We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set <span>(Omega subset mathbb {R}^2)</span> or on the torus <span>(Omega =mathbb {T}^2)</span>. We construct global-in-time weak solutions with vorticity in <span>(L^1cap L^p_{ul})</span> and in <span>(L^1cap Y^Theta _{ul})</span>, where <span>(L^p_{ul})</span> and <span>(Y^Theta _{ul})</span> are suitable uniformly-localized versions of the Lebesgue space <span>(L^p)</span> and of the Yudovich space <span>(Y^Theta )</span> respectively, with no condition at infinity for the growth function <span>(Theta )</span>. We also provide an explicit modulus of continuity for the velocity depending on the growth function <span>(Theta )</span>. We prove uniqueness of weak solutions in <span>(L^1cap Y^Theta _{ul})</span> under the assumption that <span>(Theta )</span> grows moderately at infinity. In contrast to Yudovich’s energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón–Zygmund theory or Littlewood–Paley decomposition, and actually applies not only to the Biot–Savart law, but also to more general operators whose kernels obey some natural structural assumptions.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551997","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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