Advances in Calculus of Variations最新文献

{"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}

{"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}

{"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}

{"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>&gt;</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&gt;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}

{"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}

{"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>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</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&lt;p&lt;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}

{"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>&gt;</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&gt;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}

{"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}

{"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}

{"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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;m:mo&gt;×&lt;/m:mo&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0192.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{R}^{n}timesmathbb{R}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;≥&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0220.png\" /&gt; &lt;jats:tex-math&gt;{ngeq 2}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, of the form &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;r&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;y&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;r&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0150.png\" /&gt; &lt;jats:tex-math&gt;{(r,y(r))}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; or &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;r&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;y&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;y&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0149.png\" /&gt; &lt;jats:tex-math&gt;{(r(y),y)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;r&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0232.png\" /&gt; &lt;jats:tex-math&gt;{r=|x|}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0257.png\" /&gt; &lt;jats:tex-math&gt;{xinmathbb{R}^{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, is the radially symmetric coordinate and &lt;jats:inline-formula&gt;","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}