Pub Date : 2023-04-01DOI: 10.1515/acv-2023-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/acv-2023-frontmatter2","DOIUrl":"https://doi.org/10.1515/acv-2023-frontmatter2","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135170546","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}
Abstract In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group ℍ n {{mathbb{H}^{n}}} via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in ℍ 1 {mathbb{H}^{1}} under an assumption on the location of the singular set, and in ℍ n {mathbb{H}^{n}} for n ≥ 2 {ngeq 2} in the proper class of horizontally umbilical hypersurfaces.
{"title":"A characterization of gauge balls in ℍ n by horizontal curvature","authors":"Chiara Guidi, Vittorio Martino, G. Tralli","doi":"10.1515/acv-2022-0058","DOIUrl":"https://doi.org/10.1515/acv-2022-0058","url":null,"abstract":"Abstract In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group ℍ n {{mathbb{H}^{n}}} via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in ℍ 1 {mathbb{H}^{1}} under an assumption on the location of the singular set, and in ℍ n {mathbb{H}^{n}} for n ≥ 2 {ngeq 2} in the proper class of horizontally umbilical hypersurfaces.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49659487","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}
Abstract In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau’s problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau’s problem. It gives a very good description of local behavior for soap films. The natural question of whether the product of any two Almgren minimal sets is still minimal is still open, although it seems obvious in intuition. We prove the Almgren minimality for the product of two large classes of Almgren minimal sets – the class of 1-codimensional calibrated manifolds and the class of paired calibrated sets. The general idea is to properly combine different topological conditions (separation and spanning) under different homology groups, to set up a reasonable topological condition and prove the minimality for the product under this condition, which will imply the Almgren minimality. A main difficulty comes from the codimension – algebraic coherences such as multiplicity, separation and orientation do not exist anymore for codimensions larger than 1. An unexpectedly useful thing in the present paper is the flow of the calibrations. Its most important role among all is helping us to do the decomposition of a competitor with the help of the first projections along the flows.
{"title":"On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1","authors":"Xiangyu Liang","doi":"10.1515/acv-2021-0105","DOIUrl":"https://doi.org/10.1515/acv-2021-0105","url":null,"abstract":"Abstract In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau’s problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau’s problem. It gives a very good description of local behavior for soap films. The natural question of whether the product of any two Almgren minimal sets is still minimal is still open, although it seems obvious in intuition. We prove the Almgren minimality for the product of two large classes of Almgren minimal sets – the class of 1-codimensional calibrated manifolds and the class of paired calibrated sets. The general idea is to properly combine different topological conditions (separation and spanning) under different homology groups, to set up a reasonable topological condition and prove the minimality for the product under this condition, which will imply the Almgren minimality. A main difficulty comes from the codimension – algebraic coherences such as multiplicity, separation and orientation do not exist anymore for codimensions larger than 1. An unexpectedly useful thing in the present paper is the flow of the calibrations. Its most important role among all is helping us to do the decomposition of a competitor with the help of the first projections along the flows.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45594869","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}
Abstract In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ( x ) {u(x)} is the viscosity sub-solution of the equation if and only if u ( x ) {u(x)} is so-called ( α , p ) {(alpha,p)} -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an ( α , p ) {(alpha,p)} -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.
{"title":"Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity","authors":"S. Shi, Zhichun Zhai, Lei Zhang","doi":"10.1515/acv-2021-0110","DOIUrl":"https://doi.org/10.1515/acv-2021-0110","url":null,"abstract":"Abstract In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ( x ) {u(x)} is the viscosity sub-solution of the equation if and only if u ( x ) {u(x)} is so-called ( α , p ) {(alpha,p)} -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an ( α , p ) {(alpha,p)} -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44403781","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}
Abstract Let (M,g) {(M,g)} be a smooth compact Riemannian manifold of dimension n≥3 {ngeq 3} . Let also A be a smooth symmetrical positive (0,2) {(0,2)} -tensor field in M . By the Sobolev embedding theorem, we can write that there exist K,B>0 {K,B>0} such that for any u∈H1(M) {uin H^{1}(M)} , (0.1) ∥u∥L2⋆2≤K∥∇Au∥L22+B∥u∥L12 |u|_{L^{2^{star}}}^{2}leq K|nabla_{A}u|_{L^{2}}^{2}+B|u|_{L^{1}}^{2} where H1(M) {H^{1}(M)} is the standard Sobolev space of functions in L2 {L^{2}} with one derivative in L2 {L^{2}} , |∇Au|2=
摘要设(M,g) {(M,g)}是维数n≥3n的光滑紧致黎曼流形{geq 3}。也设A是{M中的光滑对称正(0,2)}(0,2)张量场。根据Sobolev嵌入定理，我们可以写出存在K, B ＆gt;0{ K,B＆gt;0}使得对于任意u∈H 1¹(M){ u in H¹(M)，{(0.1)∥u∥L²- 2≤K¹∥∇A²∥L²|u|＿L}²^ }{{{star}}} ^{2}leq K| nabla ＿Au{|＿L²}^{2{+}}B|u|＿L{²}^{2{其中}}H 1(M) H²(M{)是}L²L²中函数的{标准{Sobolev空间在L²L²}中}有一个导数，{|∇A²u | 2 = A²(∇²)u，∇{(u}}){ | {}}{nabla ＿Au|{^}2=A({}nabla u, nabla u)和2 - - 2^ }{{star}}是H^1的{临界{Sobolev指数。本文计算了(0.1)}}中最优可能K的值，并研究了相应的尖锐不等式的有效性。
{"title":"A twist in sharp Sobolev inequalities with lower order remainder terms","authors":"Emmanuel Hebey","doi":"10.1515/acv-2022-0046","DOIUrl":"https://doi.org/10.1515/acv-2022-0046","url":null,"abstract":"Abstract Let <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> {(M,g)} be a smooth compact Riemannian manifold of dimension <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> {ngeq 3} . Let also A be a smooth symmetrical positive <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> {(0,2)} -tensor field in M . By the Sobolev embedding theorem, we can write that there exist <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>K</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {K,B>0} such that for any <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {uin H^{1}(M)} , (0.1) <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mi>u</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:msup> <m:mi>L</m:mi> <m:msup> <m:mn>2</m:mn> <m:mo>⋆</m:mo> </m:msup> </m:msup> <m:mn>2</m:mn> </m:msubsup> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>K</m:mi> <m:mo></m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:msub> <m:mo>∇</m:mo> <m:mi>A</m:mi> </m:msub> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mn>2</m:mn> </m:msubsup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>B</m:mi> <m:mo></m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mi>u</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> <m:mn>2</m:mn> </m:msubsup> </m:mrow> </m:mrow> </m:mrow> </m:math> |u|_{L^{2^{star}}}^{2}leq K|nabla_{A}u|_{L^{2}}^{2}+B|u|_{L^{1}}^{2} where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{1}(M)} is the standard Sobolev space of functions in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} with one derivative in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} , <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:msub> <m:mo>∇</m:mo> <m:mi>A</m:mi> </m:msub> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>=</m:m","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135794396","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}
Abstract Given ε 0 > 0 {{varepsilon}_{0}>0} , I ∈ ℕ ∪ { 0 } {Iinmathbb{N}cup{0}} and K 0 , H 0 ≥ 0 {K_{0},H_{0}geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ ε 0 {operatorname{Inj}(X)geq{varepsilon}_{0}} and with the supremum of absolute sectional curvature at most K 0 {K_{0}} , and let M ↬ X {Mlooparrowright X} be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] {Hin[0,H_{0}]} with index at most I. For such M ↬ X {Mlooparrowright X} , we prove a structure theorem which describes how the interesting ambient geometry of the immersion is organized locally around at most I points of M, where the norm of the second fundamental form takes on large local maximum values.
给定ε 0 > {{varepsilon}＿{0}b> 0} ， I∈∈∪ { 0 } {Iinmathbb{N}cupb｛0｝} K 0, H 0≥0 {k＿{0}，嗯……{0}geq 0} ，设X是一个完备的黎曼3流形，注入半径为Inj (X)≥ε 0 {operatorname{Inj}(x)geq{varepsilon}＿{0}} 且绝对截面曲率的最大值不超过k0 {k＿{0}} ，让M * X {mlooparrowright x} 为平均曲率为H∈[0,H 0]的完全浸没面 {hin[0,H_{0}]} 对于这样的M * * * X {mlooparrowright x} ，我们证明了一个结构定理，该定理描述了浸入的有趣环境几何是如何在M的最多I个点附近局部组织的，其中第二个基本形式的范数具有较大的局部最大值。
{"title":"Hierarchy structures in finite index CMC surfaces","authors":"William H. Meeks III, Joaquín Pérez","doi":"10.1515/acv-2022-0113","DOIUrl":"https://doi.org/10.1515/acv-2022-0113","url":null,"abstract":"Abstract Given ε 0 > 0 {{varepsilon}_{0}>0} , I ∈ ℕ ∪ { 0 } {Iinmathbb{N}cup{0}} and K 0 , H 0 ≥ 0 {K_{0},H_{0}geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ ε 0 {operatorname{Inj}(X)geq{varepsilon}_{0}} and with the supremum of absolute sectional curvature at most K 0 {K_{0}} , and let M ↬ X {Mlooparrowright X} be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] {Hin[0,H_{0}]} with index at most I. For such M ↬ X {Mlooparrowright X} , we prove a structure theorem which describes how the interesting ambient geometry of the immersion is organized locally around at most I points of M, where the norm of the second fundamental form takes on large local maximum values.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44006407","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}
Abstract We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.
{"title":"Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case","authors":"G. Dal Maso, Rodica Toader","doi":"10.1515/acv-2022-0025","DOIUrl":"https://doi.org/10.1515/acv-2022-0025","url":null,"abstract":"Abstract We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46158932","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}
Abstract We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.
{"title":"Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity","authors":"J. C. Bellido, J. Cueto, C. Mora-Corral","doi":"10.1515/acv-2022-0089","DOIUrl":"https://doi.org/10.1515/acv-2022-0089","url":null,"abstract":"Abstract We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48180322","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}
Abstract Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L q {L^{q}} - or L ∞ {L^{infty}} -bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.
{"title":"Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds","authors":"G. Barletta, A. Cianchi, V. Maz'ya","doi":"10.1515/acv-2022-0014","DOIUrl":"https://doi.org/10.1515/acv-2022-0014","url":null,"abstract":"Abstract Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L q {L^{q}} - or L ∞ {L^{infty}} -bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46741668","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":"Une mise au point","authors":"Herbert Marcuse","doi":"10.4000/variations.2195","DOIUrl":"https://doi.org/10.4000/variations.2195","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87457603","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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