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A PDE Approach to the Existence and Regularity of Surfaces of Minimum Mean Curvature Variation 最小均方差曲面的存在性和规则性的 PDE 方法
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-05 DOI: 10.1007/s00205-024-02016-5
Luis A. Caffarelli, Pablo Raúl Stinga, Hernán Vivas
{"title":"A PDE Approach to the Existence and Regularity of Surfaces of Minimum Mean Curvature Variation","authors":"Luis A. Caffarelli,&nbsp;Pablo Raúl Stinga,&nbsp;Hernán Vivas","doi":"10.1007/s00205-024-02016-5","DOIUrl":"10.1007/s00205-024-02016-5","url":null,"abstract":"<div><p>We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem </p><div><div><span>$$begin{aligned} min _mathcal {M}frac{1}{2}int _mathcal {M}|nabla _{mathcal {M}}H|^2,{text {d}}A, end{aligned}$$</span></div></div><p>where <span>(mathcal {M})</span> ranges over all <i>n</i>-dimensional manifolds in <span>(mathbb {R}^{n+1})</span> with a prescribed boundary, <span>(nabla _{mathcal {M}}H)</span> is the tangential gradient along <span>(mathcal {M})</span> of the mean curvature <i>H</i> of <span>(mathcal {M})</span> and d<i>A</i> is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results in the case of graphs. These are the first analytic results available for this problem.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Quantified Legendreness and the Regularity of Minima 量化传奇和最小值的规律性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-29 DOI: 10.1007/s00205-024-02008-5
Cristiana De Filippis, Lukas Koch, Jan Kristensen
{"title":"Quantified Legendreness and the Regularity of Minima","authors":"Cristiana De Filippis,&nbsp;Lukas Koch,&nbsp;Jan Kristensen","doi":"10.1007/s00205-024-02008-5","DOIUrl":"10.1007/s00205-024-02008-5","url":null,"abstract":"<div><p>We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and 2<i>d</i>-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples—in particular, we improve in an essentially optimal fashion Marcellini’s original results (Marcellini in Arch Rat Mech Anal 105:267–284, 1989).\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02008-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Gradient Flow for Entropy on Closed Planar Curves 封闭平面曲线上的熵梯度流
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-27 DOI: 10.1007/s00205-024-02014-7
Lachlann O’Donnell, Glen Wheeler, Valentina-Mira Wheeler
{"title":"The Gradient Flow for Entropy on Closed Planar Curves","authors":"Lachlann O’Donnell,&nbsp;Glen Wheeler,&nbsp;Valentina-Mira Wheeler","doi":"10.1007/s00205-024-02014-7","DOIUrl":"10.1007/s00205-024-02014-7","url":null,"abstract":"<div><p>In this paper we consider the steepest descent <span>(L^2)</span>-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class <span>(C^2)</span> or embedded of class <span>(W^{2,2})</span> bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02014-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Interior Regularity for Two-Dimensional Stationary Q-Valued Maps 二维静态 Q 值映射的内部正则性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-22 DOI: 10.1007/s00205-024-02011-w
Jonas Hirsch, Luca Spolaor
{"title":"Interior Regularity for Two-Dimensional Stationary Q-Valued Maps","authors":"Jonas Hirsch,&nbsp;Luca Spolaor","doi":"10.1007/s00205-024-02011-w","DOIUrl":"10.1007/s00205-024-02011-w","url":null,"abstract":"<div><p>We prove that 2-dimensional <i>Q</i>-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02011-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Objective Rates as Covariant Derivatives on the Manifold of Riemannian Metrics 作为黎曼度量漫域上协变导数的客观率
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-15 DOI: 10.1007/s00205-024-02010-x
B. Kolev, R. Desmorat
{"title":"Objective Rates as Covariant Derivatives on the Manifold of Riemannian Metrics","authors":"B. Kolev,&nbsp;R. Desmorat","doi":"10.1007/s00205-024-02010-x","DOIUrl":"10.1007/s00205-024-02010-x","url":null,"abstract":"<div><p>The subject of so-called objective derivatives in Continuum Mechanics has a long history and has generated varying views concerning their true mathematical interpretation. Several attempts have been made to provide a mathematical definition that would at least partially unify the existing notions. In this paper, we demonstrate that, under natural assumptions, all objective derivatives correspond to covariant derivatives on the infinite-dimensional manifold <span>(textrm{Met}(mathcal {B}))</span> of Riemannian metrics on the body. Furthermore, a natural Leibniz rule enables canonical extensions from covariant to contravariant tensor fields and vice versa. This makes the sometimes-used distinction between objective derivatives of “Lie type” and “co-rotational type” unnecessary. For an exhaustive list of objective derivatives found in the literature, we exhibit the corresponding covariant derivative on <span>(textrm{Met}(mathcal {B}))</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index 稳定或指数受限的 7 维最小超曲面的退化
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-03 DOI: 10.1007/s00205-024-02003-w
Nick Edelen
{"title":"Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index","authors":"Nick Edelen","doi":"10.1007/s00205-024-02003-w","DOIUrl":"10.1007/s00205-024-02003-w","url":null,"abstract":"<div><p>A 7-dimensional area-minimizing embedded hypersurface <span>(M^7)</span> will in general have a discrete singular set, and the same is true if <i>M</i> is locally stable provided <span>({mathcal {H}}^6(textrm{sing}M) = 0)</span>. We show that if <span>(M_i^7)</span> is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then <span>(M_i rightarrow M)</span> can limit to a singular <span>(M^7)</span> with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence <span>(i')</span> with controlled bi-Lipschitz maps <span>(phi _{i'})</span> taking <span>(phi _{i'}(M_{1'}) = M_{i'})</span>. As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces <i>M</i> in a closed Riemannian 8-manifold <span>((N^8, g))</span> with a priori bounds <span>({mathcal {H}}^7(M) leqq Lambda )</span> and <span>(textrm{index}(M) leqq I)</span> divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric <i>g</i> to vary, or <i>M</i> to be singular.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02003-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stable Recovery of Coefficients in an Inverse Fault Friction Problem 反断层摩擦问题中系数的稳定恢复
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-01 DOI: 10.1007/s00205-024-02009-4
Maarten V. de Hoop, Matti Lassas, Jinpeng Lu, Lauri Oksanen
{"title":"Stable Recovery of Coefficients in an Inverse Fault Friction Problem","authors":"Maarten V. de Hoop,&nbsp;Matti Lassas,&nbsp;Jinpeng Lu,&nbsp;Lauri Oksanen","doi":"10.1007/s00205-024-02009-4","DOIUrl":"10.1007/s00205-024-02009-4","url":null,"abstract":"<div><p>We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02009-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of Minimizers for the Dirac–Fock Model of Crystals 晶体的狄拉克-福克模型存在最小值
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-25 DOI: 10.1007/s00205-024-01988-8
Isabelle Catto, Long Meng, Éric Paturel, Éric Séré
{"title":"Existence of Minimizers for the Dirac–Fock Model of Crystals","authors":"Isabelle Catto,&nbsp;Long Meng,&nbsp;Éric Paturel,&nbsp;Éric Séré","doi":"10.1007/s00205-024-01988-8","DOIUrl":"10.1007/s00205-024-01988-8","url":null,"abstract":"<div><p>Whereas many different models exist in mathematics and physics for the ground states of non-relativistic crystals, the relativistic case has been much less studied, and we are not aware of any mathematical result on a fully relativistic treatment of crystals. In this paper, we introduce a mean-field relativistic energy for crystals in terms of periodic density matrices. This model is inspired both from a recent definition of the Dirac–Fock ground state for atoms and molecules, due to one of us, and from the non-relativistic Hartree–Fock model for crystals. We prove the existence of a ground state when the number of electrons per cell is not too large.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Recovery of Coefficients in Semilinear Transport Equations 恢复半线性传输方程中的系数
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-19 DOI: 10.1007/s00205-024-02007-6
Ru-Yu Lai, Gunther Uhlmann, Hanming Zhou
{"title":"Recovery of Coefficients in Semilinear Transport Equations","authors":"Ru-Yu Lai,&nbsp;Gunther Uhlmann,&nbsp;Hanming Zhou","doi":"10.1007/s00205-024-02007-6","DOIUrl":"10.1007/s00205-024-02007-6","url":null,"abstract":"<div><p>We consider the inverse problem for time-dependent semilinear transport equations. We show that time-independent coefficients of both the linear (absorption or scattering coefficients) and nonlinear terms can be uniquely determined, in a stable way, from the boundary measurements, by applying a linearization scheme and Carleman estimates for the linear transport equations. We establish results in both Euclidean and general geometry settings.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System 凯勒-西格尔系统中无限时间炸裂的存在性和稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-18 DOI: 10.1007/s00205-024-02006-7
Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei
{"title":"Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System","authors":"Juan Dávila,&nbsp;Manuel del Pino,&nbsp;Jean Dolbeault,&nbsp;Monica Musso,&nbsp;Juncheng Wei","doi":"10.1007/s00205-024-02006-7","DOIUrl":"10.1007/s00205-024-02006-7","url":null,"abstract":"<div><p>Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system </p><div><figure><div><div><picture><img></picture></div></div></figure></div><p> We consider the critical mass case <span>(int _{{mathbb {R}}^2} u_0(x), textrm{d}x = 8pi )</span>, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function <span>(u_0^*)</span> with mass <span>(8pi )</span> such that for any initial condition <span>(u_0)</span> sufficiently close to <span>(u_0^*)</span> and mass <span>(8pi )</span>, the solution <i>u</i>(<i>x</i>, <i>t</i>) of (<span>(*)</span>) is globally defined and blows-up in infinite time. As <span>(trightarrow +infty )</span> it has the approximate profile </p><div><div><span>$$begin{aligned} u(x,t) approx frac{1}{lambda ^2(t)} Uleft( frac{x-xi (t)}{lambda (t)} right) , quad U(y)= frac{8}{(1+|y|^2)^2}, end{aligned}$$</span></div></div><p>where <span>(lambda (t) approx frac{c}{sqrt{log t}})</span>, <span>(xi (t)rightarrow q)</span> for some <span>(c&gt;0)</span> and <span>(qin {mathbb {R}}^2)</span>. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02006-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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