Journal de l’École polytechnique — Mathématiques最新文献

{"title":"Large mass rigidity for a liquid drop model in 2D with kernels of finite moments","authors":"B. Merlet, Marc Pegon","doi":"10.5802/jep.178","DOIUrl":"https://doi.org/10.5802/jep.178","url":null,"abstract":". — Motivated by Gamow’s liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter P ( E ) is replaced by P ( E ) − γP ε ( E ) , with 0 < γ < 1 and P ε a nonlocal energy such that P ε ( E ) → P ( E ) as ε vanishes. We prove that unit area minimizers are disks for ε > 0 small enough. More precisely, we first show that in dimension 2 , minimizers are necessarily convex, provided that ε is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension n (cid:62) 2 ) the unit ball in R n is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass m 0 such that for any m > m 0 , the disk is the unique minimizer of area m up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass. ) − γP ε ( E ) , où 0 < γ < 1 et P ε est une énergie non locale telle que P ε ( E ) → P ( E ) lorsque ε tend vers zéro. Nous montrons que pour ε assez petit les minimiseurs à aire fixée sont les disques. Pour cela, nous établissons d’abord qu’en dimension 2 , les minimiseurs sont convexes dès que ε est suffisamment petit. Ceci implique que le bord d’un minimiseur est une petite perturbation Lipschitz d’un cercle. Puis, par un argument à la Fuglede, nous prouvons (en dimension arbitraire n (cid:62) 2 ) que si un minimiseur à volume fixé est une perturbation d’une boule au sens précédent, alors c’est une boule. Ce problème isopérimétrique est équivalent à une généralisation du modèle de goutte liquide pour le noyau atomique introduit par Gamow lorsque le potentiel répulsif non local est donné par un noyau suffisamment intégrable. Dans cette formulation, notre résultat principal indique que si le premier moment du noyau est inférieur à un seuil explicite, il existe une masse critique m 0 telle que les minimiseurs de masse prescrite m > m 0 sont les disques. Ceci contraste fortement avec le cas classique des noyaux de Riesz, où le problème n’admet pas de minimiseur au-delà d’une masse critique.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114101035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface","authors":"P. Berger","doi":"10.5802/jep.224","DOIUrl":"https://doi.org/10.5802/jep.224","url":null,"abstract":"We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (intregrable KAM island) among analytic, symplectic diffeomorphism of any closed surface. In particilar this solves a problem by F. Przytycki (1982).","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127640290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Zero-cycle groups on algebraic varieties","authors":"F. Binda, A. Krishna","doi":"10.5802/jep.183","DOIUrl":"https://doi.org/10.5802/jep.183","url":null,"abstract":"We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sato for the Chow group of 0-cycles over p-adic fields. We also use these results to deduce a torsion theorem for Suslin homology which extends a result of Bloch to open varieties.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123010647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation","authors":"T. Pauw","doi":"10.5802/jep.211","DOIUrl":"https://doi.org/10.5802/jep.211","url":null,"abstract":"Letting $A subset mathbb{R}^n$ be Borel measurable and $W_0 : A to mathbb{G}(n,m)$ Lipschitzian, we establish that begin{equation*} limsup_{r to 0^+} frac{mathcal{H}^m left[ A cap B(x,r) cap (x+ W_0(x))right]}{alpha(m)r^m} geq frac{1}{2^n}, end{equation*} for $mathcal{L}^n$-almost every $x in A$. In particular, it follows that $A$ is $mathcal{L}^n$-negligible if and only if $mathcal{H}^m(A cap (x+W_0(x))=0$, for $mathcal{L}^n$-almost every $x in A$.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121641868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantitative De Giorgi methods in kinetic theory","authors":"Jessica Guerand, C. Mouhot","doi":"10.5802/jep.203","DOIUrl":"https://doi.org/10.5802/jep.203","url":null,"abstract":"We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{\"o}lder continuity with quantitative estimates. The paper is self-contained.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125295566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantitative homogenization theory for random suspensions in steady Stokes flow","authors":"Mitia Duerinckx, A. Gloria","doi":"10.5802/jep.204","DOIUrl":"https://doi.org/10.5802/jep.204","url":null,"abstract":"Abstract. This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"4 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122650897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Product set growth in Burnside groups","authors":"Rémi Coulon, M. Steenbock","doi":"10.5802/jep.187","DOIUrl":"https://doi.org/10.5802/jep.187","url":null,"abstract":"A BSTRACT . Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups. 20F65, 20F67, 20F50, 20F06, 20F69.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133499364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Birkhoff normal forms for Hamiltonian PDEs in their energy space","authors":"J. Bernier, B. Gr'ebert","doi":"10.5802/jep.193","DOIUrl":"https://doi.org/10.5802/jep.193","url":null,"abstract":"We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that, to exchange energy, modes have to oscillate at the same frequency. Contrary to the previous existing results, we do not require the solutions to be especially smooth. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d = 1 and nonlinear Schr{\"o}dinger equations in dimension d $le$ 2.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124577919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces","authors":"Christian El Emam, Andrea Seppi","doi":"10.5802/jep.190","DOIUrl":"https://doi.org/10.5802/jep.190","url":null,"abstract":"—We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions. Résumé (Rigidité des difféomorphismes minimaux lagrangiens entre surfaces sphériques à singularités coniques) Nous démontrons que toute application minimale lagrangienne entre deux surfaces fermées sphériques à singularités coniques est une isométrie, sans aucune hypothèse sur les valeurs des multi-angles des deux surfaces. En appliquant ce résultat, nous prouvons une généralisation du théorème classique de rigidité de Liebmann, notamment l’énoncé que toute immersion dans l’espace euclidien de dimension 3 d’une surface fermée avec courbure gaussienne constante positive et avec points de ramification est un revêtement ramifié sur une sphère.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129432066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unconditional Chebyshev biases in number fields","authors":"D. Fiorilli, F. Jouve","doi":"10.5802/jep.192","DOIUrl":"https://doi.org/10.5802/jep.192","url":null,"abstract":". Prime counting functions are believed to exhibit, in various contexts, discrep-ancies beyond what famous equidistribution results predict; this phenomenon is known as Chebyshev’s bias. Rubinstein and Sarnak have developed a framework which allows to con-ditionally quantify biases in the distribution of primes in general arithmetic progressions. Their analysis has been generalized by Ng to the context of the Chebotarev density theorem, under the assumption of the Artin holomorphy conjecture, the Generalized Riemann Hypothesis, as well as a linear independence hypothesis on the zeros of Artin L -functions. In this paper we show unconditionally the occurence of extreme biases in this context. These biases lie far beyond what the strongest effective forms of the Chebotarev density theorem can predict. More precisely, we prove the existence of an infinite family of Galois extensions and associated conjugacy classes C 1 , C 2 ⊂ Gal( L/K ) of same size such that the number of prime ideals of norm up to x with Frobenius conjugacy class C 1 always exceeds that of Frobenius conjugacy class C 2 , for every large enough x . A key argument in our proof relies on features of certain subgroups of symmetric groups which enable us to circumvent the need for unproven properties of zeros of Artin L -functions.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121383207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}