Serena Guarino Lo Bianco, Domenico Angelo La Manna, B. Velichkov
We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an existence and a regularity result for minimizers of the associated variational problem. Finally, in the appendix, we give an example of a class of Steiner symmetric minimizers.
{"title":"A two-phase problem with Robin conditions on the free boundary","authors":"Serena Guarino Lo Bianco, Domenico Angelo La Manna, B. Velichkov","doi":"10.5802/jep.139","DOIUrl":"https://doi.org/10.5802/jep.139","url":null,"abstract":"We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an existence and a regularity result for minimizers of the associated variational problem. Finally, in the appendix, we give an example of a class of Steiner symmetric minimizers.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134645960","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}
We consider a transport equation by a gradient vector field with a small viscous perturbation $-epsilonDelta_g$. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit $epsilonto 0^+$ , that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation $epsilon = 0$. We also prove that the two minimal times coincides for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY86].
{"title":"On uniform observability of gradient flows in the vanishing viscosity limit","authors":"C. Laurent, Matthieu L'eautaud","doi":"10.5802/JEP.151","DOIUrl":"https://doi.org/10.5802/JEP.151","url":null,"abstract":"We consider a transport equation by a gradient vector field with a small viscous perturbation $-epsilonDelta_g$. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit $epsilonto 0^+$ , that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation $epsilon = 0$. We also prove that the two minimal times coincides for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY86].","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131137409","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}
Let $mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $overline{mathcal{L}}$. We prove that the proportion of global sections $sigma$ with $leftlVert sigma rightrVert_{infty}<1$ of $overline{mathcal{L}}^{otimes d}$ whose divisor does not have a singular point on the fiber $mathcal{X}_p$ over any prime $p
{"title":"On the Bertini regularity theorem for arithmetic varieties","authors":"Xiaozong Wang","doi":"10.5802/jep.191","DOIUrl":"https://doi.org/10.5802/jep.191","url":null,"abstract":"Let $mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $overline{mathcal{L}}$. We prove that the proportion of global sections $sigma$ with $leftlVert sigma rightrVert_{infty}<1$ of $overline{mathcal{L}}^{otimes d}$ whose divisor does not have a singular point on the fiber $mathcal{X}_p$ over any prime $p<e^{varepsilon d}$ tends to $zeta_{mathcal{X}}(1+dim mathcal{X})^{-1}$ as $drightarrow infty$.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121184287","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}
If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.
{"title":"Duality for complexes of tori over a global field of positive characteristic","authors":"C. Demarche, David Harari","doi":"10.5802/jep.129","DOIUrl":"https://doi.org/10.5802/jep.129","url":null,"abstract":"If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"161 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122167144","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}
Using homological residue fields, we define supports for big objects in tensor-triangulated categories and prove a tensor-product formula.
利用同调剩余域，我们定义了张量三角化范畴中大对象的支撑点，并证明了一个张量积公式。
{"title":"Homological support of big objects in tensor-triangulated categories","authors":"Paul Balmer","doi":"10.5802/jep.135","DOIUrl":"https://doi.org/10.5802/jep.135","url":null,"abstract":"Using homological residue fields, we define supports for big objects in tensor-triangulated categories and prove a tensor-product formula.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129923636","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}
We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.
{"title":"A rigidity result for metric measure spaces with Euclidean heat kernel","authors":"G. Carron, David Tewodrose","doi":"10.5802/jep.179","DOIUrl":"https://doi.org/10.5802/jep.179","url":null,"abstract":"We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131360720","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}
We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Our proofs are based on a correspondence principle for action sets which was developed previously by the first two authors.
{"title":"Sets of transfer times with small densities","authors":"M. Bjorklund, A. Fish, I. Shkredov","doi":"10.5802/JEP.147","DOIUrl":"https://doi.org/10.5802/JEP.147","url":null,"abstract":"We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Our proofs are based on a correspondence principle for action sets which was developed previously by the first two authors.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132925952","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}
Using geometrical correspondences induced by projections and two-steps flag varieties, and a generalization of Orlov's projective bundle theorem, we relate the Hodge structures and derived categories of subvarieties of different Grassmannians. We construct isomorphisms between Calabi-Yau subHodge structures of hyperplane sections of Gr(3,n) and those of other varieties arising from symplectic Grassmannian and/or congruences of lines or planes. Similar results hold conjecturally for Calabi-Yau subcategories: we describe in details the Hodge structures and give partial categorical results relating the K3 Fano hyperplane sections of Gr(3,10) to other Fano varieties such as the Peskine variety. Moreover, we show how these correspondences allow to construct crepant categorical resolutions of the Coble cubics.
{"title":"Nested varieties of K3 type","authors":"M. Bernardara, Enrico Fatighenti, L. Manivel","doi":"10.5802/JEP.156","DOIUrl":"https://doi.org/10.5802/JEP.156","url":null,"abstract":"Using geometrical correspondences induced by projections and two-steps flag varieties, and a generalization of Orlov's projective bundle theorem, we relate the Hodge structures and derived categories of subvarieties of different Grassmannians. We construct isomorphisms between Calabi-Yau subHodge structures of hyperplane sections of Gr(3,n) and those of other varieties arising from symplectic Grassmannian and/or congruences of lines or planes. Similar results hold conjecturally for Calabi-Yau subcategories: we describe in details the Hodge structures and give partial categorical results relating the K3 Fano hyperplane sections of Gr(3,10) to other Fano varieties such as the Peskine variety. Moreover, we show how these correspondences allow to construct crepant categorical resolutions of the Coble cubics.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130516289","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}
In this paper we consider a coupled system of pdes modelling the interaction between a two-dimensional incompressible viscous fluid and a one-dimensional elastic beam located on the upper part of the fluid domain boundary. We design a functional framework to define weak solutions in case of contact between the elastic beam and the bottom of the fluid cavity. We then prove that such solutions exist globally in time regardless a possible contact by approximating the beam equation by a damped beam and letting this additional viscosity vanishes.
{"title":"On an existence theory for a fluid-beam problem encompassing possible contacts","authors":"Jean-J'erome Casanova, C. Grandmont, M. Hillairet","doi":"10.5802/JEP.162","DOIUrl":"https://doi.org/10.5802/JEP.162","url":null,"abstract":"In this paper we consider a coupled system of pdes modelling the interaction between a two-dimensional incompressible viscous fluid and a one-dimensional elastic beam located on the upper part of the fluid domain boundary. We design a functional framework to define weak solutions in case of contact between the elastic beam and the bottom of the fluid cavity. We then prove that such solutions exist globally in time regardless a possible contact by approximating the beam equation by a damped beam and letting this additional viscosity vanishes.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122529023","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}
Given any measurable subset $omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $ell^T(omega)in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $omega$ as an observation subset: the condition $ell^T(omega)>0$ is the well known emph{Geometric Control Condition}. In this article we establish two properties of the functional $ell^T$, one is geometric and the other is probabilistic. The first geometric property is on the maximal discrepancy of $ell^T$ when taking the closure. We may have $ell^T(mathring{omega})1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $omega$ in time $T$. The second property is of probabilistic nature. We take $M=mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random subsets $omega_varepsilon^n$ by darkening each cell in this grid with a probability $varepsilon$. We prove that the random law $ell^T(omega_varepsilon^n)$ converges in probability to $varepsilon$ as $nrightarrow+infty$. As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $omega_varepsilon^n$ in time $T$.
{"title":"Geometric and probabilistic results for the observability of the wave equation","authors":"E. Humbert, Y. Privat, E. Trélat","doi":"10.5802/jep.186","DOIUrl":"https://doi.org/10.5802/jep.186","url":null,"abstract":"Given any measurable subset $omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $ell^T(omega)in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $omega$ as an observation subset: the condition $ell^T(omega)>0$ is the well known emph{Geometric Control Condition}. \u0000 \u0000In this article we establish two properties of the functional $ell^T$, one is geometric and the other is probabilistic. \u0000 \u0000The first geometric property is on the maximal discrepancy of $ell^T$ when taking the closure. We may have $ell^T(mathring{omega})<ell^T(overlineomega)$ whenever there exist rays grazing $omega$ and the discrepancy between both quantities may be equal to $1$ for some subsets $omega$. We prove that, if the metric $g$ is $C^2$ and if $omega$ satisfies a slight regularity assumption, then $ell^T(overlineomega) leq frac{1}{2} left( ell^T(mathring{omega}) + 1 right)$. \u0000We also show that our assumptions are essentially sharp; in particular, surprisingly the result is wrong if the metric $g$ is not $C^2$. \u0000As a consequence, if $omega$ is regular enough and if $ell^T(overlineomega)>1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $omega$ in time $T$. \u0000 \u0000The second property is of probabilistic nature. We take $M=mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random subsets $omega_varepsilon^n$ by darkening each cell in this grid with a probability $varepsilon$. We prove that the random law $ell^T(omega_varepsilon^n)$ converges in probability to $varepsilon$ as $nrightarrow+infty$. \u0000As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $omega_varepsilon^n$ in time $T$.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"169 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128121792","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}
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