Annales De L Institut Henri Poincare-Analyse Non Lineaire最新文献

{"title":"Point interactions for 3D sub-Laplacians","authors":"Riccardo Adami ,&nbsp;Ugo Boscain ,&nbsp;Valentina Franceschi ,&nbsp;Dario Prandi","doi":"10.1016/j.anihpc.2020.10.007","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.10.007","url":null,"abstract":"<div><p>In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold <em>M</em>, no point interaction centered at a point <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>M</mi></math></span> exists. When <em>M</em> is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><mi>M</mi><mo>∖</mo><mo>{</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> is essentially self-adjoint in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math></span><span>. A particular example is the standard sub-Laplacian on the Heisenberg group<span>. This is in stark contrast with what happens in a Riemannian manifold </span></span><em>N</em>, whose associated Laplace-Beltrami operator acting on <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><mi>N</mi><mo>∖</mo><mo>{</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> is never essentially self-adjoint in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>N</mi><mo>)</mo></math></span>, if <span><math><mi>dim</mi><mo>⁡</mo><mi>N</mi><mo>≤</mo><mn>3</mn></math></span><span><span>. We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing </span>moment of inertia, rotating around its center of mass.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91684211","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}

{"title":"Connecting planar linear chains in the spatial N-body problem","authors":"Guowei Yu","doi":"10.1016/j.anihpc.2020.10.004","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.10.004","url":null,"abstract":"<div><p><span>The family of planar linear chains are found as collision-free action minimizers of the spatial </span><em>N</em>-body problem with equal masses under <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in <span>[32]</span> for the planar <em>N</em>-body problem. In particular, the monotone constraints required in <span>[32]</span> are proven to be unnecessary, as it will be implied by the action minimization property.</p><p><span><span>For each type of topological constraints, by considering the corresponding action minimization problem in a coordinate frame rotating around the vertical axis at a constant </span>angular velocity </span><em>ω</em>, we find an entire family of simple choreographies (seen in the rotating frame), as <em>ω</em> changes from 0 to <em>N</em>. Such a family starts from one planar linear chain and ends at another (seen in the original non-rotating frame). The action minimizer is collision-free, when <span><math><mi>ω</mi><mo>=</mo><mn>0</mn></math></span> or <em>N</em>, but may contain collision for <span><math><mn>0</mn><mo>&lt;</mo><mi>ω</mi><mo>&lt;</mo><mi>N</mi></math></span>. However it can only contain binary collisions and the corresponding collision solutions are <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> block-regularizable.</p><p>These families of solutions can be seen as a generalization of Marchal's <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>12</mn></mrow></msub></math></span> family for <span><math><mi>N</mi><mo>=</mo><mn>3</mn></math></span> to arbitrary <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span>. In particular, for certain types of topological constraints, based on results from <span>[3]</span> and <span>[7]</span>, we show that when <em>ω</em> belongs to some sub-intervals of <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>N</mi><mo>]</mo></math></span>, the corresponding minimizer must be a rotating regular <em>N</em>-gon contained in the horizontal plane.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91684212","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}

{"title":"The vorticity equations in a half plane with measures as initial data","authors":"Ken Abe","doi":"10.1016/j.anihpc.2020.10.002","DOIUrl":"10.1016/j.anihpc.2020.10.002","url":null,"abstract":"<div><p><span>We consider the two-dimensional Navier-Stokes equations subject to the Dirichlet boundary condition<span> in a half plane for initial vorticity with finite measures. We study local well-posedness of the associated vorticity equations for measures with a small pure point part and global well-posedness for measures with a small total variation. Our construction is based on an </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span><span>-estimate of a solution operator for the vorticity equations associated with the Stokes equations.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90155055","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}

{"title":"Optimal gradient estimates for the perfect conductivity problem with C1,α inclusions","authors":"Yu Chen,&nbsp;Haigang Li,&nbsp;Longjuan Xu","doi":"10.1016/j.anihpc.2020.09.009","DOIUrl":"10.1016/j.anihpc.2020.09.009","url":null,"abstract":"<div><p><span>In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from </span><span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> to <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span><span>. To obtain a pointwise upper bound of the gradient, we follow an iteration technique which is first used to deal with elliptic systems in a narrow domain by Li-Li-Bao-Yin (QAM 2014). However, when the inclusions are of </span><span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>, we can not use <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msup></math></span><span> estimates for elliptic equations<span><span> any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not </span>strictly convex, we prove that blow-up does not occur any more. The establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon.</span></span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89403349","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}

{"title":"Full and partial regularity for a class of nonlinear free boundary problems","authors":"Aram Karakhanyan","doi":"10.1016/j.anihpc.2020.09.008","DOIUrl":"10.1016/j.anihpc.2020.09.008","url":null,"abstract":"<div><p><span>In this paper we classify the nonnegative global minimizers of the functional</span><span><span><span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><mi>F</mi><mo>(</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>χ</mi></mrow><mrow><mo>{</mo><mi>u</mi><mo>&gt;</mo><mn>0</mn><mo>}</mo></mrow></msub><mo>,</mo></math></span></span></span> where <em>F</em> satisfies some structural conditions and <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>D</mi></mrow></msub></math></span> is the characteristic function of a set <span><math><mi>D</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We compute the second variation of the energy and study the properties of the stability operator. The free boundary <span><math><mo>∂</mo><mo>{</mo><mi>u</mi><mo>&gt;</mo><mn>0</mn><mo>}</mo></math></span> can be seen as a rectifiable <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span><span> varifold<span><span>. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of </span>tangent cones<span> modulo<span> a set of small Hausdorff dimension. In particular, we prove that if </span></span></span></span><span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span><span> and the ellipticity constants of the quasilinear elliptic operator generated by </span><em>F</em> are close to 1 then the conical free boundary must be flat.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75487627","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}

{"title":"Homogenization of a stochastically forced Hamilton-Jacobi equation","authors":"Benjamin Seeger","doi":"10.1016/j.anihpc.2020.11.001","DOIUrl":"10.1016/j.anihpc.2020.11.001","url":null,"abstract":"<div><p>We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation<span> is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions, which are of independent interest.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.11.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77792881","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}

{"title":"Entropy theory for sectional hyperbolic flows","authors":"Maria José Pacifico ,&nbsp;Fan Yang ,&nbsp;Jiagang Yang","doi":"10.1016/j.anihpc.2020.10.001","DOIUrl":"10.1016/j.anihpc.2020.10.001","url":null,"abstract":"<div><p><span>We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. We show that for </span><span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span><span><span> flows, every sectional hyperbolic set Λ is entropy expansive, and the </span>topological entropy varies continuously with the flow. Furthermore, if Λ is Lyapunov stable, then it has positive entropy; in addition, if Λ is a chain recurrent class, then it contains a periodic orbit. As a corollary, we prove that for </span><span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> generic flows, every Lorenz-like class is an attractor.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74371853","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}

{"title":"A game of alignment: Collective behavior of multi-species","authors":"Siming He ,&nbsp;Eitan Tadmor","doi":"10.1016/j.anihpc.2020.10.003","DOIUrl":"10.1016/j.anihpc.2020.10.003","url":null,"abstract":"<div><p>We study the (hydro-)dynamics of multi-species driven by alignment. What distinguishes the different species is the protocol of their interaction with the rest of the crowd: the collective motion is described by different <em>communication kernels</em>, <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>α</mi><mi>β</mi></mrow></msub></math></span>, between the crowds in species <em>α</em> and <em>β</em>. We show that flocking of the overall crowd emerges provided the communication array between species forms a <span><em>connected graph</em></span>. In particular, the crowd within each species need <em>not</em> interact with its own kind, i.e., <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>α</mi><mi>α</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span>; different species which are engaged in such ‘game’ of alignment require a connecting path for propagation of information which will lead to the flocking of overall crowd. The same methodology applies to multi-species aggregation dynamics governed by first-order alignment: connectivity implies concentration around an emerging consensus.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83117367","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}

{"title":"On the self-similar behavior of coagulation systems with injection","authors":"Marina A. Ferreira, Eugenia Franco, J. Vel'azquez","doi":"10.4171/aihpc/61","DOIUrl":"https://doi.org/10.4171/aihpc/61","url":null,"abstract":"In this paper we prove the existence of a family of self-similar solutions for a class of coagulation equations with a constant flux of particles from the origin. These solutions are expected to describe the longtime asymptotics of Smoluchowski's coagulation equations with a time independent source of clusters concentrated in small sizes. The self-similar profiles are shown to be smooth, provided the coagulation kernel is also smooth. Moreover, the self-similar profiles are estimated from above and from below by $x^{-(gamma+3)/2}$ as $x to 0$, where $gamma<1 $ is the homogeneity of the kernel, and are proven to decay at least exponentially as $x to infty$.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80425762","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}

{"title":"Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves","authors":"M. Haragus, Mathew A. Johnson, Wesley R. Perkins, B. D. Rijk","doi":"10.4171/aihpc/65","DOIUrl":"https://doi.org/10.4171/aihpc/65","url":null,"abstract":"We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such solutions has only been established against co-periodic perturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbations which are localized, i.e., integrable on the line. Such localized perturbations naturally yield the absence of a spectral gap, so we must rely on a substantially different method with origins in the stability analysis of periodic waves in reaction-diffusion systems. The relevant linear estimates have been obtained in recent work by the first three authors through a delicate decomposition of the associated linearized solution operator. Since its most critical part just decays diffusively, the nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation. However, the modulated perturbation satisfies a quasilinear equation yielding an apparent loss of regularity. To overcome this obstacle, we incorporate tame estimates on the unmodulated perturbation, which satisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to close an independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLE against localized perturbations with precisely the same decay rates as predicted by the linear theory.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87187830","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}