Journal de Mathematiques Pures et Appliquees最新文献

{"title":"Migrating elastic flows","authors":"Tomoya Kemmochi ,&nbsp;Tatsuya Miura","doi":"10.1016/j.matpur.2024.02.003","DOIUrl":"10.1016/j.matpur.2024.02.003","url":null,"abstract":"<div><p>Huisken's problem asks whether there is an elastic flow of closed planar curves that is initially contained in the upper half-plane but ‘migrates’ to the lower half-plane at a positive time. Here we consider variants of Huisken's problem for open curves under the natural boundary condition, and construct various migrating elastic flows both analytically and numerically.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"185 ","pages":"Pages 47-62"},"PeriodicalIF":2.3,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140047412","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":"Minimal solutions of master equations for extended mean field games","authors":"Chenchen Mou ,&nbsp;Jianfeng Zhang","doi":"10.1016/j.matpur.2024.02.002","DOIUrl":"10.1016/j.matpur.2024.02.002","url":null,"abstract":"<div><p>In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is well known that, without any monotonicity conditions, mean field games typically contain multiple mean field equilibria and the wellposedness of their corresponding master equations fails. In this paper, a partial order for the set of probability measure flows is proposed to compare different mean field equilibria. The minimal and maximal mean field equilibria under this partial order are constructed and satisfy the flow property. The corresponding value functions, however, are in general discontinuous. We thus introduce a notion of weak-viscosity solutions for the master equation and verify that the value functions are indeed weak-viscosity solutions. Moreover, a comparison principle for weak-viscosity semi-solutions is established and thus these two value functions serve as the minimal and maximal weak-viscosity solutions in appropriate sense. In particular, when these two value functions coincide, the value function becomes the unique weak-viscosity solution to the master equation. The novelties of the work persist even when restricted to the standard mean field games.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"184 ","pages":"Pages 190-217"},"PeriodicalIF":2.3,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140047916","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":"Fast fusion in a two-dimensional coagulation model","authors":"Iulia Cristian ,&nbsp;Juan J.L. Velázquez","doi":"10.1016/j.matpur.2024.02.004","DOIUrl":"10.1016/j.matpur.2024.02.004","url":null,"abstract":"<div><p>In this work, we study a particular system of coagulation equations characterized by two values, namely volume <em>v</em> and surface area <em>a</em>. Compared to the standard one-dimensional models, this model incorporates additional information about the geometry of the particles. We describe the coagulation process as a combination between collision and fusion of particles. We prove that we are able to recover the standard one-dimensional coagulation model when fusion happens quickly and that we are able to recover an equation in which particles interact and form a ramified like system in time when fusion happens slowly.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"184 ","pages":"Pages 91-117"},"PeriodicalIF":2.3,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000242/pdfft?md5=f692611bb7a87f3a392cc9f2733a069b&pid=1-s2.0-S0021782424000242-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140047567","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}

{"title":"A geometrisation of N-manifolds","authors":"M. Heuer,&nbsp;M. Jotz","doi":"10.1016/j.matpur.2024.02.005","DOIUrl":"10.1016/j.matpur.2024.02.005","url":null,"abstract":"<div><p>This paper proposes a <em>geometrisation</em> of <span><math><mi>N</mi></math></span>-manifolds of degree <em>n</em> as <em>n</em>-fold vector bundles equipped with a (signed) <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-symmetry. More precisely, it proves an equivalence between the categories of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>-manifolds and the category of (signed) symmetric <em>n</em>-fold vector bundles, by finding that symmetric <em>n</em>-fold vector bundle cocycles and <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>-manifold cocycles are identical.</p><p>This extends the already known equivalences of [1]-manifolds with vector bundles, and of [2]-manifolds with involutive double vector bundles, where the involution is understood as an <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-action.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"184 ","pages":"Pages 1-70"},"PeriodicalIF":2.3,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000254/pdfft?md5=f674935c74ae1cfa9abff704eba87938&pid=1-s2.0-S0021782424000254-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140047731","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}

{"title":"A global branch approach to normalized solutions for the Schrödinger equation","authors":"Louis Jeanjean ,&nbsp;Jianjun Zhang ,&nbsp;Xuexiu Zhong","doi":"10.1016/j.matpur.2024.01.004","DOIUrl":"10.1016/j.matpur.2024.01.004","url":null,"abstract":"<div><p>We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schrödinger equation of the form<span><span><span><math><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>λ</mi><mi>u</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><mi>N</mi><mo>≥</mo><mn>1</mn><mo>.</mo></math></span></span></span> Our approach permits to handle in a unified way nonlinearities <span><math><mi>g</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> which are either mass subcritical, mass critical or mass supercritical. Among its main ingredients is the study of the asymptotic behaviors of the positive solutions as <span><math><mi>λ</mi><mo>→</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></math></span> or <span><math><mi>λ</mi><mo>→</mo><mo>+</mo><mo>∞</mo></math></span> and the existence of an unbounded continuum of solutions in <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></math></span>.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 44-75"},"PeriodicalIF":2.3,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000102/pdfft?md5=f6c2872a0f6dac1f94b8685209ca5ffc&pid=1-s2.0-S0021782424000102-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657810","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}

{"title":"The primitive equations with stochastic wind driven boundary conditions","authors":"Tim Binz ,&nbsp;Matthias Hieber ,&nbsp;Amru Hussein ,&nbsp;Martin Saal","doi":"10.1016/j.matpur.2024.01.001","DOIUrl":"10.1016/j.matpur.2024.01.001","url":null,"abstract":"<div><p>The primitive equations for geophysical flows are studied under the influence of <em>stochastic wind driven boundary conditions</em> modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math></span>-<span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>z</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>-setting. This solution is constructed in critical spaces.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 76-101"},"PeriodicalIF":2.3,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000072/pdfft?md5=30fb9d977114954181c690ca0fa1ea9e&pid=1-s2.0-S0021782424000072-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139665447","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}

{"title":"Stability of the separable solutions for a nonlinear boundary diffusion problem","authors":"Tianling Jin ,&nbsp;Jingang Xiong ,&nbsp;Xuzhou Yang","doi":"10.1016/j.matpur.2024.01.002","DOIUrl":"10.1016/j.matpur.2024.01.002","url":null,"abstract":"<div><p>In this paper, we study a nonlinear boundary diffusion equation<span> of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the stability of its separable solutions.</span></p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 1-43"},"PeriodicalIF":2.3,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658132","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 Almgren minimality of the product of a paired calibrated set with a calibrated set of codimension 1 with singularities, and new Almgren minimal cones","authors":"Xiangyu Liang","doi":"10.1016/j.matpur.2024.01.006","DOIUrl":"10.1016/j.matpur.2024.01.006","url":null,"abstract":"<div><p>In this paper, we prove that the product of a paired calibrated set and a set of codimension 1 calibrated by a coflat calibration with small singularity set is Almgren minimal. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets–Plateau's problem in the setting of sets. In particular, a direct application of the above result leads to various types of new singularities for Almgren minimal sets, e.g. the product of any paired calibrated cone (such as the cone over the <span><math><mi>d</mi><mo>−</mo><mn>2</mn></math></span> skeleton of the unit cube in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>d</mi><mo>≥</mo><mn>4</mn></math></span>) with homogeneous area minimizing hypercones (such as the Simons cone).</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 137-169"},"PeriodicalIF":2.3,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658525","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":"Nonlocal critical growth elliptic problems with jumping nonlinearities","authors":"Giovanni Molica Bisci ,&nbsp;Kanishka Perera ,&nbsp;Raffaella Servadei ,&nbsp;Caterina Sportelli","doi":"10.1016/j.matpur.2024.01.005","DOIUrl":"10.1016/j.matpur.2024.01.005","url":null,"abstract":"<div><p>In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in the presence of a jumping nonlinearity. By using variational and topological methods and applying some new linking theorems recently proved by Perera and Sportelli in <span>[19]</span>, we prove the existence of a nontrivial solution for the problem under consideration.</p><p>The results we obtain here are the nonlocal counterparts of the ones obtained in <span>[19]</span> in the context of a local equation. Due to the nonlocal nature of our problem, some additional difficulties arise, and the arguments employed in the local setting need to be improved or reconceived. In fact, the proofs of our main theorems require some refined techniques and new regularity results for weak solutions of nonlocal problems that are of independent interest.</p><p>We would like to point out that our results are specifically for a nonlocal problem with the fractional operator in integral form. However, we do not exclude the possibility that our results may have a counterpart for the spectral operator studied in <span>[27]</span>. Since nonlocal operators in integral form are being widely investigated in the current literature, especially in connection with geometric problems, we have restricted ourselves to elliptic equations driven by a fractional operator in integral form here.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 170-196"},"PeriodicalIF":2.3,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000114/pdfft?md5=f51fc8015e6f845f984b0ffcc0c1be63&pid=1-s2.0-S0021782424000114-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657933","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}

{"title":"Borel (α,β)-multitransforms and quantum Leray–Hirsch: Integral representations of solutions of quantum differential equations for P1-bundles","authors":"Giordano Cotti","doi":"10.1016/j.matpur.2024.01.003","DOIUrl":"10.1016/j.matpur.2024.01.003","url":null,"abstract":"<div><p>In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (<em>qDE</em>s) associated with the quantum cohomology of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundles on Fano varieties. It is shown that bases of solutions of the <em>qDE</em> associated with the total space of the <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundle can be reconstructed from the datum of bases of solutions of the <em>qDE</em> associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in <span>[17]</span>, called <em>Borel</em> <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span><em>-multitransforms</em>. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a <em>universal</em> feature, being independent of the specifically chosen <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of <em>qDE</em>s. As an example, we show how to integrate the <em>qDE</em> of blow-up of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> at one point via Borel multitransforms of solutions of the <em>qDE</em> of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 102-136"},"PeriodicalIF":2.3,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000096/pdfft?md5=454aefc44b3bd17a37fbb4d16963075b&pid=1-s2.0-S0021782424000096-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658186","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}