Pub Date : 2021-08-25DOI: 10.1080/03605302.2021.1983598
H. Ishii, Kaizhi Wang, Lin Wang, Jun Yan
Abstract We study the Hamilton–Jacobi equations in M and in where the Hamiltonian depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.
{"title":"Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown","authors":"H. Ishii, Kaizhi Wang, Lin Wang, Jun Yan","doi":"10.1080/03605302.2021.1983598","DOIUrl":"https://doi.org/10.1080/03605302.2021.1983598","url":null,"abstract":"Abstract We study the Hamilton–Jacobi equations in M and in where the Hamiltonian depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"417 - 452"},"PeriodicalIF":1.9,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43956121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-23DOI: 10.1080/03605302.2022.2062572
C. Judge, Sugata Mondal
Abstract We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.
{"title":"Critical points of Laplace eigenfunctions on polygons","authors":"C. Judge, Sugata Mondal","doi":"10.1080/03605302.2022.2062572","DOIUrl":"https://doi.org/10.1080/03605302.2022.2062572","url":null,"abstract":"Abstract We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1559 - 1590"},"PeriodicalIF":1.9,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47703514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-17DOI: 10.1080/03605302.2023.2202720
T. Leslie, Changhui Tan
Abstract We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker–Smale dynamics. Our approach is inspired by the work of Brenier and Grenier on the pressureless Euler equations.
{"title":"Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment system","authors":"T. Leslie, Changhui Tan","doi":"10.1080/03605302.2023.2202720","DOIUrl":"https://doi.org/10.1080/03605302.2023.2202720","url":null,"abstract":"Abstract We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker–Smale dynamics. Our approach is inspired by the work of Brenier and Grenier on the pressureless Euler equations.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"753 - 791"},"PeriodicalIF":1.9,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44876221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-12DOI: 10.1080/03605302.2022.2109487
R. Moser
Abstract Given an ∞-harmonic function on a domain consider the function If with and then it is easy to check that the streamlines of are the level sets of w and w solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of by p-harmonic functions, the use of conjugate -harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of arises as a by-product.
{"title":"The streamlines of ∞-harmonic functions obey the inverse mean curvature flow","authors":"R. Moser","doi":"10.1080/03605302.2022.2109487","DOIUrl":"https://doi.org/10.1080/03605302.2022.2109487","url":null,"abstract":"Abstract Given an ∞-harmonic function on a domain consider the function If with and then it is easy to check that the streamlines of are the level sets of w and w solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of by p-harmonic functions, the use of conjugate -harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of arises as a by-product.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"2124 - 2145"},"PeriodicalIF":1.9,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45863210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-21DOI: 10.1080/03605302.2022.2050257
R. Carles, Chunmei Su
Abstract We consider the Schrödinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty.
{"title":"Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential","authors":"R. Carles, Chunmei Su","doi":"10.1080/03605302.2022.2050257","DOIUrl":"https://doi.org/10.1080/03605302.2022.2050257","url":null,"abstract":"Abstract We consider the Schrödinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1176 - 1192"},"PeriodicalIF":1.9,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48368245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-13DOI: 10.1080/03605302.2022.2139721
Kyudong Choi, In-Jee Jeong
Abstract For the axi-symmetric incompressible Euler equations, we prove linear in time filamentation near Hill’s vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining the recent nonlinear orbital stability obtained by the first author with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical simulations by Pozrikidis in 1986.
{"title":"Filamentation near Hill’s vortex","authors":"Kyudong Choi, In-Jee Jeong","doi":"10.1080/03605302.2022.2139721","DOIUrl":"https://doi.org/10.1080/03605302.2022.2139721","url":null,"abstract":"Abstract For the axi-symmetric incompressible Euler equations, we prove linear in time filamentation near Hill’s vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining the recent nonlinear orbital stability obtained by the first author with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical simulations by Pozrikidis in 1986.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"54 - 85"},"PeriodicalIF":1.9,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47241067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-30DOI: 10.1080/03605302.2021.1998910
P. D’Ancona, R. Schnaubelt
Abstract We show global-in-time Strichartz estimates for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as and a non-trapping condition. The proof is based on smoothing estimates in weighted L 2 spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem.
{"title":"Global Strichartz estimates for an inhomogeneous Maxwell system","authors":"P. D’Ancona, R. Schnaubelt","doi":"10.1080/03605302.2021.1998910","DOIUrl":"https://doi.org/10.1080/03605302.2021.1998910","url":null,"abstract":"Abstract We show global-in-time Strichartz estimates for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as and a non-trapping condition. The proof is based on smoothing estimates in weighted L 2 spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"630 - 675"},"PeriodicalIF":1.9,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47184900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-28DOI: 10.1080/03605302.2022.2051187
D. De Silva, D. Jerison, H. Shahgholian
Abstract Given a global 1-homogeneous minimizer U 0 to the Alt-Caffarelli energy functional, with we provide a foliation of the half-space with dilations of graphs of global minimizers with analytic free boundaries at distance 1 from the origin.
{"title":"Inhomogeneous global minimizers to the one-phase free boundary problem","authors":"D. De Silva, D. Jerison, H. Shahgholian","doi":"10.1080/03605302.2022.2051187","DOIUrl":"https://doi.org/10.1080/03605302.2022.2051187","url":null,"abstract":"Abstract Given a global 1-homogeneous minimizer U 0 to the Alt-Caffarelli energy functional, with we provide a foliation of the half-space with dilations of graphs of global minimizers with analytic free boundaries at distance 1 from the origin.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1193 - 1216"},"PeriodicalIF":1.9,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41960971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-15eCollection Date: 2021-01-01DOI: 10.1080/03605302.2021.1936021
Anna Kiesenhofer, Joachim Krieger
We prove that the half-wave maps problem on with target S2 is globally well-posed for smooth initial data which are small in the critical l1 based Besov space. This is a formal analogue of the result proved by Tataru for wave maps.
{"title":"Small data global regularity for half-wave maps in <i>n</i> = 4 dimensions.","authors":"Anna Kiesenhofer, Joachim Krieger","doi":"10.1080/03605302.2021.1936021","DOIUrl":"https://doi.org/10.1080/03605302.2021.1936021","url":null,"abstract":"<p><p>We prove that the half-wave maps problem on <math> <mrow> <msup><mrow><mi>R</mi></mrow> <mrow><mn>4</mn> <mo>+</mo> <mn>1</mn></mrow> </msup> </mrow> </math> with target <i>S</i> <sup>2</sup> is globally well-posed for smooth initial data which are small in the critical <i>l</i> <sup>1</sup> based Besov space. This is a formal analogue of the result proved by Tataru for wave maps.</p>","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"46 12","pages":"2305-2324"},"PeriodicalIF":1.9,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/03605302.2021.1936021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39578400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-14DOI: 10.1080/03605302.2023.2175216
Maria Colombo, Sunghan Kim, H. Shahgholian
Abstract In this paper we consider the so-called double-phase problem where the phase transition takes place across the interface of the positive and negative phase of minimizers of the functional We prove that minimizers exist, are Hölder regular and verify in a weak sense. We also prove that their free boundary is a.e. with respect to the measure whose support is of σ-finite -dimensional Hausdorff measure.
{"title":"A transmission problem with (p, q)-Laplacian","authors":"Maria Colombo, Sunghan Kim, H. Shahgholian","doi":"10.1080/03605302.2023.2175216","DOIUrl":"https://doi.org/10.1080/03605302.2023.2175216","url":null,"abstract":"Abstract In this paper we consider the so-called double-phase problem where the phase transition takes place across the interface of the positive and negative phase of minimizers of the functional We prove that minimizers exist, are Hölder regular and verify in a weak sense. We also prove that their free boundary is a.e. with respect to the measure whose support is of σ-finite -dimensional Hausdorff measure.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"315 - 349"},"PeriodicalIF":1.9,"publicationDate":"2021-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48503754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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