Pub Date : 2024-11-15DOI: 10.1016/j.jde.2024.10.046
Maciej Tadej
This paper explores a non-linear, non-local model describing the evolution of a single species. We investigate scenarios where the spatial domain is either an arbitrary bounded and open subset of the n-dimensional Euclidean space or a periodic environment modelled by n-dimensional torus. The analysis includes the study of spectrum of the linear, bounded operator in the considered equation, which is a scaled, non-local analogue of classical Laplacian with Neumann boundaries. In particular we show the explicit formulas for eigenvalues and eigenfunctions. Moreover we show the asymptotic behaviour of eigenvalues. Within the context of the non-linear evolution problem, we establish the existence of an invariant region, give a criterion for convergence to the mean mass, and construct spatially heterogeneous steady states.
本文探讨了描述单一物种进化的非线性、非局部模型。我们研究了空间域是 n 维欧几里得空间的任意有界开放子集或以 n 维环状体为模型的周期性环境的情形。分析包括对所考虑方程中的线性有界算子谱的研究，该算子是具有诺伊曼边界的经典拉普拉斯算子的缩放非局部类似物。我们特别展示了特征值和特征函数的明确公式。此外，我们还展示了特征值的渐近行为。在非线性演化问题的背景下，我们确定了不变区域的存在，给出了向平均质量收敛的标准，并构建了空间异质稳态。
{"title":"Long time behaviour of solutions to non-local and non-linear dispersal problems","authors":"Maciej Tadej","doi":"10.1016/j.jde.2024.10.046","DOIUrl":"10.1016/j.jde.2024.10.046","url":null,"abstract":"<div><div>This paper explores a non-linear, non-local model describing the evolution of a single species. We investigate scenarios where the spatial domain is either an arbitrary bounded and open subset of the <em>n</em>-dimensional Euclidean space or a periodic environment modelled by <em>n</em>-dimensional torus. The analysis includes the study of spectrum of the linear, bounded operator in the considered equation, which is a scaled, non-local analogue of classical Laplacian with Neumann boundaries. In particular we show the explicit formulas for eigenvalues and eigenfunctions. Moreover we show the asymptotic behaviour of eigenvalues. Within the context of the non-linear evolution problem, we establish the existence of an invariant region, give a criterion for convergence to the mean mass, and construct spatially heterogeneous steady states.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2043-2064"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652798","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 : 2024-11-15DOI: 10.1016/j.jde.2024.11.002
Cesar S. Eschenazi , Wanderson J. Lambert , Marlon M. López-Flores , Dan Marchesin , Carlos F.B. Palmeira , Bradley J. Plohr
In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and rarefaction waves. In the present paper, we construct, in detail, the three-dimensional wave manifold for a system of two conservation laws with quadratic flux functions. Using adapted coordinates, we derive explicit formulae for important surfaces and curves within the wave manifold and display them graphically. The surfaces subdivide the manifold into regions according to shock type, such as ones corresponding to the Lax admissibility criterion. The curves parametrize rarefaction, shock, and composite waves appearing in contiguous wave patterns. Whereas wave curves overlap in state space, they are disentangled within the wave manifold. We solve a Riemann problem by constructing a wave curve associated with the slow characteristic speed family, generating a surface from it using shock curves, and intersecting this surface with a fast family wave curve. This construction is applied to solve Riemann problems for several illustrative cases.
{"title":"Solving Riemann problems with a topological tool","authors":"Cesar S. Eschenazi , Wanderson J. Lambert , Marlon M. López-Flores , Dan Marchesin , Carlos F.B. Palmeira , Bradley J. Plohr","doi":"10.1016/j.jde.2024.11.002","DOIUrl":"10.1016/j.jde.2024.11.002","url":null,"abstract":"<div><div>In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and rarefaction waves. In the present paper, we construct, in detail, the three-dimensional wave manifold for a system of two conservation laws with quadratic flux functions. Using adapted coordinates, we derive explicit formulae for important surfaces and curves within the wave manifold and display them graphically. The surfaces subdivide the manifold into regions according to shock type, such as ones corresponding to the Lax admissibility criterion. The curves parametrize rarefaction, shock, and composite waves appearing in contiguous wave patterns. Whereas wave curves overlap in state space, they are disentangled within the wave manifold. We solve a Riemann problem by constructing a wave curve associated with the slow characteristic speed family, generating a surface from it using shock curves, and intersecting this surface with a fast family wave curve. This construction is applied to solve Riemann problems for several illustrative cases.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2134-2174"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652716","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 : 2024-11-15DOI: 10.1016/j.jde.2024.11.007
Liqin Zhao, Zheng Si, Ranran Jia
In this paper, we focus on providing the exact bounds for the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by the curve under perturbation of arbitrary polynomials of with degree n can have, where . By the first two order Melnikov functions, we achieve that , for and for any n. The results are novel and improve the previous results in the literature.
在本文中，我们重点给出了平面片断线性微分系统的最大极限循环数 Z(3,n)的精确边界，在 n∈N 时，该系统在 x,y 的度数为 n 的任意多项式的扰动下，有两个区域被曲线 y=x3 分隔。通过一阶二阶梅利尼科夫函数，我们得到了 3≤n≤88 时 Z(3,2)=12, Z(3,n)=2n+1 和任意 n 时 Z(3,n)≥2n+1 的结果。
{"title":"Up to the first two order Melnikov analysis for the exact cyclicity of planar piecewise linear vector fields with nonlinear switching curve","authors":"Liqin Zhao, Zheng Si, Ranran Jia","doi":"10.1016/j.jde.2024.11.007","DOIUrl":"10.1016/j.jde.2024.11.007","url":null,"abstract":"<div><div>In this paper, we focus on providing the exact bounds for the maximum number of limit cycles <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span> that planar piecewise linear differential systems with two zones separated by the curve <span><math><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> under perturbation of arbitrary polynomials of <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> with degree <em>n</em> can have, where <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. By the first two order Melnikov functions, we achieve that <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>12</mn></math></span>, <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></span> for <span><math><mn>3</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>88</mn></math></span> and <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></span> for any <em>n</em>. The results are novel and improve the previous results in the literature.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2255-2292"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652717","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 : 2024-11-15DOI: 10.1016/j.jde.2024.11.006
Jun-ichi Segata
The purpose of this paper is to study large time behavior of solution to the cubic nonlinear Schrödinger equation on the tadpole graph which is a ring attached to a semi-infinite line subject to the Kirchhoff conditions at the junction. Note that the cubic nonlinearity belongs borderline between short and long range scatterings on the whole line. We show that if the initial data has some symmetry on the graph which excludes the standing wave solutions, then the asymptotic behavior of solution to this equation is given by the solution to linear equation with logarithmic phase correction by the nonlinear effect.
{"title":"Asymptotic behavior in time of solution for the cubic nonlinear Schrödinger equation on the tadpole graph","authors":"Jun-ichi Segata","doi":"10.1016/j.jde.2024.11.006","DOIUrl":"10.1016/j.jde.2024.11.006","url":null,"abstract":"<div><div>The purpose of this paper is to study large time behavior of solution to the cubic nonlinear Schrödinger equation on the tadpole graph which is a ring attached to a semi-infinite line subject to the Kirchhoff conditions at the junction. Note that the cubic nonlinearity belongs borderline between short and long range scatterings on the whole line. We show that if the initial data has some symmetry on the graph which excludes the standing wave solutions, then the asymptotic behavior of solution to this equation is given by the solution to linear equation with logarithmic phase correction by the nonlinear effect.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1977-1999"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652796","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 : 2024-11-15DOI: 10.1016/j.jde.2024.11.008
Jifeng Chu , Gang Meng , Feng Wang , Meirong Zhang
The aim of this paper is to study the dependence of all nodes on integrable potentials, for one-dimensional p-Laplacian with separated boundary conditions, including the complete continuity of nodes in potentials with the weak topology, and the continuous Fréchet differentiability of nodes in potentials. We present the precise formula for the Fréchet derivatives of nodes in potentials. These results are natural but nontrivial generalizations of those for Sturm-Liouville operators, with quite different proofs due to the nonlinearity of the p-Laplacian.
{"title":"Complete continuity and Fréchet derivatives of nodes in potentials for one-dimensional p-Laplacian","authors":"Jifeng Chu , Gang Meng , Feng Wang , Meirong Zhang","doi":"10.1016/j.jde.2024.11.008","DOIUrl":"10.1016/j.jde.2024.11.008","url":null,"abstract":"<div><div>The aim of this paper is to study the dependence of all nodes on integrable potentials, for one-dimensional <em>p</em>-Laplacian with separated boundary conditions, including the complete continuity of nodes in potentials with the weak topology, and the continuous Fréchet differentiability of nodes in potentials. We present the precise formula for the Fréchet derivatives of nodes in potentials. These results are natural but nontrivial generalizations of those for Sturm-Liouville operators, with quite different proofs due to the nonlinearity of the <em>p</em>-Laplacian.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1960-1976"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652795","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 : 2024-11-15DOI: 10.1016/j.jde.2024.10.041
Jie Liu, Shanshan Chen
In this paper, we consider a two-species competition patch model in advective heterogeneous environments, where the two species are ecologically identical except for their dispersal rates. It is shown that there exist two critical values such that the species with slower dispersal rate wins the competition if the drift rate is smaller than one critical value, whereas the species with faster dispersal rate is selected if the drift rate is larger than the other critical value. Moreover, treating one species as a resident species and the other one as a mutant species, and viewing dispersal rates as strategies, we show that the dispersal rate of the resident species can be an evolutionarily stable strategy for some intermediate drift rate between the above two critical values.
{"title":"Global dynamics and evolutionarily stable strategies in a two-species competition patch model","authors":"Jie Liu, Shanshan Chen","doi":"10.1016/j.jde.2024.10.041","DOIUrl":"10.1016/j.jde.2024.10.041","url":null,"abstract":"<div><div>In this paper, we consider a two-species competition patch model in advective heterogeneous environments, where the two species are ecologically identical except for their dispersal rates. It is shown that there exist two critical values such that the species with slower dispersal rate wins the competition if the drift rate is smaller than one critical value, whereas the species with faster dispersal rate is selected if the drift rate is larger than the other critical value. Moreover, treating one species as a resident species and the other one as a mutant species, and viewing dispersal rates as strategies, we show that the dispersal rate of the resident species can be an evolutionarily stable strategy for some intermediate drift rate between the above two critical values.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2175-2220"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652916","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 : 2024-11-15DOI: 10.1016/j.jde.2024.10.039
Chuchu Chen , Jialin Hong , Lihai Ji , Ge Liang
This paper studies the existence and uniqueness of the invariant measure for a class of stochastic Maxwell equations and proposes a novel kind of ergodic numerical approximations to inherit the intrinsic properties. The key to proving the ergodicity lies in the uniform regularity estimates of the exact and numerical solutions with respect to time, which are established by analyzing some important physical quantities. By introducing an auxiliary process, we show that the mean-square convergence order of the discontinuous Galerkin full discretization is in the temporal direction and in the spatial direction, which provides the convergence order of the numerical invariant measure to the exact one in -Wasserstein distance.
{"title":"Invariant measures of stochastic Maxwell equations and ergodic numerical approximations","authors":"Chuchu Chen , Jialin Hong , Lihai Ji , Ge Liang","doi":"10.1016/j.jde.2024.10.039","DOIUrl":"10.1016/j.jde.2024.10.039","url":null,"abstract":"<div><div>This paper studies the existence and uniqueness of the invariant measure for a class of stochastic Maxwell equations and proposes a novel kind of ergodic numerical approximations to inherit the intrinsic properties. The key to proving the ergodicity lies in the uniform regularity estimates of the exact and numerical solutions with respect to time, which are established by analyzing some important physical quantities. By introducing an auxiliary process, we show that the mean-square convergence order of the discontinuous Galerkin full discretization is <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> in the temporal direction and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> in the spatial direction, which provides the convergence order of the numerical invariant measure to the exact one in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-Wasserstein distance.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1899-1959"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652794","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 : 2024-11-15DOI: 10.1016/j.jde.2024.10.037
Yanni Zeng , Kun Zhao
This paper considers the global dynamics of classical solutions to an initial-boundary value problem of the system of viscous balance laws arising from chemotaxis in one space dimension: The system of equations is supplemented with time-dependent influx boundary condition for u and homogeneous Dirichlet boundary condition for v. Under suitable assumptions on the dynamic boundary data, it is shown that classical solutions with generic initial data exist globally in time. Moreover, the solutions are shown to converge to the constant equilibrium , as . There is no smallness assumption on the initial data. This is the first rigorous mathematical study of the model subject to dynamic Neumann boundary condition, and generalizes previous works in content and technicality.
{"title":"Global stability of a system of viscous balance laws arising from chemotaxis with dynamic boundary flux","authors":"Yanni Zeng , Kun Zhao","doi":"10.1016/j.jde.2024.10.037","DOIUrl":"10.1016/j.jde.2024.10.037","url":null,"abstract":"<div><div>This paper considers the global dynamics of classical solutions to an initial-boundary value problem of the system of viscous balance laws arising from chemotaxis in one space dimension:<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>u</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mo>(</mo><mi>u</mi><mo>+</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>.</mo></math></span></span></span> The system of equations is supplemented with <em>time-dependent influx</em> boundary condition for <em>u</em> and homogeneous Dirichlet boundary condition for <em>v</em>. Under suitable assumptions on the dynamic boundary data, it is shown that classical solutions with generic initial data exist globally in time. Moreover, the solutions are shown to converge to the constant equilibrium <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>, as <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span>. There is no smallness assumption on the initial data. This is the first rigorous mathematical study of the model subject to dynamic Neumann boundary condition, and generalizes previous works in content and technicality.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2221-2254"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652915","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 : 2024-11-15DOI: 10.1016/j.jde.2024.11.001
Wan-Tong Li , Ming-Zhen Xin , Xiao-Qiang Zhao
This paper is concerned with the spatio-temporal dynamics of nonlocal dispersal systems with monostable and time-space periodic nonlinearity. Firstly, when the dispersal kernels are all light-tailed, we obtain the existence and variational characterization of the linear spreading speed; while the accelerated propagation happens if one species has a long-tailed dispersal kernel, and the accelerated spreading rate can be determined by the principle eigenvalue of the linearized system and the tail of the maximum of kernels. Secondly, we establish the existence and non-existence of traveling waves and semi-transition-waves in cooperative case and non-cooperative, respectively. Lastly, we apply these analytic results to a man-environment-man model and conduct some numerical simulations.
{"title":"Spatio-temporal dynamics of nonlocal dispersal systems in time-space periodic habitats","authors":"Wan-Tong Li , Ming-Zhen Xin , Xiao-Qiang Zhao","doi":"10.1016/j.jde.2024.11.001","DOIUrl":"10.1016/j.jde.2024.11.001","url":null,"abstract":"<div><div>This paper is concerned with the spatio-temporal dynamics of nonlocal dispersal systems with monostable and time-space periodic nonlinearity. Firstly, when the dispersal kernels are all light-tailed, we obtain the existence and variational characterization of the linear spreading speed; while the accelerated propagation happens if one species has a long-tailed dispersal kernel, and the accelerated spreading rate can be determined by the principle eigenvalue of the linearized system and the tail of the maximum of kernels. Secondly, we establish the existence and non-existence of traveling waves and semi-transition-waves in cooperative case and non-cooperative, respectively. Lastly, we apply these analytic results to a man-environment-man model and conduct some numerical simulations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2000-2042"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652797","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 : 2024-11-15DOI: 10.1016/j.jde.2024.11.005
Huaiyu Jian, Ruixuan Zhu
We study the first derivative estimates for solutions to Monge-Ampère equations in terms of modulus of continuity. As a result, we establish the optimal global log-Lipschitz continuity for the gradient of solutions to the Monge-Ampère equation with natural boundary condition.
{"title":"Continuity estimates for the gradient of solutions to the Monge-Ampère equation with natural boundary condition","authors":"Huaiyu Jian, Ruixuan Zhu","doi":"10.1016/j.jde.2024.11.005","DOIUrl":"10.1016/j.jde.2024.11.005","url":null,"abstract":"<div><div>We study the first derivative estimates for solutions to Monge-Ampère equations in terms of modulus of continuity. As a result, we establish the optimal global log-Lipschitz continuity for the gradient of solutions to the Monge-Ampère equation with natural boundary condition.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2065-2084"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652799","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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