Pub Date : 2024-11-05DOI: 10.1016/j.na.2024.113685
Giulio Pascale
We investigate, under a volume constraint and among sets contained in a Euclidean half-space, the minimization problem of an energy functional given by the sum of a capillarity perimeter, a nonlocal interaction term and a gravitational potential energy. The capillarity perimeter assigns a constant weight to the portion of the boundary touching the boundary of the half-space. The nonlocal term is represented by a double integral of a positive kernel , while the gravitational term is represented by the integral of a positive potential .
We first establish existence of volume-constrained minimizers in the small mass regime, together with several qualitative properties of minimizers. The existence result holds for rather general choices of kernels in the nonlocal interaction term, including attractive–repulsive ones. When the nonlocal kernel with , we also obtain nonexistence of volume constrained minimizers in the large mass regime. Finally, we prove a generalized existence result of minimizers holding for all masses and general nonlocal interaction terms, meaning that the infimum of the problem is realized by a finite disjoint union of sets thought located at “infinite distance” one from the other.
These results stem from an application of quantitative isoperimetric inequalities for the capillarity problem in a half-space.
我们研究了在欧几里得半空间所含集合的体积约束条件下，由毛细周长、非局部相互作用项和重力势能之和给出的能量函数的最小化问题。毛细周长为接触半空间边界的边界部分赋予一个恒定权重。非局部项由正内核 g 的双积分表示，而引力项由正势能 G 的积分表示。我们首先确定了小质量体系中体积受限最小值的存在性，以及最小值的几个定性性质。存在性结果适用于非局部相互作用项中的核的一般选择，包括吸引力-反弹力核。当非局部核 g(x)=1/|x|β 且β∈(0,2]时，我们还得到了大质量体系中体积受限最小化子的不存在性。最后，我们证明了对所有质量和一般非局部相互作用项都适用的最小化子的广义存在性结果，这意味着问题的下极值是由认为彼此位于 "无限距离 "的集合的有限不相交联盟实现的。
{"title":"Existence and nonexistence of minimizers for classical capillarity problems in presence of nonlocal repulsion and gravity","authors":"Giulio Pascale","doi":"10.1016/j.na.2024.113685","DOIUrl":"10.1016/j.na.2024.113685","url":null,"abstract":"<div><div>We investigate, under a volume constraint and among sets contained in a Euclidean half-space, the minimization problem of an energy functional given by the sum of a capillarity perimeter, a nonlocal interaction term and a gravitational potential energy. The capillarity perimeter assigns a constant weight to the portion of the boundary touching the boundary of the half-space. The nonlocal term is represented by a double integral of a positive kernel <span><math><mi>g</mi></math></span>, while the gravitational term is represented by the integral of a positive potential <span><math><mi>G</mi></math></span>.</div><div>We first establish existence of volume-constrained minimizers in the small mass regime, together with several qualitative properties of minimizers. The existence result holds for rather general choices of kernels in the nonlocal interaction term, including attractive–repulsive ones. When the nonlocal kernel <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>/</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>β</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></math></span>, we also obtain nonexistence of volume constrained minimizers in the large mass regime. Finally, we prove a generalized existence result of minimizers holding for all masses and general nonlocal interaction terms, meaning that the infimum of the problem is realized by a finite disjoint union of sets thought located at “infinite distance” one from the other.</div><div>These results stem from an application of quantitative isoperimetric inequalities for the capillarity problem in a half-space.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113685"},"PeriodicalIF":1.3,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586277","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}
Necessary and sufficient conditions are offered for Sobolev type spaces built on rearrangement-invariant spaces to be continuously embedded into (generalized) Campanato and Morrey spaces on open subsets of the -dimensional Euclidean space. As a consequence, the optimal target and domain spaces in the relevant embeddings are identified. Our general criteria are implemented to derive sharp embeddings in the class of Orlicz-Sobolev spaces.
为建立在重排不变空间上的索波列夫类型空间连续嵌入到 n 维欧几里得空间开放子集上的（广义）坎帕纳托和莫雷空间提供了必要和充分条件。因此，相关嵌入中的最佳目标空间和域空间得以确定。我们的一般标准可用于推导奥尔利茨-索博廖夫空间类中的尖锐嵌入。
{"title":"Higher-order Sobolev embeddings into spaces of Campanato and Morrey type","authors":"Paola Cavaliere , Andrea Cianchi , Luboš Pick , Lenka Slavíková","doi":"10.1016/j.na.2024.113678","DOIUrl":"10.1016/j.na.2024.113678","url":null,"abstract":"<div><div>Necessary and sufficient conditions are offered for Sobolev type spaces built on rearrangement-invariant spaces to be continuously embedded into (generalized) Campanato and Morrey spaces on open subsets of the <span><math><mi>n</mi></math></span>-dimensional Euclidean space. As a consequence, the optimal target and domain spaces in the relevant embeddings are identified. Our general criteria are implemented to derive sharp embeddings in the class of Orlicz-Sobolev spaces.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113678"},"PeriodicalIF":1.3,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586276","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 : 2024-11-05DOI: 10.1016/j.na.2024.113682
M.S. Shahrokhi-Dehkordi
<div><div>Consider a bounded Lipschitz domain <span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> and the <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></math></span>-energy functional <span><span><span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></mrow></msub><mrow><mo>[</mo><mi>u</mi><mo>;</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow><mo>≔</mo><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mrow><mo>|</mo></mrow><mo>∇</mo><mi>u</mi><mo>∧</mo><mo>∇</mo><msup><mrow><mi>u</mi><mrow><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi><mo>,</mo></mrow></math></span></span></span>with <span><math><mrow><mrow><mi>p</mi><mo>∈</mo><mo>]</mo></mrow><mn>1</mn><mo>,</mo><mrow><mi>∞</mi><mo>]</mo></mrow></mrow></math></span>, defined over the space of admissible Sobolev maps <span><span><span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mo>≔</mo><mrow><mo>{</mo><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mi>p</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mo>:</mo><mi>u</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>∂</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></mfrac></mrow><mo>}</mo></mrow><mo>.</mo></mrow></math></span></span></span>In this paper, we investigate the multiplicity and uniqueness of extremals and strong local minimisers of the <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></math></span>-energy functional <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></mrow></msub><mrow><mo>[</mo><mi>⋅</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow></mrow></math></span> in <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Our focus is on the space of admissible Sobolev maps and a topological class of maps known as spherical twists in connection with the Euler–Lagrange equations associated with the <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></math></span>-energy functional over <span><math><mro
{"title":"Radially symmetric σ2,p-harmonic maps from n-dimensional annuli into sphere","authors":"M.S. Shahrokhi-Dehkordi","doi":"10.1016/j.na.2024.113682","DOIUrl":"10.1016/j.na.2024.113682","url":null,"abstract":"<div><div>Consider a bounded Lipschitz domain <span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> and the <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></math></span>-energy functional <span><span><span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></mrow></msub><mrow><mo>[</mo><mi>u</mi><mo>;</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow><mo>≔</mo><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mrow><mo>|</mo></mrow><mo>∇</mo><mi>u</mi><mo>∧</mo><mo>∇</mo><msup><mrow><mi>u</mi><mrow><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi><mo>,</mo></mrow></math></span></span></span>with <span><math><mrow><mrow><mi>p</mi><mo>∈</mo><mo>]</mo></mrow><mn>1</mn><mo>,</mo><mrow><mi>∞</mi><mo>]</mo></mrow></mrow></math></span>, defined over the space of admissible Sobolev maps <span><span><span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mo>≔</mo><mrow><mo>{</mo><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mi>p</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mo>:</mo><mi>u</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>∂</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></mfrac></mrow><mo>}</mo></mrow><mo>.</mo></mrow></math></span></span></span>In this paper, we investigate the multiplicity and uniqueness of extremals and strong local minimisers of the <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></math></span>-energy functional <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></mrow></msub><mrow><mo>[</mo><mi>⋅</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow></mrow></math></span> in <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Our focus is on the space of admissible Sobolev maps and a topological class of maps known as spherical twists in connection with the Euler–Lagrange equations associated with the <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msub></math></span>-energy functional over <span><math><mro","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113682"},"PeriodicalIF":1.3,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586278","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 : 2024-11-04DOI: 10.1016/j.na.2024.113684
Bang Van Tran , Ngoan Tien Ha , Tho Huu Nguyen , Tien Trong Phan
To solve the Dirichlet problem for nonsymmetric augmented -Hessian type equations, , we first of all solve this problem for corresponding symmetric augmented -Hessian type ones. Then, by using the Banach fixed point theorem, we prove the existence of -admissible solution in of the problem, provided that the skew-symmetric matrices entering the equations are sufficiently small in some sense. Some necessary conditions for existence and sufficient conditions for uniqueness of this kind of solution are given.
{"title":"The Dirichlet problem for nonsymmetric augmented k-Hessian type equations","authors":"Bang Van Tran , Ngoan Tien Ha , Tho Huu Nguyen , Tien Trong Phan","doi":"10.1016/j.na.2024.113684","DOIUrl":"10.1016/j.na.2024.113684","url":null,"abstract":"<div><div>To solve the Dirichlet problem for nonsymmetric augmented <span><math><mi>k</mi></math></span>-Hessian type equations, <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></math></span>, we first of all solve this problem for corresponding symmetric augmented <span><math><mi>k</mi></math></span>-Hessian type ones. Then, by using the Banach fixed point theorem, we prove the existence of <span><math><mi>δ</mi></math></span>-admissible solution in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> of the problem, provided that the skew-symmetric matrices entering the equations are sufficiently small in some sense. Some necessary conditions for existence and sufficient conditions for uniqueness of this kind of solution are given.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113684"},"PeriodicalIF":1.3,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578738","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 : 2024-11-02DOI: 10.1016/j.na.2024.113683
Adolfo Arroyo-Rabasa
We introduce a general difference quotient representation for non-local operators associated with a first-order linear operator. We establish new local to non-local estimates and strong localization principles in various spaces of functions, measures and distributions, which fully generalize those known for gradients. Under suitable assumptions, we also establish the invariance of quasiconvexity within the proposed local-nonlocal setting. Applications to the fine properties of -gradient measures are further discussed.
我们介绍了与一阶线性算子相关的非局部算子的一般差商表示。我们在各种函数、度量和分布空间中建立了新的本地到非本地估计和强本地化原则，这些原则完全概括了梯度的已知原则。在适当的假设条件下，我们还在提议的局部-非局部设置中建立了类凸不变性。我们还进一步讨论了 A 梯度量的精细特性的应用。
{"title":"Functional and variational aspects of nonlocal operators associated with linear PDEs","authors":"Adolfo Arroyo-Rabasa","doi":"10.1016/j.na.2024.113683","DOIUrl":"10.1016/j.na.2024.113683","url":null,"abstract":"<div><div>We introduce a general difference quotient representation for non-local operators associated with a first-order linear operator. We establish new local to non-local estimates and strong localization principles in various spaces of functions, measures and distributions, which fully generalize those known for gradients. Under suitable assumptions, we also establish the invariance of quasiconvexity within the proposed local-nonlocal setting. Applications to the fine properties of <span><math><mi>A</mi></math></span>-gradient measures are further discussed.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113683"},"PeriodicalIF":1.3,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571566","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 : 2024-11-02DOI: 10.1016/j.na.2024.113689
Priscila L. da Silva
In this paper we consider a Novikov equation, recently shown to describe pseudospherical surfaces, to extend some recent results of regularity of its solutions. By making use of the global well-posedness in Sobolev spaces, for analytic initial data in Gevrey spaces we prove some new estimates for the solution in order to use the Kato–Masuda Theorem and obtain a lower bound for the radius of spatial analyticity. After that, we use embeddings between spaces to then conclude that the unique solution is, in fact, globally analytic in both variables. Finally, the global analyticity of the solution is used to prove that it endows the strip with a global analytic metric associated to pseudospherical surfaces obtained in Sales Filho and Freire (2022).
{"title":"Global analytic solutions of a pseudospherical Novikov equation","authors":"Priscila L. da Silva","doi":"10.1016/j.na.2024.113689","DOIUrl":"10.1016/j.na.2024.113689","url":null,"abstract":"<div><div>In this paper we consider a Novikov equation, recently shown to describe pseudospherical surfaces, to extend some recent results of regularity of its solutions. By making use of the global well-posedness in Sobolev spaces, for analytic initial data in Gevrey spaces we prove some new estimates for the solution in order to use the Kato–Masuda Theorem and obtain a lower bound for the radius of spatial analyticity. After that, we use embeddings between spaces to then conclude that the unique solution is, in fact, globally analytic in both variables. Finally, the global analyticity of the solution is used to prove that it endows the strip <span><math><mrow><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>×</mo><mi>R</mi></mrow></math></span> with a global analytic metric associated to pseudospherical surfaces obtained in Sales Filho and Freire (2022).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113689"},"PeriodicalIF":1.3,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571406","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 : 2024-11-02DOI: 10.1016/j.na.2024.113692
Xiaobiao Jia , Shanshan Ma
In this paper, we consider the Liouville theorem for -convex solutions to Sum Hessian equations in half spaces. The key is to show the Pogorelov type estimate up to the flat boundary.
在本文中，我们考虑了半空间中 Sum Hessian 方程 k 个凸解的 Liouville 定理。关键在于证明波哥列洛夫式估计直到平边界。
{"title":"The Liouville theorem for Sum Hessian equations in half spaces","authors":"Xiaobiao Jia , Shanshan Ma","doi":"10.1016/j.na.2024.113692","DOIUrl":"10.1016/j.na.2024.113692","url":null,"abstract":"<div><div>In this paper, we consider the Liouville theorem for <span><math><mi>k</mi></math></span>-convex solutions to Sum Hessian equations in half spaces. The key is to show the Pogorelov type estimate up to the flat boundary.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113692"},"PeriodicalIF":1.3,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571496","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 : 2024-11-02DOI: 10.1016/j.na.2024.113688
Xueke Pu , Wenli Zhou
In this paper, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is established under the assumption of initial data with additional regularity . Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, with the convergence rate , in the cases of initial data with and initial data , respectively, as the aspect ratio goes to zero.
{"title":"The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain","authors":"Xueke Pu , Wenli Zhou","doi":"10.1016/j.na.2024.113688","DOIUrl":"10.1016/j.na.2024.113688","url":null,"abstract":"<div><div>In this paper, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is established under the assumption of initial data <span><math><mrow><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></math></span> with additional regularity <span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>z</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math></span>. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, with the convergence rate <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>min</mo><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mi>β</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>γ</mi><mo>−</mo><mn>2</mn><mo>}</mo></mrow><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><mn>2</mn><mo><</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo><</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, in the cases of initial data <span><math><mrow><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></math></span> with <span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>z</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math></span> and initial data <span><math><mrow><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, respectively, as the aspect ratio <span><math><mi>λ</mi></math></span> goes to zero.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113688"},"PeriodicalIF":1.3,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571407","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 : 2024-10-31DOI: 10.1016/j.na.2024.113691
Fabian Bäuerlein
We consider vector valued weak solutions with of degenerate or singular parabolic systems of type where denotes an open set in for and a finite time. Assuming that the vector field is not of Uhlenbeck-type structure, satisfies -growth assumptions and is Hölder continuous for every , we show that the gradient is partially Hölder continuous, provided the vector field degenerates like that of the -Laplacian for small gradients.
我们考虑矢量值弱解 u:ΩT→RN，N∈N 的∂tu-diva(z,u,Du)=0inΩT=Ω×(0,T)类型的退化或奇异抛物线系统，其中Ω表示 Rn 中的开集，n≥1，T>0 为有限时间。假定向量场 a 不是乌伦贝克型结构，满足 p 生长假设，且 (z,u)↦a(z,u,ξ) 对于每个 ξ∈RNn 都是霍尔德连续的，我们证明梯度 Du 部分是霍尔德连续的，条件是向量场像 p-Laplacian 的梯度一样退化为小梯度。
{"title":"Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients","authors":"Fabian Bäuerlein","doi":"10.1016/j.na.2024.113691","DOIUrl":"10.1016/j.na.2024.113691","url":null,"abstract":"<div><div>We consider vector valued weak solutions <span><math><mrow><mi>u</mi><mo>:</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi></mrow></math></span> of degenerate or singular parabolic systems of type <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>div</mi><mspace></mspace><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>Ω</mi></math></span> denotes an open set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span> a finite time. Assuming that the vector field <span><math><mi>a</mi></math></span> is not of Uhlenbeck-type structure, satisfies <span><math><mi>p</mi></math></span>-growth assumptions and <span><math><mrow><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>↦</mo><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> is Hölder continuous for every <span><math><mrow><mi>ξ</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mi>n</mi></mrow></msup></mrow></math></span>, we show that the gradient <span><math><mrow><mi>D</mi><mi>u</mi></mrow></math></span> is partially Hölder continuous, provided the vector field degenerates like that of the <span><math><mi>p</mi></math></span>-Laplacian for small gradients.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113691"},"PeriodicalIF":1.3,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553966","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 : 2024-10-30DOI: 10.1016/j.na.2024.113681
Allan Freitas , Márcio S. Santos , Joyce S. Sindeaux
In this paper, we work with the existence and uniqueness of free boundary constant mean curvature surfaces in rotational domains. These are domains whose boundary is generated by a rotation of a graph. We classify the free boundary CMC surfaces as topological disks or annulus under some conditions on the function that generates the graph and a gap condition on the umbilicity tensor. Also, we construct some examples of free boundary CMC surfaces in the rotational ellipsoid that, in particular, satisfy our gap condition.
{"title":"Gap results and existence of free boundary CMC surfaces in rotational domains","authors":"Allan Freitas , Márcio S. Santos , Joyce S. Sindeaux","doi":"10.1016/j.na.2024.113681","DOIUrl":"10.1016/j.na.2024.113681","url":null,"abstract":"<div><div>In this paper, we work with the existence and uniqueness of free boundary constant mean curvature surfaces in rotational domains. These are domains whose boundary is generated by a rotation of a graph. We classify the free boundary CMC surfaces as topological disks or annulus under some conditions on the function that generates the graph and a gap condition on the umbilicity tensor. Also, we construct some examples of free boundary CMC surfaces in the rotational ellipsoid that, in particular, satisfy our gap condition.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113681"},"PeriodicalIF":1.3,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554046","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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