在本文中，我们考虑具有临界指数的耦合薛定谔系统： - Δ u i + λ V i ( x ) + a i u i = ∑ j = 1 d β i j u j 3 u i u i in R 3 , u i ∈ H 1 ( R N ) , i = 1,2 , ..., d , $$begin{cases}-{Delta}{u}_{i}+left(lambda {V}_{i}left(xright)+{a}_{i}right){u}_{i}=sum _{j=1}^{d}{beta }_{ij}{leftvert {u}_{j}rightvert }^{3}leftvert {u}_{i}rightvert {u}_{i}quad 、text{in},{mathbb{R}}^{3},quad hfill {u}_{i}in {H}^{1}left({mathbb{R}}^{N}right),quad i=1,2,dots ,d,quad hfill end{cases}$$ 其中 d ≥ 2, β ii >；0 for every i, β ij = β ji when i ≠ j, λ >；0 是一个参数，且 0 ≤ V i ∈ L loc ∞ R N $0le {V}_{i}in {L}_{text{loc、}}^{infty }left({mathbb{R}}^{N}right)$ 有一个共同的底部 int V i - 1 ( 0 ) ${V}_{i}^{-1}left(0right)$ 由 ℓ 0 ≥ 1 ${ell }_{0}left({ell }_{0}ge 1right)$ 连接的组件 Ω k k = 1 ℓ 0 ${left{{Omega}}_{k}right}}_{k=1}^{ell }_{0}}$ 组成、其中 int V i - 1 ( 0 ) ${V}_{i}^{-1}left(0right)$ 是零集 V i - 1 ( 0 ) = x∈ R N ∣ V i ( x ) = 0 ${V}_{i}^{-1}left(0right)=left{xin {mathbb{R}}^{N}mid {V}_{i}left(xright)=0right}$ of V i。我们研究了这个系统的最小能量正解的存在，这些正解被困在 λ > 0 大的弱合作情况 β i j > 的零集附近 int V i - 1 ( 0 ) ${V}_{i}^{-1}left(0right)$ ；0 s m a l l $left({beta }_{ij}{ >}0 mathrm{s}mathrm{m}mathrm{a}mathrm{l}mathrm{l}right)$ 而对于纯竞争情况 β i j ≤ 0 $left({beta }_{ij}le 0right)$ 。此外，当 d = 2 时，我们构建了一个单凸块完全非难解，该解在一个规定分量 Ω k k = 1 ℓ ${left{{{Omega}}_{k}right}}_{k=1}^{ell }$ 时为大λ。
The Willmore flow is the negative gradient flow of the Willmore energy. In this paper, we consider a special kind of solutions to Willmore flow, which we call solitons, and investigate their geometric properties.
{"title":"Solitons to the Willmore flow","authors":"Pak Tung Ho, Juncheol Pyo","doi":"10.1515/ans-2023-0150","DOIUrl":"https://doi.org/10.1515/ans-2023-0150","url":null,"abstract":"The Willmore flow is the negative gradient flow of the Willmore energy. In this paper, we consider a special kind of solutions to Willmore flow, which we call solitons, and investigate their geometric properties.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"11 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142184674","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}
In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ(ρ) = ρθ, λ(ρ) = (θ − 1)ρθ in RN ${mathbb{R}}^{N}$ . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” J. Differ. Equ., vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ > 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for θ=23 $theta =frac{2}{3}$ , N = 3 and γ > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, γ=53 $gamma =frac{5}{3}$ , respectively.
本文考虑了 R N ${mathbb{R}}^{N}$ 中粘度系数为 μ(ρ) = ρ θ , λ(ρ) = (θ - 1)ρ θ 的径向对称可压缩纳维-斯托克斯方程的自由边界问题。在连续密度边界条件下，我们纠正了 (Z. H. Guo and Z. P. Xin, "Analytical solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients and free boundaries," J. Differ. Equ.Equ., vol. 253, no. 1, pp.此外，我们还构建了 θ = 2 3 $theta =frac{2}{3}$ , N = 3 和 γ > 1 的解析解，并利用一些新技术证明了自由边界随时间线性增长。当 θ = 1 时，在无应力边界条件下，我们分别为 N = 2, γ = 2 和 N = 3, γ = 5 3 $gamma =frac{5}{3}$ 构造了一些解析解。
{"title":"Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries","authors":"Jianwei Dong, Manwai Yuen","doi":"10.1515/ans-2023-0146","DOIUrl":"https://doi.org/10.1515/ans-2023-0146","url":null,"abstract":"In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form <jats:italic>μ</jats:italic>(<jats:italic>ρ</jats:italic>) = <jats:italic>ρ</jats:italic> <jats:sup> <jats:italic>θ</jats:italic> </jats:sup>, <jats:italic>λ</jats:italic>(<jats:italic>ρ</jats:italic>) = (<jats:italic>θ</jats:italic> − 1)<jats:italic>ρ</jats:italic> <jats:sup> <jats:italic>θ</jats:italic> </jats:sup> in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{N}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0146_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula>. Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” <jats:italic>J. Differ. Equ.</jats:italic>, vol. 253, no. 1, pp. 1–19, 2012) for <jats:italic>N</jats:italic> = 3, <jats:italic>θ</jats:italic> = <jats:italic>γ</jats:italic> > 1 and improve the spreading rate of the free boundary, where <jats:italic>γ</jats:italic> is the adiabatic exponent. Moreover, we construct an analytical solution for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>θ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> <jats:tex-math> $theta =frac{2}{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0146_ineq_002.png\"/> </jats:alternatives> </jats:inline-formula>, <jats:italic>N</jats:italic> = 3 and <jats:italic>γ</jats:italic> > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When <jats:italic>θ</jats:italic> = 1, under the stress free boundary condition, we construct some analytical solutions for <jats:italic>N</jats:italic> = 2, <jats:italic>γ</jats:italic> = 2 and <jats:italic>N</jats:italic> = 3, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>γ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>5</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> <jats:tex-math> $gamma =frac{5}{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0146_ineq_003.png\"/> </jats:alternatives> </jats:inline-formula>, respectively.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"28 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507103","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}
In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer’s disease). We prove the existence, uniqueness, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.
在这项研究中，我们证明了一组具有传输边界条件的扩散-凝结 Smoluchowski 型方程的双尺度均匀化结果。该系统旨在描述脑组织中病态 tau 蛋白的聚集和扩散，这一过程与多种 tau 病（如阿尔茨海默病）的发生和演变有关。我们证明了在微观尺度（即单个神经元尺度）上得出的模型方程的解的存在性、唯一性、正向性和有界性。然后，我们研究了同质化过程向与微观模型渐近一致的宏观模型解的收敛性。
{"title":"Homogenization of Smoluchowski-type equations with transmission boundary conditions","authors":"Bruno Franchi, Silvia Lorenzani","doi":"10.1515/ans-2023-0143","DOIUrl":"https://doi.org/10.1515/ans-2023-0143","url":null,"abstract":"In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer’s disease). We prove the existence, uniqueness, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"17 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507108","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}
This paper is concerned with the periodic solutions for a coupled system of wave equations with x-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T=2a−1b(a,bare positive integers)$T=frac{2a-1}{b}left(a,b text{are,positive,integers}right)$ and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf ηϱ(x) > 0 is not required.
本文关注的是具有 x 依赖系数的耦合波方程系统的周期解。当两个波同时在非等向介质中传播时，自然会产生这样的模型。在本文中，对于具有 T = 2 a - 1 b （a , b 为正整数）$T=frac{2a-1}{b}left(a,b text{are,positive,integers}right)$ 形式的周期和某些类型的边界条件，我们利用变分法得到了时间周期解的存在性，并分析了当非线性为超线性和单调时，耦合参数归零时的渐近行为。其中，不需要条件 ess inf η ϱ (x) > 0。
{"title":"Periodic solutions for a coupled system of wave equations with x-dependent coefficients","authors":"Jiayu Deng, Shuguan Ji","doi":"10.1515/ans-2023-0144","DOIUrl":"https://doi.org/10.1515/ans-2023-0144","url":null,"abstract":"This paper is concerned with the periodic solutions for a coupled system of wave equations with <jats:italic>x</jats:italic>-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>T</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>a</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>b</m:mi> </m:mrow> </m:mfrac> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mspace width=\"0.3333em\"/> <m:mspace width=\"0.28em\"/> <m:mtext>are positive integers</m:mtext> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math>$T=frac{2a-1}{b}left(a,b text{are,positive,integers}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0144_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf <jats:italic>η</jats:italic> <jats:sub> <jats:italic>ϱ</jats:italic> </jats:sub>(<jats:italic>x</jats:italic>) > 0 is not required.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"31 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529522","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}
For a probability P in Rd${mathbb{R}}^{d}$ its center outward distribution function F ±, introduced in V. Chernozhukov, A. Galichon, M. Hallin, and M. Henry (“Monge–Kantorovich depth, quantiles, ranks and signs,” Ann. Stat., vol. 45, no. 1, pp. 223–256, 2017) and M. Hallin, E. del Barrio, J. Cuesta-Albertos, and C. Matrán (“Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach,” Ann. Stat., vol. 49, no. 2, pp. 1139–1165, 2021), is a new and successful concept of multivariate distribution function based on mass transportation theory. This work proves, for a probability P with density locally bounded away from zero and infinity in its support, the continuity of the center-outward map on the interior of the support of P and the continuity of its inverse, the quantile, Q ±. This relaxes the convexity assumption in E. del Barrio, A. González-Sanz, and M. Hallin (“A note on the regularity of optimal-transport-based center-outward distribution and quantile functions,” J. Multivariate Anal., vol. 180, p. 104671, 2020). Some important consequences of this continuity are Glivenko–Cantelli type theorems and characterisation of weak convergence by the stability of the center-outward map.
对于 R d ${mathbb{R}}^{d}$ 中的概率 P，其中心向外分布函数 F ± 在 V. Chernozhukov、A. Galichon、M. Hallin 和 M. Henry（"Monge-Kantorovich 深度、定量、等级和符号"，《统计年鉴》，第 45 卷，第 1 期，第 223-256 页，2017 年）以及 M. Hallin、E. del Barrio、J. Cuesta-Albertos 和 C. Matr.统计》，第 45 卷，第 1 期，第 223-256 页，2017 年）和 M. Hallin、E. del Barrio、J. Cuesta-Albertos 和 C. Matrán（"维度 d 中的分布和量化函数、等级和符号：一种度量运输方法"，《统计》，第 49 卷，第 1 期，第 223-256 页，2017 年）。Stat., vol. 49, no. 2, pp.这项工作证明了，对于密度局部离零有界且在其支持中为无穷大的概率 P，P 支持内部的中心向外映射的连续性及其倒数 Q ± 的连续性。这放宽了 E. del Barrio、A. González-Sanz 和 M. Hallin（"基于最优传输的中心向外分布和量值函数的正则性说明"，《多变量分析》，第 180 卷，第 104671 页，2020 年）中的凸性假设。这种连续性的一些重要后果是格利文科-康特利类型定理以及通过中心向外映射的稳定性来描述弱收敛性。
{"title":"Regularity of center-outward distribution functions in non-convex domains","authors":"Eustasio del Barrio, Alberto González-Sanz","doi":"10.1515/ans-2023-0140","DOIUrl":"https://doi.org/10.1515/ans-2023-0140","url":null,"abstract":"For a probability <jats:italic>P</jats:italic> in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>${mathbb{R}}^{d}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0140_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> its center outward distribution function F <jats:sub>±</jats:sub>, introduced in V. Chernozhukov, A. Galichon, M. Hallin, and M. Henry (“Monge–Kantorovich depth, quantiles, ranks and signs,” <jats:italic>Ann. Stat.</jats:italic>, vol. 45, no. 1, pp. 223–256, 2017) and M. Hallin, E. del Barrio, J. Cuesta-Albertos, and C. Matrán (“Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach,” <jats:italic>Ann. Stat.</jats:italic>, vol. 49, no. 2, pp. 1139–1165, 2021), is a new and successful concept of multivariate distribution function based on mass transportation theory. This work proves, for a probability <jats:italic>P</jats:italic> with density locally bounded away from zero and infinity in its support, the continuity of the center-outward map on the interior of the support of <jats:italic>P</jats:italic> and the continuity of its inverse, the quantile, Q <jats:sub>±</jats:sub>. This relaxes the convexity assumption in E. del Barrio, A. González-Sanz, and M. Hallin (“A note on the regularity of optimal-transport-based center-outward distribution and quantile functions,” <jats:italic>J. Multivariate Anal.</jats:italic>, vol. 180, p. 104671, 2020). Some important consequences of this continuity are Glivenko–Cantelli type theorems and characterisation of weak convergence by the stability of the center-outward map.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"86 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507179","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}
In this paper, we prove the existence of a bounded linear extension operator T:L2,p(E)→L2,p(R2)$T:{L}^{2,p}left(Eright)to {L}^{2,p}left({mathbb{R}}^{2}right)$ when 1 < p < 2, where E⊂R2$Esubset {mathbb{R}}^{2}$ is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman–Klartag (“Linear extension operators for Sobolev spaces on radially symmetric binary trees,” Adv. Nonlinear Stud., vol. 23, no. 1, p. 20220075, 2023) on the existence of linear extension operators for radially symmetric binary trees.
本文证明了有界线性扩展算子 T : L 2 , p ( E ) → L 2 , p ( R 2 ) $T 的存在性：当 1 < p < 2 时，{L}^{2,p}left(E/right)to {L}^{2,p}left({mathbb{R}}^{2}right)$ ，其中 E ⊂ R 2 $Esubset {mathbb{R}}^{2}$ 是一个具有分形结构的离散集合。我们的证明利用了 Fefferman-Klartag （"径向对称二叉树上 Sobolev 空间的线性扩展算子"，《非线性研究》，第 23 卷第 1 期，第 20220075 页，2023 年）关于径向对称二叉树的线性扩展算子存在性的定理。
{"title":"An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2","authors":"Jacob Carruth, Arie Israel","doi":"10.1515/ans-2023-0126","DOIUrl":"https://doi.org/10.1515/ans-2023-0126","url":null,"abstract":"In this paper, we prove the existence of a bounded linear extension operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>T</m:mi> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math>$T:{L}^{2,p}left(Eright)to {L}^{2,p}left({mathbb{R}}^{2}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0126_ineq_002.png\"/> </jats:alternatives> </jats:inline-formula> when 1 < <jats:italic>p</jats:italic> < 2, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>E</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>$Esubset {mathbb{R}}^{2}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0126_ineq_003.png\"/> </jats:alternatives> </jats:inline-formula> is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman–Klartag (“Linear extension operators for Sobolev spaces on radially symmetric binary trees,” <jats:italic>Adv. Nonlinear Stud.</jats:italic>, vol. 23, no. 1, p. 20220075, 2023) on the existence of linear extension operators for radially symmetric binary trees.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"24 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196839","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}
Marjorie Drake, Charles Fefferman, Kevin Ren, Anna Skorobogatova
In this paper, we establish the existence of a bounded, linear extension operator T:L2,p(E)→L2,p(R2) $T :{L}^{2,p}left(Eright)to {L}^{2,p}left({mathbb{R}}^{2}right)$ when 1 < p < 2 and E is a finite subset of R2 ${mathbb{R}}^{2}$ contained in a line.
在本文中，当 1 < p < 2 且 E 是 R 2 的有限子集 ${mathbb{R}}^{2}$ 包含在一条直线中时，我们建立了有界线性扩展算子 T : L 2 , p ( E ) → L 2 , p ( R 2 ) $T :{L}^{2,p}left(Eright)/to {L}^{2,p}left({mathbb{R}}^{2}right)$ 的存在性。
{"title":"Sobolev extension in a simple case","authors":"Marjorie Drake, Charles Fefferman, Kevin Ren, Anna Skorobogatova","doi":"10.1515/ans-2023-0132","DOIUrl":"https://doi.org/10.1515/ans-2023-0132","url":null,"abstract":"In this paper, we establish the existence of a bounded, linear extension operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>T</m:mi> <m:mspace width=\"0.17em\"/> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math> $T :{L}^{2,p}left(Eright)to {L}^{2,p}left({mathbb{R}}^{2}right)$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0132_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> when 1 < <jats:italic>p</jats:italic> < 2 and <jats:italic>E</jats:italic> is a finite subset of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{2}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0132_ineq_002.png\"/> </jats:alternatives> </jats:inline-formula> contained in a line.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"83 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150512","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}
Barbara Kaltenbacher, Mostafa Meliani, Vanja Nikolić
In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.
{"title":"Limiting behavior of quasilinear wave equations with fractional-type dissipation","authors":"Barbara Kaltenbacher, Mostafa Meliani, Vanja Nikolić","doi":"10.1515/ans-2023-0139","DOIUrl":"https://doi.org/10.1515/ans-2023-0139","url":null,"abstract":"In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"106 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933581","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}
Eleonora Amoroso, Gabriele Bonanno, Giuseppina D’Aguì, Patrick Winkert
This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.
{"title":"Two solutions for Dirichlet double phase problems with variable exponents","authors":"Eleonora Amoroso, Gabriele Bonanno, Giuseppina D’Aguì, Patrick Winkert","doi":"10.1515/ans-2023-0134","DOIUrl":"https://doi.org/10.1515/ans-2023-0134","url":null,"abstract":"This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"41 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933590","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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