Calculus of Variations and Partial Differential Equations最新文献_第7页

Uniqueness when the $$L_p$$ curvature is close to be a constant for $$pin [0,1)$$ 当 $$L_p$$ 曲率在 $$pin [0,1)$$ 时接近常数时的唯一性

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-27 DOI: 10.1007/s00526-024-02763-z

Károly J. Böröczky, Christos Saroglou

{"title":"Uniqueness when the $$L_p$$ curvature is close to be a constant for $$pin [0,1)$$","authors":"Károly J. Böröczky, Christos Saroglou","doi":"10.1007/s00526-024-02763-z","DOIUrl":"https://doi.org/10.1007/s00526-024-02763-z","url":null,"abstract":"For fixed positive integer n, (pin [0,1)), (ain (0,1)), we prove that if a function (g:{mathbb {S}}^{n-1}rightarrow {mathbb {R}}) is sufficiently close to 1, in the (C^a) sense, then there exists a unique convex body K whose (L_p) curvature function equals g. This was previously established for (n=3), (p=0) by Chen et al. (Adv Math 411(A):108782, 2022) and in the symmetric case by Chen et al. (Adv Math 368:107166, 2020). Related, we show that if (p=0) and (n=4) or (nle 3) and (pin [0,1)), and the (L_p) curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies (lambda ^{-1}le gle lambda ), for some (lambda >1), then (max _{xin {mathbb {S}}^{n-1}}h_K(x)le C(p,lambda )), for some constant (C(p,lambda )>0) that depends only on p and (lambda ). This also extends a result from Chen et al. [10]. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the (L_p) surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the (L_p)-Minkowksi problem, for (-n<p<0).","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"174 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515269","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}

引用次数: 0

Existence results for the higher-order Q-curvature equation 高阶 Q 曲率方程的存在结果

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-27 DOI: 10.1007/s00526-024-02757-x

Saikat Mazumdar, Jérôme Vétois

引用次数: 0

Uniqueness of weak solutions of the Plateau flow 高原流弱解的唯一性

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-27 DOI: 10.1007/s00526-024-02760-2

Christopher Wright

引用次数: 0

The compressible Euler system with nonlocal pressure: global existence and relaxation 具有非局部压力的可压缩欧拉系统：全局存在与松弛

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-25 DOI: 10.1007/s00526-024-02774-w

Raphael Danchin, Piotr Bogusław Mucha

{"title":"The compressible Euler system with nonlocal pressure: global existence and relaxation","authors":"Raphael Danchin, Piotr Bogusław Mucha","doi":"10.1007/s00526-024-02774-w","DOIUrl":"https://doi.org/10.1007/s00526-024-02774-w","url":null,"abstract":"We here investigate a modification of the compressible barotropic Euler system with friction, involving a fuzzy nonlocal pressure term in place of the conventional one. This nonlocal term is parameterized by (varepsilon > 0) and formally tends to the classical pressure when (varepsilon ) approaches zero. The central challenge is to establish that this system is a reliable approximation of the classical compressible Euler system. We establish the global existence and uniqueness of regular solutions in the neighborhood of the static state with density 1 and null velocity. Our results are demonstrated independently of the parameter (varepsilon ,) which enable us to prove the convergence of solutions to those of the classical Euler system. Another consequence is the rigorous justification of the convergence of the mass equation to various versions of the porous media equation in the asymptotic limit where the friction tends to infinity. Note that our results are demonstrated in the whole space, which necessitates to use the (L^1(mathbb {R}_+; dot{B}^sigma _{2,1}(mathbb {R}^d))) spaces framework.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551996","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}

引用次数: 0

Regularity for quasi-minima of the Alt–Caffarelli functional 阿尔特-卡法雷利函数准极小值的规律性

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-25 DOI: 10.1007/s00526-024-02773-x

Daniel M. Pellegrino, Eduardo V. Teixeira

引用次数: 0

Periodic solution for Hamiltonian type systems with critical growth 具有临界增长的哈密尔顿型系统的周期解

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-24 DOI: 10.1007/s00526-024-02770-0

Yuxia Guo, Shengyu Wu, Shusen Yan

引用次数: 0

An obstacle problem for the p-elastic energy p 弹性能量的障碍问题

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-24 DOI: 10.1007/s00526-024-02752-2

Anna Dall’Acqua, Marius Müller, Shinya Okabe, Kensuke Yoshizawa

引用次数: 0

Multiplicity results for constant Q-curvature conformal metrics 恒Q曲率共形度量的多重性结果

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-24 DOI: 10.1007/s00526-024-02762-0

Salomón Alarcón, Jimmy Petean, Carolina Rey

{"title":"Multiplicity results for constant Q-curvature conformal metrics","authors":"Salomón Alarcón, Jimmy Petean, Carolina Rey","doi":"10.1007/s00526-024-02762-0","DOIUrl":"https://doi.org/10.1007/s00526-024-02762-0","url":null,"abstract":"In this paper we provide a positive lower bound for the number of metrics of constant Q-curvature which are conformal to a Riemannian product of the form ((Mtimes X, g+delta h)), where (delta >0) is a small positive constant, (M, g) is a closed (compact without boundary) n-dimensional Riemannian manifold and (X, h) a closed m-dimensional (positive) Einstein manifold. We assume that (mge 3) and (nge 2) or, if (m=2), that (nge 7). More specifically, we study the constant Q-curvature equation on the Riemannian product ((Mtimes X, g+delta h)), which becomes, by restricting the equation to functions which depend only on the M-variable, a subcritical equation on (M, g) driven by a fourth order operator, known as the Paneitz operator. Then we prove that, for (delta >0) small enough, the equation has at least (textrm{Cat}(M)) positive solutions, where (textrm{Cat}(M)) is the Lusternik-Schnirelmann category of M. This implies that there are at least (textrm{Cat}(M)) metrics of constant Q-curvature in the conformal class of the Riemannian product ((Mtimes X, g+delta h)).","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515270","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}

引用次数: 0

Existence of singular isoperimetric regions in 8-dimensional manifolds 8 维流形中奇异等周区域的存在性

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-20 DOI: 10.1007/s00526-024-02748-y

Gongping Niu

引用次数: 0

On the critical exponent $$p_c$$ of the 3D quasilinear wave equation $$-big (1+(partial _tphi )^pbig )partial _t^2phi +Delta phi =0$$ with short pulse initial data: II—shock formation 关于具有短脉冲初始数据的三维准线性波方程 $$-big (1+(partial _tphi )^pbig )partial _t^2phi +Delta phi =0$$ 的临界指数 $$p_c$$：II-shock formation

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-18 DOI: 10.1007/s00526-024-02753-1

Lu Yu, Yin Huicheng

{"title":"On the critical exponent $$p_c$$ of the 3D quasilinear wave equation $$-big (1+(partial _tphi )^pbig )partial _t^2phi +Delta phi =0$$ with short pulse initial data: II—shock formation","authors":"Lu Yu, Yin Huicheng","doi":"10.1007/s00526-024-02753-1","DOIUrl":"https://doi.org/10.1007/s00526-024-02753-1","url":null,"abstract":"In the previous paper (Ding et al. in J Differ Equ 385:183–253, 2024), for the 3D quasilinear wave equation (-big (1+(partial _tphi )^pbig )partial _t^2phi +Delta phi =0) with short pulse initial data ((phi ,partial _tphi )(1,x)=big (delta ^{2-varepsilon _{0}}phi _0 (frac{r-1}{delta },omega ),delta ^{1-varepsilon _{0}}phi _1(frac{r-1}{delta },omega )big )), where (pin mathbb {N}), (0<varepsilon _{0}<1), under the outgoing constraint condition ((partial _t+partial _r)^kphi (1,x)=O(delta ^{2-varepsilon _{0}-kmax {0,1-(1-varepsilon _{0})p}})) for (k=1,2), the authors establish the global existence of smooth large solution (phi ) when (p>p_c) with (p_c=frac{1}{1-varepsilon _{0}}). In the present paper, under the same outgoing constraint condition, when (1le ple p_c), we will show that the smooth solution (phi ) may blow up and further the outgoing shock is formed in finite time.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551999","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}

引用次数: 0