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Blow up analysis for a parabolic MEMS problem, I: Hölder estimate 抛物线 MEMS 问题的爆炸分析，I：荷尔德估计
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-08-05 DOI: 10.1007/s00526-024-02804-7
Kelei Wang, Guangzeng Yi
{"title":"Blow up analysis for a parabolic MEMS problem, I: Hölder estimate","authors":"Kelei Wang, Guangzeng Yi","doi":"10.1007/s00526-024-02804-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02804-7","url":null,"abstract":"<p>This is the first in a series of papers devoted to the blow up analysis for the quenching phenomena in a parabolic MEMS equation. In this paper, we first give an optimal Hölder estimate for solutions to this equation by using the blow up method and some Liouville theorems on stationary two-valued caloric functions, and then establish a convergence theory for sequences of uniformly Hölder continuous solutions. These results are also used to prove a stratification theorem on the rupture set <span>({u=0})</span>.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937438","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
Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions 无穷远处完全极小曲面的几何及其反转的威尔莫尔指数
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-08-05 DOI: 10.1007/s00526-024-02792-8
Jonas Hirsch, Rob Kusner, Elena Mäder-Baumdicker
{"title":"Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions","authors":"Jonas Hirsch, Rob Kusner, Elena Mäder-Baumdicker","doi":"10.1007/s00526-024-02792-8","DOIUrl":"https://doi.org/10.1007/s00526-024-02792-8","url":null,"abstract":"<p>We study complete minimal surfaces in <span>(mathbb {R}^n)</span> with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy <span>(mathcal {W}: =frac{1}{4} int |vec H|^2)</span>. In codimension one, we prove that the <span>(mathcal {W})</span>-Morse index for any inverted minimal sphere or real projective plane with <i>m</i> such ends is exactly <span>(m-3=frac{mathcal {W}}{4pi }-3)</span>. We also consider several geometric properties—for example, the property that all <i>m</i> asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the <span>(mathcal {W})</span>-Morse index of their inverted surfaces.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937534","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
Roles of density-related diffusion and signal-dependent motilities in a chemotaxis–consumption system 密度相关扩散和信号依赖运动在趋化消耗系统中的作用
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-08-05 DOI: 10.1007/s00526-024-02802-9
Genglin Li, Yuan Lou
{"title":"Roles of density-related diffusion and signal-dependent motilities in a chemotaxis–consumption system","authors":"Genglin Li, Yuan Lou","doi":"10.1007/s00526-024-02802-9","DOIUrl":"https://doi.org/10.1007/s00526-024-02802-9","url":null,"abstract":"<p>This study examines an initial-boundary value problem involving the system </p><span>$$begin{aligned} left{ begin{array}{l} u_t = Delta big (u^mphi (v)big ), [1mm] v_t = Delta v-uv. [1mm] end{array} right. qquad (star ) end{aligned}$$</span><p>in a smoothly bounded domain <span>(Omega subset mathbb {R}^n)</span> with no-flux boundary conditions, where <span>(m, nge 1)</span>. The motility function <span>(phi in C^0([0,infty )) cap C^3((0,infty )))</span> is positive on <span>((0,infty ))</span> and satisfies </p><span>$$begin{aligned} liminf _{xi searrow 0} frac{phi (xi )}{xi ^{alpha }}&gt;0 qquad hbox { and }qquad limsup _{xi searrow 0} frac{|phi '(xi )|}{xi ^{alpha -1}}&lt;infty , end{aligned}$$</span><p>for some <span>(alpha &gt;0)</span>. Through distinct approaches, we establish that, for sufficiently regular initial data, in two- and higher-dimensional contexts, if <span>(alpha in [1,2m))</span>, then <span>((star ))</span> possesses global weak solutions, while in one-dimensional settings, the same conclusion holds for <span>(alpha &gt;0)</span>, and notably, the solution remains uniformly bounded when <span>(alpha ge 1)</span>. Furthermore, for the one-dimensional case where <span>(alpha ge 1)</span>, the bounded solution additionally possesses the convergence property that </p><span>$$begin{aligned} u(cdot ,t)overset{*}{rightharpoonup } u_{infty } hbox {in } L^{infty }(Omega ) hbox { and } v(cdot ,t)rightarrow 0 hbox { in },,W^{1,infty }(Omega ) qquad hbox {as } trightarrow infty , end{aligned}$$</span><p>with <span>(u_{infty }in L^{infty }(Omega ))</span>. Further conditions on the initial data enable the identification of admissible initial data for which <span>(u_{infty })</span> exhibits spatial heterogeneity. </p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"22 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937538","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
Infinitely many nodal solutions of Kirchhoff-type equations with asymptotically cubic nonlinearity without oddness hypothesis 具有渐近立方非线性的基尔霍夫型方程的无限多节点解，无奇异性假设
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-31 DOI: 10.1007/s00526-024-02805-6
Fuyi Li, Cui Zhang, Zhanping Liang
{"title":"Infinitely many nodal solutions of Kirchhoff-type equations with asymptotically cubic nonlinearity without oddness hypothesis","authors":"Fuyi Li, Cui Zhang, Zhanping Liang","doi":"10.1007/s00526-024-02805-6","DOIUrl":"https://doi.org/10.1007/s00526-024-02805-6","url":null,"abstract":"<p>In this paper, we consider the existence and asymptotic behavior of infinitely many nodal solutions of Kirchhoff-type equations with an asymptotically cubic nonlinear term without oddness assumptions. Combining variational methods and convex analysis techniques, we show, for any positive integer <i>k</i>, the existence of a radial nodal solution that changes sign exactly <i>k</i> times. Meanwhile, we prove that the energy of such solution is an increasing function of <i>k</i>. Moreover, the asymptotic behavior of these solutions are also studied upon varying the parameters. By using different analytical approaches, the question of the existence of infinite solutions to some elliptic nonlinear equations is addressed without invoking oddness assumptions. At the same time, we propose a method to overcome the difficulties caused by the complicated competition between the nonlocal term and the asymptotically cubic nonlinearity.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"95 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250277","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
Optimal coordinates for Ricci-flat conifolds Ricci-flat conifolds 的最佳坐标
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-22 DOI: 10.1007/s00526-024-02780-y
Klaus Kröncke, Áron Szabó
{"title":"Optimal coordinates for Ricci-flat conifolds","authors":"Klaus Kröncke, Áron Szabó","doi":"10.1007/s00526-024-02780-y","DOIUrl":"https://doi.org/10.1007/s00526-024-02780-y","url":null,"abstract":"<p>We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold (<i>M</i>, <i>g</i>) which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold <span>((M^n,g))</span> is of order <i>n</i> and thereby close a small gap in a paper by Cheeger and Tian.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"19 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741637","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
Non-uniqueness for the compressible Euler–Maxwell equations 可压缩欧拉-麦克斯韦方程的非唯一性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-20 DOI: 10.1007/s00526-024-02798-2
Shunkai Mao, Peng Qu
{"title":"Non-uniqueness for the compressible Euler–Maxwell equations","authors":"Shunkai Mao, Peng Qu","doi":"10.1007/s00526-024-02798-2","DOIUrl":"https://doi.org/10.1007/s00526-024-02798-2","url":null,"abstract":"<p>We consider the Cauchy problem for the isentropic compressible Euler–Maxwell equations under general pressure laws in a three-dimensional periodic domain. For any smooth initial electron density away from the vacuum and smooth equilibrium-charged ion density, we could construct infinitely many <span>(alpha )</span>-Hölder continuous entropy solutions emanating from the same initial data for <span>(alpha &lt;frac{1}{7})</span>. Especially, the electromagnetic field belongs to the Hölder class <span>(C^{1,alpha })</span>. Furthermore, we provide a continuous entropy solution satisfying the entropy inequality strictly. The proof relies on the convex integration scheme. Due to the constrain of the Maxwell equations, we propose a method of Mikado potential and construct new building blocks.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"167 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741803","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
A De Lellis–Müller type estimate on the Minkowski lightcone 关于闵科夫斯基光锥的德莱里斯-缪勒式估计
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-20 DOI: 10.1007/s00526-024-02784-8
Markus Wolff
{"title":"A De Lellis–Müller type estimate on the Minkowski lightcone","authors":"Markus Wolff","doi":"10.1007/s00526-024-02784-8","DOIUrl":"https://doi.org/10.1007/s00526-024-02784-8","url":null,"abstract":"<p>We prove an analogue statement to an estimate by De Lellis–Müller in <span>(mathbb {R}^3)</span> on the standard Minkowski lightcone. More precisely, we show that under some additional assumptions, any spacelike cross section of the standard lightcone is <span>(W^{2,2})</span>-close to a round surface provided the trace-free part of a scalar second fundamental form <i>A</i> is sufficiently small in <span>(L^2)</span>. To determine the correct intrinsically round cross section of reference, we define an associated 4-vector, which transforms equivariantly under Lorentz transformations in the restricted Lorentz group. A key step in the proof consists of a geometric, scaling invariant estimate, and we give two different proofs. One utilizes a recent characterization of singularity models of null mean curvature flow along the standard lightcone by the author, while the other is heavily inspired by an almost-Schur lemma by De Lellis–Topping.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"47 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741638","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
On potentials whose level sets are orbits 关于水平集为轨道的势
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-20 DOI: 10.1007/s00526-024-02790-w
Philippe Bolle, Marco Mazzucchelli, Andrea Venturelli
{"title":"On potentials whose level sets are orbits","authors":"Philippe Bolle, Marco Mazzucchelli, Andrea Venturelli","doi":"10.1007/s00526-024-02790-w","DOIUrl":"https://doi.org/10.1007/s00526-024-02790-w","url":null,"abstract":"<p>A level orbit of a mechanical Hamiltonian system is a solution of Newton equation that is contained in a level set of the potential energy. In 2003, Mark Levi asked for a characterization of the smooth potential energy functions on the plane with the property that any point on the plane lies on a level orbit; we call such functions Levi potentials. The basic examples are the radial monotone increasing smooth functions. In this paper we show that any Levi potential that is analytic or has totally path-disconnected critical set must be radial. Nevertheless, we show that every compact convex subset of the plane is the critical set of a Levi potential. A crucial observation for these theorems is that, outside the critical set, the family of level sets of a Levi potential forms a solution of the inverse curvature flow.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"84 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741804","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
On Ricci flows with closed and smooth tangent flows 关于具有封闭平稳切线流的利玛窦流
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-16 DOI: 10.1007/s00526-024-02778-6
Pak-Yeung Chan, Zilu Ma, Yongjia Zhang
{"title":"On Ricci flows with closed and smooth tangent flows","authors":"Pak-Yeung Chan, Zilu Ma, Yongjia Zhang","doi":"10.1007/s00526-024-02778-6","DOIUrl":"https://doi.org/10.1007/s00526-024-02778-6","url":null,"abstract":"<p>In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler (Structure theory of non-collapsed limits of Ricci flows, 2020. arXiv:2009.03243). The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a question proposed in Chow et al. (The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical surveys and monographs, vol 163. AMS, Providence, 2010), (3) a dichotomy theorem that characterizes ancient Ricci flows admitting a closed and smooth backward sequential limit.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"11 12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718855","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 in the two-phase Bernoulli problem for the p-Laplace operator p 拉普拉斯算子两相伯努利问题中的正则性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-16 DOI: 10.1007/s00526-024-02789-3
Masoud Bayrami, Morteza Fotouhi
{"title":"Regularity in the two-phase Bernoulli problem for the p-Laplace operator","authors":"Masoud Bayrami, Morteza Fotouhi","doi":"10.1007/s00526-024-02789-3","DOIUrl":"https://doi.org/10.1007/s00526-024-02789-3","url":null,"abstract":"<p>We show that any minimizer of the well-known ACF functional (for the <i>p</i>-Laplacian) constitutes a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, which boils down to <span>(C^{1,eta })</span> regularity of the flat part of the free boundary. This result, in turn, is used to prove the Lipschitz regularity of minimizers by a dichotomy argument. It is noteworthy that the analysis of branch points is also included.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"37 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718856","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
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