We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in Rn+1 ${mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.
我们考虑了 R n + 1 ${{mathbb{R}}^{n+1}$ 中凸超曲面的一类非均质收缩流，并证明了流在有限时间内消亡到某一点之前的存在性和正则性。
{"title":"Non-homogeneous fully nonlinear contracting flows of convex hypersurfaces","authors":"Pengfei Guan, Jiuzhou Huang, Jiawei Liu","doi":"10.1515/ans-2022-0077","DOIUrl":"https://doi.org/10.1515/ans-2022-0077","url":null,"abstract":"We consider a general class of non-homogeneous contracting flows of convex hypersurfaces 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:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{n+1}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2022-0077_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula>, and prove the existence and regularity of the flow before extincting to a point in finite time.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019653","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 a class of functionals subject to a duality restriction. The functional is of the form J(Ω,Ω*)=∫Ωf+∫Ω*g $mathcal{J}left({Omega},{{Omega}}^{{ast}}right)={int }_{{Omega}}f+{int }_{{{Omega}}^{{ast}}}g$ , where f, g are given nonnegative functions in a manifold. The duality is a relation α(x, y) ≤ 0 ∀ x ∈ Ω, y ∈ Ω*, for a suitable function α. This model covers several geometric and physical applications. In this paper we review two topological methods introduced in the study of the functional, and discuss possible extensions of the methods to related problems.
{"title":"On the functional ∫Ωf + ∫Ω*g","authors":"Qiang Guang, Qi-Rui Li, Xu-Jia Wang","doi":"10.1515/ans-2023-0105","DOIUrl":"https://doi.org/10.1515/ans-2023-0105","url":null,"abstract":"In this paper, we consider a class of functionals subject to a duality restriction. The functional is of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi mathvariant=\"script\">J</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> <m:mi>f</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mrow> </m:msub> <m:mi>g</m:mi> </m:math> <jats:tex-math> $mathcal{J}left({Omega},{{Omega}}^{{ast}}right)={int }_{{Omega}}f+{int }_{{{Omega}}^{{ast}}}g$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0105_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula>, where <jats:italic>f</jats:italic>, <jats:italic>g</jats:italic> are given nonnegative functions in a manifold. The duality is a relation <jats:italic>α</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>) ≤ 0 ∀ <jats:italic>x</jats:italic> ∈ Ω, <jats:italic>y</jats:italic> ∈ Ω*, for a suitable function <jats:italic>α</jats:italic>. This model covers several geometric and physical applications. In this paper we review two topological methods introduced in the study of the functional, and discuss possible extensions of the methods to related problems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"54 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019548","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}
We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel Spruck, which had been obtained via Reilly’s identities.
{"title":"Comparison formulas for total mean curvatures of Riemannian hypersurfaces","authors":"Mohammad Ghomi","doi":"10.1515/ans-2022-0081","DOIUrl":"https://doi.org/10.1515/ans-2022-0081","url":null,"abstract":"We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel Spruck, which had been obtained via Reilly’s identities.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"171 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019555","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}
We show certain rigidity for minimizers of generalized Colding–Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow.
{"title":"Rigidity properties of Colding–Minicozzi entropies","authors":"Jacob Bernstein","doi":"10.1515/ans-2022-0082","DOIUrl":"https://doi.org/10.1515/ans-2022-0082","url":null,"abstract":"We show certain rigidity for minimizers of generalized Colding–Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"44 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019659","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}
Abstract This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.
{"title":"Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis","authors":"Shanbing Li, Jianhua Wu","doi":"10.1515/ans-2022-0060","DOIUrl":"https://doi.org/10.1515/ans-2022-0060","url":null,"abstract":"Abstract This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42094199","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}
Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ ∗ > 0 {kappa }^{ast }gt 0 such that the elliptic problem for an inhomogeneous elliptic equation − Δ u + u = u p + κ μ -Delta u+u={u}^{p}+kappa mu in R N {{bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0 < κ ≤ κ ∗ 0lt kappa le {kappa }^{ast } . Here, N ≥ 2 Nge 2 , μ mu is a nontrivial nonnegative Radon measure in R N {{bf{R}}}^{N} with a compact support, and p > 1 pgt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.
{"title":"Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity","authors":"Kazuhiro Ishige, S. Okabe, Tokushi Sato","doi":"10.1515/ans-2022-0073","DOIUrl":"https://doi.org/10.1515/ans-2022-0073","url":null,"abstract":"Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ ∗ > 0 {kappa }^{ast }gt 0 such that the elliptic problem for an inhomogeneous elliptic equation − Δ u + u = u p + κ μ -Delta u+u={u}^{p}+kappa mu in R N {{bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0 < κ ≤ κ ∗ 0lt kappa le {kappa }^{ast } . Here, N ≥ 2 Nge 2 , μ mu is a nontrivial nonnegative Radon measure in R N {{bf{R}}}^{N} with a compact support, and p > 1 pgt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47802723","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}
Abstract The primary objective of this article is to analyze the existence of infinitely many radial p p - k k -convex solutions to the boundary blow-up p p - k k -Hessian problem σ k ( λ ( D i ( ∣ D u ∣ p − 2 D j u ) ) ) = H ( ∣ x ∣ ) f ( u ) in Ω , u = + ∞ on ∂ Ω . {sigma }_{k}left(lambda left({D}_{i}left({| Du| }^{p-2}{D}_{j}u)))=Hleft(| x| )fleft(u)hspace{0.33em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}Omega ,hspace{0.33em}u=+infty hspace{0.33em}hspace{0.1em}text{on}hspace{0.1em}hspace{0.33em}partial Omega . Here, k ∈ { 1 , 2 , … , N } kin left{1,2,ldots ,Nright} , σ k ( λ ) {sigma }_{k}left(lambda ) is the k k -Hessian operator, and Ω Omega is a ball in R N ( N ≥ 2 ) {{mathbb{R}}}^{N}hspace{0.33em}left(Nge 2) . Our methods are mainly based on the sub- and super-solutions method.
摘要本文的主要目的是分析边界爆破p p-k k-Hessian问题σk（λ（DI（ŞD uŞp−2 D j u））=Ω中的H（ŞxÜ）f（u），在ŞΩ上的u=+∞的无穷多径向p-k k-凸解的存在性。｛sigma｝_｛k｝left（lambda left({D}_{i} left（｛| Du |｝^｛p-2｝{D}_{j}u)))=Hleft（|x|）fleft{0.33em}u=+inftyhspace｛0.33em｝space｛0.1em｝text｛on｝spage｛0.1em｝sspace｛0.33em｝partialOmega。这里，k∈｛1，2，…，N｝kinleft｛1,2，ldots，Nright｝，σk（λ）｛sigma｝_｛k｝left（lambda）是k-Hessian算子，ΩOmega是R N（N≥2）｛mathbb｛R｝｝中的球^｛N｝space｛0.33em｝lift（Nge2）。我们的方法主要基于亚解和超解方法。
{"title":"The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation","authors":"M. Feng, Xuemei Zhang","doi":"10.1515/ans-2022-0074","DOIUrl":"https://doi.org/10.1515/ans-2022-0074","url":null,"abstract":"Abstract The primary objective of this article is to analyze the existence of infinitely many radial p p - k k -convex solutions to the boundary blow-up p p - k k -Hessian problem σ k ( λ ( D i ( ∣ D u ∣ p − 2 D j u ) ) ) = H ( ∣ x ∣ ) f ( u ) in Ω , u = + ∞ on ∂ Ω . {sigma }_{k}left(lambda left({D}_{i}left({| Du| }^{p-2}{D}_{j}u)))=Hleft(| x| )fleft(u)hspace{0.33em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}Omega ,hspace{0.33em}u=+infty hspace{0.33em}hspace{0.1em}text{on}hspace{0.1em}hspace{0.33em}partial Omega . Here, k ∈ { 1 , 2 , … , N } kin left{1,2,ldots ,Nright} , σ k ( λ ) {sigma }_{k}left(lambda ) is the k k -Hessian operator, and Ω Omega is a ball in R N ( N ≥ 2 ) {{mathbb{R}}}^{N}hspace{0.33em}left(Nge 2) . Our methods are mainly based on the sub- and super-solutions method.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45752357","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}
Abstract In this article, we consider the following Schrödinger-Poisson problem: − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 , left{begin{array}{ll}-{varepsilon }^{2}Delta u+V(y)u+Phi (y)u={| u| }^{p-1}u,& yin {{mathbb{R}}}^{3}, -Delta Phi (y)={u}^{2},& yin {{mathbb{R}}}^{3},end{array}right. where ε > 0 varepsilon gt 0 is a small parameter, 1 < p < 5 1lt plt 5 , and V ( y ) V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V ( y ) V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.
摘要本文考虑以下Schrödinger-Poisson问题:−ε 2 Δ u + V (y) u + Φ (y) u =∣u∣p−1 u, y∈R 3，−Δ Φ (y) = u 2, y∈R 3, left ｛begin{array}{ll}-{varepsilon }^{2}Delta u+V(y)u+Phi (y)u={| u| }^{p-1}u,& yin {{mathbb{R}}}^{3}, -Delta Phi (y)={u}^{2},& yin {{mathbb{R}}}^{3},end{array}right。其中ε > 0 varepsilongt 0为小参数，1 < p < 51 1 lt p lt 5, V(y) V(y)为势函数。我们通过Lyapunov-Schmidt约简方法构造了集中在V(y) V(y)临界点的多峰解。利用爆破分析和局部Pohozaev恒等式，证明了所构造的多峰解是不退化的。据我们所知，这似乎是Schödinger-Poisson系统上的第一个非简并性结果。
{"title":"Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem","authors":"Lin Chen, Hui Ding, Benniao Li, Jianghua Ye","doi":"10.1515/ans-2022-0079","DOIUrl":"https://doi.org/10.1515/ans-2022-0079","url":null,"abstract":"Abstract In this article, we consider the following Schrödinger-Poisson problem: − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 , left{begin{array}{ll}-{varepsilon }^{2}Delta u+V(y)u+Phi (y)u={| u| }^{p-1}u,& yin {{mathbb{R}}}^{3}, -Delta Phi (y)={u}^{2},& yin {{mathbb{R}}}^{3},end{array}right. where ε > 0 varepsilon gt 0 is a small parameter, 1 < p < 5 1lt plt 5 , and V ( y ) V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V ( y ) V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46698882","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}
Abstract On a smooth bounded domain we study the Trudinger-Moser functional E α ( u ) ≔ ∫ Ω ( e α u 2 − 1 ) d x , u ∈ H 1 ( Ω ) {E}_{alpha }left(u):= mathop{int }limits_{Omega }({e}^{alpha {u}^{2}}-1){rm{d}}x,hspace{1.0em}uin {H}^{1}left(Omega ) for α ∈ ( 0 , 2 π ) alpha in left(0,2pi ) and its restriction E α ∣ Σ λ {E}_{alpha }{| }_{{Sigma }_{lambda }} , where Σ λ ≔ u ∈ H 1 ( Ω ) ∣ ∫ Ω ( ∣ ∇ u ∣ 2 + λ u 2 ) d x = 1 {Sigma }_{lambda }:= left{uin {H}^{1}left(Omega )| {int }_{Omega }(| nabla u{| }^{2}+lambda {u}^{2}){rm{d}}x=1right} for λ > 0 lambda gt 0 . By applying the asymptotic analysis and the variational method, we obtain asymptotic behavior of critical points of E α ∣ Σ λ {E}_{alpha }{| }_{{Sigma }_{lambda }} both as λ → 0 lambda to 0 and as λ → + ∞ lambda to +infty . In particular, we prove that when α alpha is sufficiently small, maximizers for sup u ∈ Σ λ E α ( u ) {sup }_{uin {Sigma }_{lambda }}{E}_{alpha }left(u) tend to 0 in C ( Ω ¯ ) Cleft(overline{Omega }) as λ → + ∞ lambda to +infty .
{"title":"Asymptotic properties of critical points for subcritical Trudinger-Moser functional","authors":"Masato Hashizume","doi":"10.1515/ans-2022-0042","DOIUrl":"https://doi.org/10.1515/ans-2022-0042","url":null,"abstract":"Abstract On a smooth bounded domain we study the Trudinger-Moser functional E α ( u ) ≔ ∫ Ω ( e α u 2 − 1 ) d x , u ∈ H 1 ( Ω ) {E}_{alpha }left(u):= mathop{int }limits_{Omega }({e}^{alpha {u}^{2}}-1){rm{d}}x,hspace{1.0em}uin {H}^{1}left(Omega ) for α ∈ ( 0 , 2 π ) alpha in left(0,2pi ) and its restriction E α ∣ Σ λ {E}_{alpha }{| }_{{Sigma }_{lambda }} , where Σ λ ≔ u ∈ H 1 ( Ω ) ∣ ∫ Ω ( ∣ ∇ u ∣ 2 + λ u 2 ) d x = 1 {Sigma }_{lambda }:= left{uin {H}^{1}left(Omega )| {int }_{Omega }(| nabla u{| }^{2}+lambda {u}^{2}){rm{d}}x=1right} for λ > 0 lambda gt 0 . By applying the asymptotic analysis and the variational method, we obtain asymptotic behavior of critical points of E α ∣ Σ λ {E}_{alpha }{| }_{{Sigma }_{lambda }} both as λ → 0 lambda to 0 and as λ → + ∞ lambda to +infty . In particular, we prove that when α alpha is sufficiently small, maximizers for sup u ∈ Σ λ E α ( u ) {sup }_{uin {Sigma }_{lambda }}{E}_{alpha }left(u) tend to 0 in C ( Ω ¯ ) Cleft(overline{Omega }) as λ → + ∞ lambda to +infty .","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44892795","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}
Abstract We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons.
{"title":"Propagation of symmetries for Ricci shrinkers","authors":"T. Colding, William P. Minicozzi II","doi":"10.1515/ans-2022-0071","DOIUrl":"https://doi.org/10.1515/ans-2022-0071","url":null,"abstract":"Abstract We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43971442","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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