Pub Date : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a1
Kin Ming Hui, Soojung Kim
Let $n geq 3$, $0 lt m lt frac{n-2}{n}$ and $T gt 0$. We construct positive solutions to the fast diffusion equation $u_t = Delta u^m$ in $mathbb{R}^n times (0, T)$, which vanish at time $T$. By introducing a scaling parameter $beta$ inspired by $href{https://dx.doi.org/10.4310/CAG.2019.v27.n8.a4}{textrm{[DKS]}}$, we study the second-order asymptotics of the self-similar solutions associated with $delta$ at spatial infinity. We also investigate the asymptotic behavior of the solutions to the fast diffusion equation near the vanishing time $T$, provided that the initial value of the solution is close to the initial value of some self-similar solution and satisfies some proper decay condition at infinity. Depending on the range of the parameter $delta$, we prove that the rescaled solution converges either to a self-similar profile or to zero as $t nearrow T$. The former implies asymptotic stabilization towards a self-similar solution, and the latter is a new vanishing phenomenon even for the case $n geq 3$ and $m = frac{n-2}{n+2}$ which corresponds to the Yamabe flow on $mathbb{R}^n$ with metric $g = u^frac{4}{n+2} dx^2$.
{"title":"Vanishing time behavior of solutions to the fast diffusion equation","authors":"Kin Ming Hui, Soojung Kim","doi":"10.4310/cag.2023.v31.n2.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a1","url":null,"abstract":"Let $n geq 3$, $0 lt m lt frac{n-2}{n}$ and $T gt 0$. We construct positive solutions to the fast diffusion equation $u_t = Delta u^m$ in $mathbb{R}^n times (0, T)$, which vanish at time $T$. By introducing a scaling parameter $beta$ inspired by $href{https://dx.doi.org/10.4310/CAG.2019.v27.n8.a4}{textrm{[DKS]}}$, we study the second-order asymptotics of the self-similar solutions associated with $delta$ at spatial infinity. We also investigate the asymptotic behavior of the solutions to the fast diffusion equation near the vanishing time $T$, provided that the initial value of the solution is close to the initial value of some self-similar solution and satisfies some proper decay condition at infinity. Depending on the range of the parameter $delta$, we prove that the rescaled solution converges either to a self-similar profile or to zero as $t nearrow T$. The former implies asymptotic stabilization towards a self-similar solution, and the latter is a new vanishing phenomenon even for the case $n geq 3$ and $m = frac{n-2}{n+2}$ which corresponds to the Yamabe flow on $mathbb{R}^n$ with metric $g = u^frac{4}{n+2} dx^2$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"195 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a8
Bingyi Chen, Stephen S.-T. Yau
In this article, we use the Bergman function, which is introduced by the second author in $href{ https://dx.doi.org/10.4310/MRL.2004.v11.n6.a8}{[textrm{Ya}]}$, to study the equivalence problem of bounded complete Reinhardt domains in the singular variety $widetilde{V} = lbrace (u_1, u_2, u_3, u_4) in mathbb{C}^4 vert u_1 u_4 = u_2 u_3 rbrace$.
本文利用第二作者在 $href{ https://dx.doi.org/10.4310/MRL.2004.v11.n6.a8}{[textrm{Ya}]}$中介绍的伯格曼函数，研究奇异簇 $widetilde{V} = lbrace (u_1, u_2, u_3, u_4) in mathbb{C}^4 vert u_1 u_4 = u_2 u_3 rbrace$中的有界完整莱因哈特域的等价问题。
{"title":"Bergman functions and the equivalence problem of singular domains","authors":"Bingyi Chen, Stephen S.-T. Yau","doi":"10.4310/cag.2023.v31.n2.a8","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a8","url":null,"abstract":"In this article, we use the Bergman function, which is introduced by the second author in $href{ https://dx.doi.org/10.4310/MRL.2004.v11.n6.a8}{[textrm{Ya}]}$, to study the equivalence problem of bounded complete Reinhardt domains in the singular variety $widetilde{V} = lbrace (u_1, u_2, u_3, u_4) in mathbb{C}^4 vert u_1 u_4 = u_2 u_3 rbrace$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a3
Nicola Garofalo, Giulio Tralli
In this note we prove the following theorem in any Carnot group of step two $mathbb{G}$:[lim_{s nearrow 1/2} (1 - 2s) mathfrak{P}_{H,s} (E) = frac{4}{sqrt{pi}} mathfrak{P}_H (E).]Here, $mathfrak{P}_H (E)$ represents the horizontal perimeter of a measurable set $E subset mathbb{G}$, whereas the nonlocal horizontal perimeter $mathfrak{P}_{H,s} (E)$ is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain–Brezis–Mironescu and Dávila.
{"title":"A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two","authors":"Nicola Garofalo, Giulio Tralli","doi":"10.4310/cag.2023.v31.n2.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a3","url":null,"abstract":"In this note we prove the following theorem in any Carnot group of step two $mathbb{G}$:[lim_{s nearrow 1/2} (1 - 2s) mathfrak{P}_{H,s} (E) = frac{4}{sqrt{pi}} mathfrak{P}_H (E).]Here, $mathfrak{P}_H (E)$ represents the horizontal perimeter of a measurable set $E subset mathbb{G}$, whereas the nonlocal horizontal perimeter $mathfrak{P}_{H,s} (E)$ is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain–Brezis–Mironescu and Dávila.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"52 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a6
William Worden
Building off ideas developed by Agol, we construct a family of hyperbolic knots $K_n$ whose complements contain no closed incompressible surfaces and have Heegaard genus exactly $n$. These are the first known examples of small knots having large Heegaard genus. Using work of Futer and Purcell, we are able to bound the crossing number for each $K_n$ in terms of $n$.
{"title":"Small knots of large Heegaard genus","authors":"William Worden","doi":"10.4310/cag.2023.v31.n2.a6","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a6","url":null,"abstract":"Building off ideas developed by Agol, we construct a family of hyperbolic knots $K_n$ whose complements contain no closed incompressible surfaces and have Heegaard genus exactly $n$. These are the first known examples of small knots having large Heegaard genus. Using work of Futer and Purcell, we are able to bound the crossing number for each $K_n$ in terms of $n$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-17DOI: 10.4310/cag.2022.v30.n9.a1
Claudianor O. Alves, Minbo Yang
In this paper we study the existence and multiplicity of solutions for the following class of strongly indefinite problems[(P)_k qquadbegin{cases}-Delta u + V(x)u=A(x/k)f(u) ; textrm{in} ; mathbb{R}^N, u ∈ H^1(mathbb{R}^N),end{cases}]where $N geq 1$, $k in mathbb{N}$ is a positive parameter, $f : mathbb{R } to mathbb{R}$ is a continuous function with subcritical growth, and $V, A : mathbb{R} to mathbb{R}$ are continuous functions verifying some technical conditions. Assuming that $V$ is a $mathbb{Z}^N$-periodic function, $0 notin sigma (-Delta+V)$ the spectrum of $(-Delta+V)$, we show how the ”shape” of the graph of function $A$ affects the number of nontrivial solutions.
本文研究了以下一类强不定问题[(P)_k qquadbegin{cases}-Delta u + V(x)u=A(x/k)f(u) ; textrm{in} ; mathbb{R}^N, u ∈ H^1(mathbb{R}^N),end{cases}]的解的存在性和多重性，其中$N geq 1$, $k in mathbb{N}$是正参数，$f : mathbb{R } to mathbb{R}$是次临界增长的连续函数，$V, A : mathbb{R} to mathbb{R}$是验证某些技术条件的连续函数。假设$V$是一个$mathbb{Z}^N$ -周期函数，$0 notin sigma (-Delta+V)$是$(-Delta+V)$的谱，我们展示了函数$A$图的“形状”如何影响非平凡解的数量。
{"title":"Existence and multiplicity of solutions for a class of indefinite variational problems","authors":"Claudianor O. Alves, Minbo Yang","doi":"10.4310/cag.2022.v30.n9.a1","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n9.a1","url":null,"abstract":"In this paper we study the existence and multiplicity of solutions for the following class of strongly indefinite problems[(P)_k qquadbegin{cases}-Delta u + V(x)u=A(x/k)f(u) ; textrm{in} ; mathbb{R}^N, u ∈ H^1(mathbb{R}^N),end{cases}]where $N geq 1$, $k in mathbb{N}$ is a positive parameter, $f : mathbb{R } to mathbb{R}$ is a continuous function with subcritical growth, and $V, A : mathbb{R} to mathbb{R}$ are continuous functions verifying some technical conditions. Assuming that $V$ is a $mathbb{Z}^N$-periodic function, $0 notin sigma (-Delta+V)$ the spectrum of $(-Delta+V)$, we show how the ”shape” of the graph of function $A$ affects the number of nontrivial solutions.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/cag.2023.v31.n1.a3
Ling Xiao
In this paper, we study the flow of closed, starshaped hypersurfaces in $mathbb{R}^{n+1}$ with speed $r^alphasigma_2^{1/2},$ where $sigma_2^{1/2}$ is the normalized square root of the scalar curvature, $alphageq 2,$ and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $alpha<2,$ a counterexample is given for the above convergence.
{"title":"Asymptotic convergence for modified scalar curvature flow","authors":"Ling Xiao","doi":"10.4310/cag.2023.v31.n1.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n1.a3","url":null,"abstract":"In this paper, we study the flow of closed, starshaped hypersurfaces in $mathbb{R}^{n+1}$ with speed $r^alphasigma_2^{1/2},$ where $sigma_2^{1/2}$ is the normalized square root of the scalar curvature, $alphageq 2,$ and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $alpha<2,$ a counterexample is given for the above convergence.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/cag.2023.v31.n1.a4
Francesco Della Pietra, Nunzia Gavitone, Chao Xia
In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken-Ilmanen. By using approximation method involving Finsler-p-Laplacian, we prove the existence and uniqueness of weak solutions.
{"title":"Motion of level sets by inverse anisotropic mean curvature","authors":"Francesco Della Pietra, Nunzia Gavitone, Chao Xia","doi":"10.4310/cag.2023.v31.n1.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n1.a4","url":null,"abstract":"In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken-Ilmanen. By using approximation method involving Finsler-p-Laplacian, we prove the existence and uniqueness of weak solutions.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135600110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/cag.2023.v31.n1.a2
Or Hershkovits, Brian White
We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint as long as one of the subsolutions is compact; previously, this was only known for Euclidean space (with no ambient vectorfield). The new version (March 2020) incorporates improvements suggested by the CAG referee.
{"title":"Avoidance for set-theoretic solutions of mean-curvature-type flows","authors":"Or Hershkovits, Brian White","doi":"10.4310/cag.2023.v31.n1.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n1.a2","url":null,"abstract":"We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint as long as one of the subsolutions is compact; previously, this was only known for Euclidean space (with no ambient vectorfield). The new version (March 2020) incorporates improvements suggested by the CAG referee.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/cag.2023.v31.n1.a5
Ryosuke Takahashi
Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper, we will construct a moduli space consisting of the following date ${(Sigma, psi)}$ where $Sigma$ is a $C^1$-embedding $S^1$ curve in $M$, $psi$ is a $mathbb{Z}/2$-harmonic spinor vanishing only on $Sigma$ and $|psi|_{L^2_1}=1$. We will prove that this moduli space can be parametrized by the space $mathcal{X}=$ all Riemannian metrics on M locally as the kernel of a Fredholm operator.
{"title":"The moduli space of $S^1$-type zero loci for $mathbb{Z}/2$-harmonic spinors in dimension $3$","authors":"Ryosuke Takahashi","doi":"10.4310/cag.2023.v31.n1.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n1.a5","url":null,"abstract":"Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper, we will construct a moduli space consisting of the following date ${(Sigma, psi)}$ where $Sigma$ is a $C^1$-embedding $S^1$ curve in $M$, $psi$ is a $mathbb{Z}/2$-harmonic spinor vanishing only on $Sigma$ and $|psi|_{L^2_1}=1$. We will prove that this moduli space can be parametrized by the space $mathcal{X}=$ all Riemannian metrics on M locally as the kernel of a Fredholm operator.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"274 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135600098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/cag.2023.v31.n1.a1
S. Berhanu
. We establish results on unique continuation at the boundary for the solutions of ∆ u = f, f harmonic, and the biharmonic equation ∆ 2 u = 0. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [HK1], [HK2], and [HKMP] and by M. S. Baouendi and L. P. Rothschild in [BR1] and [BR2].
{"title":"Boundary unique continuation for the Laplace equation and the biharmonic operator","authors":"S. Berhanu","doi":"10.4310/cag.2023.v31.n1.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n1.a1","url":null,"abstract":". We establish results on unique continuation at the boundary for the solutions of ∆ u = f, f harmonic, and the biharmonic equation ∆ 2 u = 0. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [HK1], [HK2], and [HKMP] and by M. S. Baouendi and L. P. Rothschild in [BR1] and [BR2].","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"121 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135600099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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