摘要 本文论述下列分数 p p -拉普拉斯方程组的解的存在性： ( - Δ p ) s u = ∣ u ∣ p s * - 2 u + γ α p s * ∣ u ∣ α - 2 u ∣ v ∣ β in Ω 、 ( - Δ p ) s v = ∣ v ∣ p s * - 2 v + γ β p s * ∣ v ∣ β - 2 v ∣ u ∣ α in Ω 、 left（-{Delta }_{p}）}^{s}u={| u| }^{p}_{s}^{* }-2}u+frac{gamma alpha }{p}_{s}^{* }}{ u| }^{alpha -2}u{| v| }^{beta }hspace{0.33em}hspace{0.33em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}Omega ,hspace{1.0em} {left(-{Delta }_{p})}^{s}v={| v| }^{p}_{s}^{* }-2}v+frac{gamma beta }{{p}_{s}^{* }}{| v| }^{beta -2}v{| u| }^{alpha }hspace{0.33em}hspace{0.33em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}Omega ,hspace{1.0em}end{array}right.其中 s∈ ( 0 , 1 ) sin left(0,1) , p∈ ( 1 , ∞ ) pin left(1,infty ) with N > s p Ngt sp , α , β >；1 alpha ,beta 1 such that α + β = p s * ≔ N p N - s p alpha +beta ={p}_{s}^{* }：= frac{Np}{N-sp} 和 Ω = R N Omega ={{mathbb{R}}}^{N} 或 R N 中的光滑有界域 {{mathbb{R}}}^{N} 。当 Ω = R N Omega ={{mathbb{R}}}^{N} 且 γ = 1 gamma =1 时，我们证明在一定的 τ > 条件下，上述系统的任何基态解都具有 ( λ U , τ λ V ) left(lambda U,tau lambda V) 的形式；0 tau gt 0 且 U U 和 V V 是 ( - Δ p ) s u = ∣ u ∣ p s * - 2 u {left(-{Delta }_{p})}^{s}u={| u| }^{p}_{s}^{* }-2}u in R N {{mathbb{R}}}^{N} 的两个正基态解。对于所有 γ > 0 gamma gt 0，我们确定了上述系统在球中的正径向解的存在性。当 Ω = R N Omega ={mathbb{R}}}^{N} 时，我们也建立了上述系统在不同 γ gamma 范围内的正径向解的存在性。
Abstract In this article, we prove a Carleman inequality on a product manifold M × R Mtimes {mathbb{R}} . As applications, we prove that (1) a periodic harmonic function on R 2 {{mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R 3 {{mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rate in one direction must be a hyperplane, and (3) a periodic translator in R 3 {{mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.
{"title":"A Carleman inequality on product manifolds and applications to rigidity problems","authors":"Ao Sun","doi":"10.1515/ans-2022-0048","DOIUrl":"https://doi.org/10.1515/ans-2022-0048","url":null,"abstract":"Abstract In this article, we prove a Carleman inequality on a product manifold M × R Mtimes {mathbb{R}} . As applications, we prove that (1) a periodic harmonic function on R 2 {{mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R 3 {{mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rate in one direction must be a hyperplane, and (3) a periodic translator in R 3 {{mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.","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":"41947519","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 study the sharp profiles of positive solutions to the diffusive logistic equation. By employing parameters and analyzing the corresponding perturbation equations, we find the effects of boundary and spatial heterogeneity on the positive solutions. The main results exhibit the sharp effects between boundary conditions and linear/nonlinear spatial heterogeneities on positive solutions.
{"title":"Sharp profiles for diffusive logistic equation with spatial heterogeneity","authors":"Yan-Hua Xing, Jian-Wen Sun","doi":"10.1515/ans-2022-0061","DOIUrl":"https://doi.org/10.1515/ans-2022-0061","url":null,"abstract":"Abstract In this article, we study the sharp profiles of positive solutions to the diffusive logistic equation. By employing parameters and analyzing the corresponding perturbation equations, we find the effects of boundary and spatial heterogeneity on the positive solutions. The main results exhibit the sharp effects between boundary conditions and linear/nonlinear spatial heterogeneities on positive solutions.","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":"43485971","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 concerns the L p {L}_{p} Minkowski problem for q-capacity. We consider the case p ≥ 1 pge 1 and 1 < q < n 1lt qlt n in the smooth category by a kind of curvature flow, which converges smoothly to the solution of a Monge-Ampére type equation. We show the existence of smooth solution to the problem for p ≥ n pge n . We also provide a proof for the weak solution to the problem when p ≥ 1 pge 1 , which has been obtained by Zou and Xiong.
研究q-capacity的L p {L}_{p} Minkowski问题。考虑一类曲率流在光滑范畴中p≥1 pge 1和1 < q < n 1lt qlt n的情况，该情况平滑地收敛于一个monge - ampsamre型方程的解。我们证明了p≥n pge n时问题光滑解的存在性。我们也证明了当p≥1 pge 1时问题的弱解，这是邹和熊已经得到的。
{"title":"A curvature flow to the Lp Minkowski-type problem of q-capacity","authors":"Xinying Liu, Weimin Sheng","doi":"10.1515/ans-2022-0040","DOIUrl":"https://doi.org/10.1515/ans-2022-0040","url":null,"abstract":"Abstract This article concerns the L p {L}_{p} Minkowski problem for q-capacity. We consider the case p ≥ 1 pge 1 and 1 < q < n 1lt qlt n in the smooth category by a kind of curvature flow, which converges smoothly to the solution of a Monge-Ampére type equation. We show the existence of smooth solution to the problem for p ≥ n pge n . We also provide a proof for the weak solution to the problem when p ≥ 1 pge 1 , which has been obtained by Zou and Xiong.","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":"42564126","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 Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 pgt 1 and the symmetric case of 0 < p < 1 0lt plt 1 .
弦测度是在R n {{mathbb{R}}}^{n}中新发现的凸体的平移不变几何测度，是对aleksandrov - fenchell - jessen面积测度的补充。它们由凸体和随机线的弦积分构成。在L p {L} {p} Brunn-Minkowski理论中，规定L p {L} {p}弦测度称为L p {L} {p}弦Minkowski问题，其中包括L p {L} {p} Minkowski问题作为一个特例。本文解决了p bbb1p gt 1时的L p {L}_{p}弦Minkowski问题以及0 < p < 1 0lt plt 1的对称情况。
{"title":"The Lp chord Minkowski problem","authors":"Dongmeng Xi, Deane Yang, Gaoyong Zhang, Yiming Zhao","doi":"10.1515/ans-2022-0041","DOIUrl":"https://doi.org/10.1515/ans-2022-0041","url":null,"abstract":"Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 pgt 1 and the symmetric case of 0 < p < 1 0lt plt 1 .","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":"41964404","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 make rigorous and old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.
{"title":"Pinched hypersurfaces are compact","authors":"T. Bourni, Mathew T. Langford, Stephen Lynch","doi":"10.1515/ans-2022-0046","DOIUrl":"https://doi.org/10.1515/ans-2022-0046","url":null,"abstract":"Abstract We make rigorous and old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.","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":"43428372","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 note, we establish the global C3,α {C}^{3,alpha } regularity for potential functions in optimal transportation between hypercubes in Rn {{mathbb{R}}}^{n} for n≥3 nge 3 . When n=2 n=2 , the result was proved by Jhaveri. The C3,α {C}^{3,alpha } regularity is also optimal due to a counterexample in the study by Jhaveri.
{"title":"Regularity of optimal mapping between hypercubes","authors":"Shibing Chen, Jiakun Liu, Xu-Jia Wang","doi":"10.1515/ans-2023-0087","DOIUrl":"https://doi.org/10.1515/ans-2023-0087","url":null,"abstract":"Abstract In this note, we establish the global <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C}^{3,alpha } regularity for potential functions in optimal transportation between hypercubes in <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:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{mathbb{R}}}^{n} for <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> nge 3 . When <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:math> n=2 , the result was proved by Jhaveri. The <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C}^{3,alpha } regularity is also optimal due to a counterexample in the study by Jhaveri.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"301 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":"135557389","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 discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.
{"title":"Polynomial sequences in discrete nilpotent groups of step 2","authors":"A. Ionescu, Á. Magyar, Mariusz Mirek, T. Szarek","doi":"10.1515/ans-2023-0085","DOIUrl":"https://doi.org/10.1515/ans-2023-0085","url":null,"abstract":"Abstract We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46876396","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 Given r 0 > 0 {r}_{0}gt 0 , I ∈ N ∪ { 0 } Iin {mathbb{N}}cup left{0right} , and K 0 , H 0 ≥ 0 {K}_{0},{H}_{0}ge 0 , let X X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ r 0 hspace{0.1em}text{Inj}hspace{0.1em}left(X)ge {r}_{0} and with the supremum of absolute sectional curvature at most K 0 {K}_{0} , and let M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] Hin left[0,{H}_{0}] and with index at most I I . We will obtain geometric estimates for such an M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X as a consequence of the hierarchy structure theorem. The hierarchy structure theorem (Theorem 2.2) will be applied to understand global properties of M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X , especially results related to the area and diameter of M M . By item E of Theorem 2.2, the area of such a noncompact M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X is infinite. We will improve this area result by proving the following when M M is connected; here, g ( M ) gleft(M) denotes the genus of the orientable cover of M M : (1) There exists C 1 = C 1 ( I , r 0 , K 0 , H 0 ) > 0 {C}_{1}={C}_{1}left(I,{r}_{0},{K}_{0},{H}_{0})gt 0 , such that Area ( M ) ≥ C 1 ( g ( M ) + 1 ) {rm{Area}}left(M)ge {C}_{1}left(gleft(M)+1) . (2) There exist C > 0 Cgt 0 , G ( I ) ∈ N Gleft(I)in {mathbb{N}} independent of r 0 , K 0 , H 0 {r}_{0},{K}_{0},{H}_{0} , and also C C independent of I I such that if g ( M ) ≥ G ( I ) gleft(M)ge Gleft(I) , then Area ( M ) ≥ C ( max 1 , 1 r 0 , K 0 , H 0 ) 2 ( g ( M ) + 1 ) {rm{Area}}left(M)ge frac{C}{{left(max left{1,frac{1}{{r}_{0}},sqrt{{K}_{0}},{H}_{0}right}right)}^{2}}left(gleft(M)+1) . (3) If the scalar curvature ρ rho of X X satisfies 3 H 2 + 1 2 ρ ≥ c 3{H}^{2}+frac{1}{2}rho ge c in X X for some c > 0 cgt 0 , then there exist A , D > 0 A,Dgt 0 depending on c , I , r 0 , K 0 , H 0 c,I,{r}_{0},{K}_{0},{H}_{0} such that Area ( M ) ≤ A {rm{Area}}left(M)le A and Diameter ( M ) ≤ D {rm{Diameter}}left(M)le D . Hence, M M is compact and, by item 1, g ( M ) ≤ A / C 1 − 1 gleft(M)le Ahspace{0.1em}text{/}hspace{0.1em}{C}_{1}-1 .
给定r 0 bb0 0 {r}＿{0}gt 0, I∈n∪ { 0 } Iin {mathbb{N}}cup left｛0right｝， K 0, H 0≥0 {k}＿{0}，{h}＿{0}ge 0时，设X X为注入半径Inj (X)≥r0的完备黎曼3流形 hspace{0.1em}text{Inj}hspace{0.1em}left(x)ge {r}＿{0} 且绝对截面曲率的最大值不超过k0 {k}＿{0} ，让M * X * Mhspace{0.33em}looparrowright hspace{0.33em}X为平均曲率H∈[0,h0] H的完全浸没面in left[0,{H}_{0}] 且索引不超过I。我们将得到这样一个M * X * M的几何估计hspace{0.33em}looparrowright hspace{0.33em}X作为层次结构定理的结果。层次结构定理(定理2.2)将被应用于理解M的全局属性hspace{0.33em}looparrowright hspace{0.33em}X，特别是与mm的面积和直径有关的结果。根据定理2.2的E项，给出了一个非紧矩阵M的面积hspace{0.33em}looparrowright hspace{0.33em}X是无限的。我们将通过证明以下几点来改进这个区域的结果:这里是g (M) gleft(M)表示M M可定向覆盖物的属:(1)存在c1 = c1 (I, r 0, K 0, H 0) > {c}＿{1}＝{c}＿{1}left(一);{r}＿{0}，{k}＿{0}，{h}＿{0}）gt 0，使得Area (M)≥c1 (g (M) + 1) {rm{Area}}left(m)ge {c}＿{1}left(g)left(m)+1)(2)存在C > 0 Cgt 0, g (I)∈n gleft(i)in {mathbb{N}} 与r0 k0 h0无关 {r}＿{0}，{k}＿{0}，{h}＿{0} ，并且C C独立于I I，使得g (M)≥g (I) gleft(m)ge gleft(I)，则Area (M)≥C (max 1,1r 0, K 0, H 0) 2 (g (M) + 1) {rm{Area}}left(m)ge frac{C}{{left(max left{1,frac{1}{{r}_{0}},sqrt{{K}_{0}},{H}_{0}right}right)}^{2}}left(g)left(m)+1)(3)若标量曲率ρ rho (X X)满足3h2 + 1 2 ρ≥c3{h}＾{2}+frac{1}{2}rho ge c在X X中，c在X X中，c在X X中gt 0，那么存在A,D, bb0 0 A,Dgt 0取决于c I r 0 K 0 H 0 c I，{r}＿{0}，{k}＿{0}，{h}＿{0} 使Area (M)≤A {rm{Area}}left(m)le A、直径(M)≤D {rm{Diameter}}left(m)le D。因此，M M是紧致的，并且根据第1项，g (M)≤A / C 1−1 gleft(m)le ahspace{0.1em}text{/}hspace{0.1em}{C}＿{1}-1。
{"title":"Geometry of CMC surfaces of finite index","authors":"W. Meeks, Joaquín Pérez","doi":"10.1515/ans-2022-0063","DOIUrl":"https://doi.org/10.1515/ans-2022-0063","url":null,"abstract":"Abstract Given r 0 > 0 {r}_{0}gt 0 , I ∈ N ∪ { 0 } Iin {mathbb{N}}cup left{0right} , and K 0 , H 0 ≥ 0 {K}_{0},{H}_{0}ge 0 , let X X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ r 0 hspace{0.1em}text{Inj}hspace{0.1em}left(X)ge {r}_{0} and with the supremum of absolute sectional curvature at most K 0 {K}_{0} , and let M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] Hin left[0,{H}_{0}] and with index at most I I . We will obtain geometric estimates for such an M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X as a consequence of the hierarchy structure theorem. The hierarchy structure theorem (Theorem 2.2) will be applied to understand global properties of M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X , especially results related to the area and diameter of M M . By item E of Theorem 2.2, the area of such a noncompact M ↬ X Mhspace{0.33em}looparrowright hspace{0.33em}X is infinite. We will improve this area result by proving the following when M M is connected; here, g ( M ) gleft(M) denotes the genus of the orientable cover of M M : (1) There exists C 1 = C 1 ( I , r 0 , K 0 , H 0 ) > 0 {C}_{1}={C}_{1}left(I,{r}_{0},{K}_{0},{H}_{0})gt 0 , such that Area ( M ) ≥ C 1 ( g ( M ) + 1 ) {rm{Area}}left(M)ge {C}_{1}left(gleft(M)+1) . (2) There exist C > 0 Cgt 0 , G ( I ) ∈ N Gleft(I)in {mathbb{N}} independent of r 0 , K 0 , H 0 {r}_{0},{K}_{0},{H}_{0} , and also C C independent of I I such that if g ( M ) ≥ G ( I ) gleft(M)ge Gleft(I) , then Area ( M ) ≥ C ( max 1 , 1 r 0 , K 0 , H 0 ) 2 ( g ( M ) + 1 ) {rm{Area}}left(M)ge frac{C}{{left(max left{1,frac{1}{{r}_{0}},sqrt{{K}_{0}},{H}_{0}right}right)}^{2}}left(gleft(M)+1) . (3) If the scalar curvature ρ rho of X X satisfies 3 H 2 + 1 2 ρ ≥ c 3{H}^{2}+frac{1}{2}rho ge c in X X for some c > 0 cgt 0 , then there exist A , D > 0 A,Dgt 0 depending on c , I , r 0 , K 0 , H 0 c,I,{r}_{0},{K}_{0},{H}_{0} such that Area ( M ) ≤ A {rm{Area}}left(M)le A and Diameter ( M ) ≤ D {rm{Diameter}}left(M)le D . Hence, M M is compact and, by item 1, g ( M ) ≤ A / C 1 − 1 gleft(M)le Ahspace{0.1em}text{/}hspace{0.1em}{C}_{1}-1 .","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48063299","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 Let u u be a nontrivial harmonic function in a domain D ⊂ R d Dsubset {{mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } left{Xin overline{D}:uleft(X)=0=| nabla uleft(X)| right} , has finite ( d − 2 ) left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{mathcal{ {mathcal H} }}}^{d-2} -measures.
摘要设u u是域D⊂R D Dsubet｛mathbb｛R｝｝^｛D｝中的一个非平凡调和函数，它在边界的开集上消失。在最近的一篇文章中，我们证明了如果D D是C1｛C｝^｛1｝-Dini域，那么，在开集内，u u的奇异集，定义为｛X∈D｝：u（X）=0=Şu（X。在本文中，我们通过构造一大类奇异集为无穷大的非Dini（但几乎是Dini）域，证明了C1{C}^{1}-Dini域的假设是尖锐的ℋ d−2｛mathcal｛｛math cal H｝｝｝｝^｛d-2｝-度量。
{"title":"Examples of non-Dini domains with large singular sets","authors":"C. Kenig, Zihui Zhao","doi":"10.1515/ans-2022-0058","DOIUrl":"https://doi.org/10.1515/ans-2022-0058","url":null,"abstract":"Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d Dsubset {{mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } left{Xin overline{D}:uleft(X)=0=| nabla uleft(X)| right} , has finite ( d − 2 ) left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{mathcal{ {mathcal H} }}}^{d-2} -measures.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42651856","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：