Advanced Nonlinear Studies最新文献

{"title":"A system of equations involving the fractional <i>p</i>-Laplacian and doubly critical nonlinearities","authors":"Mousomi Bhakta, Kanishka Perera, Firoj Sk","doi":"10.1515/ans-2023-0103","DOIUrl":"https://doi.org/10.1515/ans-2023-0103","url":null,"abstract":"Abstract This article deals with existence of solutions to the following fractional &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:math&gt; p -Laplacian system of equations: &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"&gt; &lt;m:mfenced open=\"{\" close=\"\"&gt; &lt;m:mrow&gt; &lt;m:mtable displaystyle=\"true\"&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mi&gt;γ&lt;/m:mi&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mspace width=\"0.33em\" /&gt; &lt;m:mspace width=\"0.33em\" /&gt; &lt;m:mstyle&gt; &lt;m:mspace width=\"0.1em\" /&gt; &lt;m:mtext&gt;in&lt;/m:mtext&gt; &lt;m:mspace width=\"0.1em\" /&gt; &lt;/m:mstyle&gt; &lt;m:mspace width=\"0.33em\" /&gt; &lt;m:mi mathvariant=\"normal\"&gt;Ω&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mspace width=\"1.0em\" /&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mi&gt;γ&lt;/m:mi&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mspace width=\"0.33em\" /&gt; &lt;m:mspace width=\"0.33em\" /&gt; &lt;m:mstyle&gt; &lt;m:mspace width=\"0.1em\" /&gt; &lt;m:mtext&gt;in&lt;/m:mtext&gt; &lt;m:mspace width=\"0.1em\" /&gt; &lt;/m:mstyle&gt; &lt;m:ms","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"39 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":"135441735","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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}