Pub Date : 2024-08-17DOI: 10.1016/j.na.2024.113643
Esther Cabezas-Rivas , Salvador Moll , Vicent Pallardó-Julià
We consider manifold constrained weak solutions of quasilinear uniformly elliptic systems of divergence type with a source term that grows at most quadratically with respect to the gradient of the solution. As we impose that the solution lies on a Riemannian manifold, the classical smallness condition for regularity can be relaxed to an inequality relating strict convexity of the squared distance and growth of the leading order term in the tangent component of the source. As a key tool for the proof of a partial regularity result, we derive a fully intrinsic Caccioppoli inequality which may be of independent interest. Finally we show how the systems under consideration have a variational nature and arise in the context of - or -harmonic maps.
{"title":"Partial regularity for manifold constrained quasilinear elliptic systems","authors":"Esther Cabezas-Rivas , Salvador Moll , Vicent Pallardó-Julià","doi":"10.1016/j.na.2024.113643","DOIUrl":"10.1016/j.na.2024.113643","url":null,"abstract":"<div><p>We consider manifold constrained weak solutions of quasilinear uniformly elliptic systems of divergence type with a source term that grows at most quadratically with respect to the gradient of the solution. As we impose that the solution lies on a Riemannian manifold, the classical smallness condition for regularity can be relaxed to an inequality relating strict convexity of the squared distance and growth of the leading order term in the tangent component of the source. As a key tool for the proof of a partial regularity result, we derive a fully intrinsic Caccioppoli inequality which may be of independent interest. Finally we show how the systems under consideration have a variational nature and arise in the context of <span><math><mi>F</mi></math></span>- or <span><math><mi>V</mi></math></span>-harmonic maps.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"249 ","pages":"Article 113643"},"PeriodicalIF":1.3,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001627/pdfft?md5=474491586a35eaf7075af1bd65557db2&pid=1-s2.0-S0362546X24001627-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141998381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-13DOI: 10.1016/j.na.2024.113641
Philipp Sürig
We consider on Riemannian manifolds the nonlinear evolution equation where . This equation is also known as a doubly non-linear parabolic equation or Trudinger’s equation. We prove that weak subsolutions of this equation have a sub-Gaussian upper bound and prove that this upper bound is sharp for a specific class of manifolds including .
{"title":"Sharp sub-Gaussian upper bounds for subsolutions of Trudinger’s equation on Riemannian manifolds","authors":"Philipp Sürig","doi":"10.1016/j.na.2024.113641","DOIUrl":"10.1016/j.na.2024.113641","url":null,"abstract":"<div><p>We consider on Riemannian manifolds the nonlinear evolution equation <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>1</mn><mo>/</mo><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>p</mi><mo>></mo><mn>1</mn></mrow></math></span>. This equation is also known as a doubly non-linear parabolic equation or Trudinger’s equation. We prove that weak subsolutions of this equation have a sub-Gaussian upper bound and prove that this upper bound is sharp for a specific class of manifolds including <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"249 ","pages":"Article 113641"},"PeriodicalIF":1.3,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001603/pdfft?md5=d58d3972144f1e8175ec28d9bd63d444&pid=1-s2.0-S0362546X24001603-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141979829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov–Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov–Hausdorff closeness to a flat torus and an integral bound on , the smallest eigenvalue of the Ricci tensor in , imply the existence of a harmonic splitting map. Combining these results with Stern’s inequality, we provide a new Gromov–Hausdorff stability theorem for flat 3-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.
{"title":"Gromov–Hausdorff stability of tori under Ricci and integral scalar curvature bounds","authors":"Shouhei Honda , Christian Ketterer , Ilaria Mondello , Raquel Perales , Chiara Rigoni","doi":"10.1016/j.na.2024.113629","DOIUrl":"10.1016/j.na.2024.113629","url":null,"abstract":"<div><p>We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov–Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov–Hausdorff closeness to a flat torus and an integral bound on <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>M</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, the smallest eigenvalue of the Ricci tensor <span><math><msub><mrow><mo>ric</mo></mrow><mrow><mi>x</mi></mrow></msub></math></span> in <span><math><mi>x</mi></math></span>, imply the existence of a harmonic splitting map. Combining these results with Stern’s inequality, we provide a new Gromov–Hausdorff stability theorem for flat 3-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"249 ","pages":"Article 113629"},"PeriodicalIF":1.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001482/pdfft?md5=ba09939bdd60c2c66bc8258ccc472db4&pid=1-s2.0-S0362546X24001482-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141979828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-09DOI: 10.1016/j.na.2024.113628
Yubo Duan , Yawei Wei
In this paper, we consider the nonlinear equation involving the fractional p-Laplacian with sign-changing potential. This model draws inspiration from De Giorgi Conjecture. There are two main results in this paper. Firstly, we obtain that the solution is radially symmetric within the bounded domain, by applying the moving plane method. Secondly, by exploiting the idea of the sliding method, we construct the appropriate auxiliary functions to prove that the solution is monotone increasing in some direction in the unbounded domain. The different properties of the solution in bounded and unbounded domains are mainly attributed to the inherent non-locality of the fractional p-Laplacian.
在本文中，我们考虑了涉及符号变化势的分数 p-拉普拉奇的非线性方程。这一模型的灵感来自 De Giorgi 猜想。本文有两个主要结果。首先，我们通过应用移动平面法，得到了解在有界域内是径向对称的。其次，利用滑动法的思想，我们构造了适当的辅助函数，证明解在无界域中的某个方向上是单调递增的。有界域和无界域解的不同性质主要归因于分数 p-Laplacian 固有的非位置性。
{"title":"Properties of fractional p-Laplace equations with sign-changing potential","authors":"Yubo Duan , Yawei Wei","doi":"10.1016/j.na.2024.113628","DOIUrl":"10.1016/j.na.2024.113628","url":null,"abstract":"<div><p>In this paper, we consider the nonlinear equation involving the fractional p-Laplacian with sign-changing potential. This model draws inspiration from De Giorgi Conjecture. There are two main results in this paper. Firstly, we obtain that the solution is radially symmetric within the bounded domain, by applying the moving plane method. Secondly, by exploiting the idea of the sliding method, we construct the appropriate auxiliary functions to prove that the solution is monotone increasing in some direction in the unbounded domain. The different properties of the solution in bounded and unbounded domains are mainly attributed to the inherent non-locality of the fractional p-Laplacian.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"249 ","pages":"Article 113628"},"PeriodicalIF":1.3,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141963882","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}
Pub Date : 2024-08-09DOI: 10.1016/j.na.2024.113632
Xianghong Chen , Dashan Fan , Ziyao Liu
<div><p>We study a family of multiplier operators <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>γ</mi></mrow></msub><mfenced><mrow><mi>t</mi><mi>⋅</mi></mrow></mfenced></mrow></msub><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math></span> on compact manifolds <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, which is an analogue of the spherical average <span><math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math></span> on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We establish the almost everywhere convergence of <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>γ</mi></mrow></msub><mfenced><mrow><mi>t</mi><mi>⋅</mi></mrow></mfenced></mrow></msub><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math></span> as <span><math><mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></math></span>. The result is an extension of a Stein’s theorem on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Let <span><math><msubsup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup></math></span> be an analogue of <span><math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mspace></mspace></mrow></math></span>on the <span><math><mrow><mi>n</mi><mo>−</mo></mrow></math></span>torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. As a consequence, we obtain that <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></msub><msubsup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> almost everywhere if <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>γ</mi></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>(</mo><mi>L</mi><mi>o</mi><msup><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msup><mi>L</mi><mo>)</mo></mrow></mrow><mrow><mi>θ</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>θ</mi><mo>></mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>γ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>γ</mi></mrow></mfrac></mrow></math></span>, <span><math><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo><</mo><mi>γ</mi><mo><</mo><mn>1</mn></mrow></math></sp
{"title":"Convergence of Multiplier Operators on Compact Manifolds","authors":"Xianghong Chen , Dashan Fan , Ziyao Liu","doi":"10.1016/j.na.2024.113632","DOIUrl":"10.1016/j.na.2024.113632","url":null,"abstract":"<div><p>We study a family of multiplier operators <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>γ</mi></mrow></msub><mfenced><mrow><mi>t</mi><mi>⋅</mi></mrow></mfenced></mrow></msub><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math></span> on compact manifolds <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, which is an analogue of the spherical average <span><math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math></span> on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We establish the almost everywhere convergence of <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>γ</mi></mrow></msub><mfenced><mrow><mi>t</mi><mi>⋅</mi></mrow></mfenced></mrow></msub><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math></span> as <span><math><mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></math></span>. The result is an extension of a Stein’s theorem on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Let <span><math><msubsup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup></math></span> be an analogue of <span><math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mspace></mspace></mrow></math></span>on the <span><math><mrow><mi>n</mi><mo>−</mo></mrow></math></span>torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. As a consequence, we obtain that <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></msub><msubsup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> almost everywhere if <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>γ</mi></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>(</mo><mi>L</mi><mi>o</mi><msup><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msup><mi>L</mi><mo>)</mo></mrow></mrow><mrow><mi>θ</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>θ</mi><mo>></mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>γ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>γ</mi></mrow></mfrac></mrow></math></span>, <span><math><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo><</mo><mi>γ</mi><mo><</mo><mn>1</mn></mrow></math></sp","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"249 ","pages":"Article 113632"},"PeriodicalIF":1.3,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959951","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}
Pub Date : 2024-08-09DOI: 10.1016/j.na.2024.113630
Sumiya Baasandorj , Sun-Sig Byun , Jehan Oh
We provide global a priori second derivative -estimates for a class of singular fully nonlinear elliptic equations with right hand side terms of .
我们为一类右手项为 Ln 的奇异全非线性椭圆方程提供了全局先验二阶导数 Lδ 估计值。
{"title":"Second derivative Lδ-estimates for a class of singular fully nonlinear elliptic equations","authors":"Sumiya Baasandorj , Sun-Sig Byun , Jehan Oh","doi":"10.1016/j.na.2024.113630","DOIUrl":"10.1016/j.na.2024.113630","url":null,"abstract":"<div><p>We provide global a priori second derivative <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>δ</mi></mrow></msup></math></span>-estimates for a class of singular fully nonlinear elliptic equations with right hand side terms of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"249 ","pages":"Article 113630"},"PeriodicalIF":1.3,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141963883","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}
Pub Date : 2024-08-02DOI: 10.1016/j.na.2024.113623
Elisa Davoli , Chiara Gavioli , Luca Lombardini
We introduce a fractional variant of the Cahn–Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as minimizing movements scheme, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the -Laplacian.
In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.
{"title":"Existence results for Cahn–Hilliard-type systems driven by nonlocal integrodifferential operators with singular kernels","authors":"Elisa Davoli , Chiara Gavioli , Luca Lombardini","doi":"10.1016/j.na.2024.113623","DOIUrl":"10.1016/j.na.2024.113623","url":null,"abstract":"<div><p>We introduce a fractional variant of the Cahn–Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as <em>minimizing movements scheme</em>, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the <span><math><mi>q</mi></math></span>-Laplacian.</p><p>In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"248 ","pages":"Article 113623"},"PeriodicalIF":1.3,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141939683","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}
Pub Date : 2024-08-02DOI: 10.1016/j.na.2024.113631
Wenjing Wu
In this paper, we prove a Talenti-type comparison theorem for the -Laplacian with Dirichlet boundary conditions on open subsets of a normalized space with and . The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov–Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the -Laplacian and a related quantitative stability result.
{"title":"A Talenti-type comparison theorem for the p-Laplacian on RCD(K,N) spaces and some applications","authors":"Wenjing Wu","doi":"10.1016/j.na.2024.113631","DOIUrl":"10.1016/j.na.2024.113631","url":null,"abstract":"<div><p>In this paper, we prove a Talenti-type comparison theorem for the <span><math><mi>p</mi></math></span>-Laplacian with Dirichlet boundary conditions on open subsets of a normalized <span><math><mrow><mi>RCD</mi><mrow><mo>(</mo><mi>K</mi><mo>,</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span> space with <span><math><mrow><mi>K</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>N</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>. The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov–Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the <span><math><mi>p</mi></math></span>-Laplacian and a related quantitative stability result.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"248 ","pages":"Article 113631"},"PeriodicalIF":1.3,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951368","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}
Pub Date : 2024-08-01DOI: 10.1016/j.na.2024.113618
María del Mar González , Liviu I. Ignat , Dragoş Manea , Sergiu Moroianu
We study a non-local evolution equation on the hyperbolic space . We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions.
We also consider a non-local, non-linear convection–diffusion equation on governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain–Brezis–Mironescu.
{"title":"Concentration limit for non-local dissipative convection–diffusion kernels on the hyperbolic space","authors":"María del Mar González , Liviu I. Ignat , Dragoş Manea , Sergiu Moroianu","doi":"10.1016/j.na.2024.113618","DOIUrl":"10.1016/j.na.2024.113618","url":null,"abstract":"<div><p>We study a non-local evolution equation on the hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions.</p><p>We also consider a non-local, non-linear convection–diffusion equation on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain–Brezis–Mironescu.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"248 ","pages":"Article 113618"},"PeriodicalIF":1.3,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001378/pdfft?md5=3cb60f3b75226cdf25776978782952f6&pid=1-s2.0-S0362546X24001378-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141939684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1016/j.na.2024.113624
Arick Shao , Bruno Vergara
We consider heat operators on a bounded domain , with a critically singular potential diverging as the inverse square of the distance to . Although null boundary controllability for such operators was recently proved in all dimensions in Enciso et al. (2023) , it crucially assumed (i) was convex, (ii) the control must be prescribed along all of , and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of , (ii) allow for the control to be localized near any , and (iii) treat the full range of strength parameters for the singular potential. Moreover, we lower the regularity required for and the lower-order coefficients. The key novelty is a local Carleman estimate near , with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of .
{"title":"Approximate boundary controllability for parabolic equations with inverse square infinite potential wells","authors":"Arick Shao , Bruno Vergara","doi":"10.1016/j.na.2024.113624","DOIUrl":"10.1016/j.na.2024.113624","url":null,"abstract":"<div><p>We consider heat operators on a bounded domain <span><math><mrow><mi>Ω</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, with a critically singular potential diverging as the inverse square of the distance to <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>. Although null boundary controllability for such operators was recently proved in all dimensions in Enciso et al. (2023) , it crucially assumed (i) <span><math><mi>Ω</mi></math></span> was convex, (ii) the control must be prescribed along all of <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of <span><math><mi>Ω</mi></math></span>, (ii) allow for the control to be localized near any <span><math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>∂</mi><mi>Ω</mi></mrow></math></span>, and (iii) treat the full range of strength parameters for the singular potential. Moreover, we lower the regularity required for <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span> and the lower-order coefficients. The key novelty is a local Carleman estimate near <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"248 ","pages":"Article 113624"},"PeriodicalIF":1.3,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951371","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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