Nonlinear Analysis-Theory Methods & Applications最新文献

{"title":"Gromov–Hausdorff stability of tori under Ricci and integral scalar curvature bounds","authors":"Shouhei Honda ,&nbsp;Christian Ketterer ,&nbsp;Ilaria Mondello ,&nbsp;Raquel Perales ,&nbsp;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":null,"pages":null},"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}

{"title":"Properties of fractional p-Laplace equations with sign-changing potential","authors":"Yubo Duan ,&nbsp;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":null,"pages":null},"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}

{"title":"Convergence of Multiplier Operators on Compact Manifolds","authors":"Xianghong Chen ,&nbsp;Dashan Fan ,&nbsp;Ziyao Liu","doi":"10.1016/j.na.2024.113632","DOIUrl":"10.1016/j.na.2024.113632","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We study a family of multiplier operators &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; on compact manifolds &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, which is an analogue of the spherical average &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. We establish the almost everywhere convergence of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; as &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The result is an extension of a Stein’s theorem on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; be an analogue of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;on the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;torus &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. As a consequence, we obtain that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; almost everywhere if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/sp","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"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}

{"title":"Second derivative Lδ-estimates for a class of singular fully nonlinear elliptic equations","authors":"Sumiya Baasandorj ,&nbsp;Sun-Sig Byun ,&nbsp;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":null,"pages":null},"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}

{"title":"Existence results for Cahn–Hilliard-type systems driven by nonlocal integrodifferential operators with singular kernels","authors":"Elisa Davoli ,&nbsp;Chiara Gavioli ,&nbsp;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":null,"pages":null},"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}

{"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>&gt;</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":null,"pages":null},"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}

{"title":"Concentration limit for non-local dissipative convection–diffusion kernels on the hyperbolic space","authors":"María del Mar González ,&nbsp;Liviu I. Ignat ,&nbsp;Dragoş Manea ,&nbsp;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":null,"pages":null},"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}

{"title":"Approximate boundary controllability for parabolic equations with inverse square infinite potential wells","authors":"Arick Shao ,&nbsp;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":null,"pages":null},"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}

{"title":"Regularizing effect for a class of Maxwell–Schrödinger systems","authors":"Ayana Pinheiro de Castro Santana ,&nbsp;Luís Henrique de Miranda","doi":"10.1016/j.na.2024.113625","DOIUrl":"10.1016/j.na.2024.113625","url":null,"abstract":"<div><p>In this paper we prove existence and regularity of weak solutions for the following system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><mtext>div</mtext><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><mspace></mspace><mi>Ω</mi><mspace></mspace></mtd></mtr><mtr><mtd><mo>−</mo><mtext>div</mtext><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∇</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><mspace></mspace><mi>Ω</mi><mspace></mspace></mtd></mtr><mtr><mtd><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>u</mi><mo>=</mo><mi>v</mi><mo>=</mo><mn>0</mn><mspace></mspace><mspace></mspace><mtext>on</mtext><mspace></mspace><mspace></mspace><mi>∂</mi><mi>Ω</mi><mo>,</mo><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mi>Ω</mi></math></span> is an open bounded subset of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, for <span><math><mrow><mi>N</mi><mo>&gt;</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>m</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mi>M</mi></math></span> is a matrix with Lipschitz coefficients, <span><math><mrow><mi>m</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span> and <span><math><mi>g</mi></math></span>, <span><math><mi>h</mi></math></span> are two Carathéodory functions. We prove that under appropriate conditions on <span><math><mi>g</mi></math></span> and <span><math><mi>h</mi></math></span>, there exist solutions which escape the predicted regularity by the classical Stampacchia’s theory causing the so-called regularizing effect.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951370","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 classification on a class of weakly weighted Einstein–Finsler metrics","authors":"Hongmei Zhu ,&nbsp;Peijuan Rao","doi":"10.1016/j.na.2024.113621","DOIUrl":"10.1016/j.na.2024.113621","url":null,"abstract":"<div><p>In this paper, we study generalized weighted Ricci curvatures, which include the <span><math><mi>N</mi></math></span>-Ricci curvature and the projective Ricci curvature with totally different geometric meanings. We completely classify a class of weakly weighted Einstein-Finsler metrics.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868485","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}