Nonlinear Analysis-Theory Methods & Applications最新文献

{"title":"On the persistence properties for the fractionary BBM equation with low dispersion in weighted Sobolev spaces","authors":"Germán Fonseca,&nbsp;Oscar Riaño,&nbsp;Guillermo Rodriguez-Blanco","doi":"10.1016/j.na.2024.113653","DOIUrl":"10.1016/j.na.2024.113653","url":null,"abstract":"<div><p>We consider the initial value problem associated to the low dispersion fractionary Benjamin–Bona–Mahony equation, fBBM. Our aim is to establish local persistence results in weighted Sobolev spaces and to obtain unique continuation results that imply that those results above are sharp. Hence, arbitrary polynomial type decay is not preserved by the fBBM flow.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X2400172X/pdfft?md5=deedf93280597ef7b37c6bea9b954b83&pid=1-s2.0-S0362546X2400172X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142137191","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":"Stability of the logarithmic Sobolev inequality and uncertainty principle for the Tsallis entropy","authors":"Takeshi Suguro","doi":"10.1016/j.na.2024.113644","DOIUrl":"10.1016/j.na.2024.113644","url":null,"abstract":"<div><p>We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001639/pdfft?md5=6bfcdd2737c232c0665680eef2bf811d&pid=1-s2.0-S0362546X24001639-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142094884","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":"Injectivity of polynomial maps and foliations in the real plane","authors":"Francisco Braun ,&nbsp;Filipe Fernandes ,&nbsp;Bruna Oréfice-Okamoto","doi":"10.1016/j.na.2024.113645","DOIUrl":"10.1016/j.na.2024.113645","url":null,"abstract":"<div><p>We develop tools to count the connected components of the fibers of a polynomial submersion in two real variables <span><math><mi>p</mi></math></span>. As a consequence, we get a necessary condition for a real number to be a bifurcation value of <span><math><mi>p</mi></math></span>. We further present new methods to verify that <span><math><mi>p</mi></math></span> has no Jacobian mates. These results are applied to prove that a polynomial local self-diffeomorphism of the real plane having one coordinate function with degree less than 6 is globally injective. As a byproduct, we completely classify the foliations defined by polynomial submersions of degree less than 6.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001640/pdfft?md5=b3a4aff57d2e2c1abcfc70f1614479b1&pid=1-s2.0-S0362546X24001640-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076349","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":"Neumann problems for nonlinear elliptic equations with lower order terms","authors":"M.F. Betta ,&nbsp;O. Guibé ,&nbsp;A. Mercaldo","doi":"10.1016/j.na.2024.113626","DOIUrl":"10.1016/j.na.2024.113626","url":null,"abstract":"<div><p>In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mi>λ</mi><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo>−</mo><mo>div</mo><mrow><mo>(</mo><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>=</mo><mi>f</mi><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow></mfenced><mi>⋅</mi><munder><mrow><mi>n</mi></mrow><mo>̲</mo></munder><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mtext>on</mtext><mspace></mspace><mi>∂</mi><mi>Ω</mi><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mi>Ω</mi></math></span> is a bounded domain of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, with Lipschitz boundary, <span><math><mrow><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mi>N</mi></mrow></math></span> , <span><math><munder><mrow><mi>n</mi></mrow><mo>̲</mo></munder></math></span> is the outer unit normal to <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>, <span><math><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, the datum <span><math><mi>f</mi></math></span> belongs to the dual space of <span><math><mrow><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> or to Lebesgue space <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. Finally the coefficients <span><math><mi>b</mi></math></span>, <span><math><mi>c</mi></math></span> belong to appropriate Lebesgue spaces or Lorentz spaces.</p><p>Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients <span><math><mi>b</mi></math></span> and <span><math><mi>c</mi></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001457/pdfft?md5=ce03c34a8fe445e869b1bd2082487f52&pid=1-s2.0-S0362546X24001457-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076602","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":"Coupled Elliptic systems with sublinear growth","authors":"J. Arratia,&nbsp;P. Ubilla","doi":"10.1016/j.na.2024.113627","DOIUrl":"10.1016/j.na.2024.113627","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Consider the coupled elliptic system &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;/mtd&gt;&lt;mtd&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;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;/mtd&gt;&lt;mtd&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;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;u&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;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&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;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;as&lt;/mtext&gt;&lt;/mtd&gt;&lt;mtd&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;∞&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;We observe that in 2008, A. Ambrosetti, G. Cerami and D. Ruiz proved the existence of positive bound and ground states in the case &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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; tends to one at infinity. In this work we complement their result, because we show that the previous system has no solutions when &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, as well as we establish sharp hypotheses on the powers &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; the parameter &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and the weights &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;, &lt;span&gt;&lt;ma","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001469/pdfft?md5=01ed59f01a98b70d3c1e8964544d608f&pid=1-s2.0-S0362546X24001469-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076603","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":"k-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds","authors":"Xiaojuan Chen,&nbsp;Qiang Tu,&nbsp;Ni Xiang","doi":"10.1016/j.na.2024.113640","DOIUrl":"10.1016/j.na.2024.113640","url":null,"abstract":"<div><p>In this paper, we consider Weingarten curvature equations for <span><math><mi>k</mi></math></span>-convex hypersurfaces with <span><math><mrow><mi>n</mi><mo>&lt;</mo><mn>2</mn><mi>k</mi></mrow></math></span> in a warped product manifold <span><math><mrow><mover><mrow><mi>M</mi></mrow><mo>¯</mo></mover><mo>=</mo><mi>I</mi><msub><mrow><mo>×</mo></mrow><mrow><mi>λ</mi></mrow></msub><mi>M</mi></mrow></math></span>. Based on the conjecture proposed by Ren–Wang in Ren and Wang (2020), which is valid for <span><math><mrow><mi>k</mi><mo>≥</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span>, we derive curvature estimates for equation <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>ν</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface <span><math><mi>Σ</mi></math></span> satisfying the above equation by the degree theory under some sufficient conditions.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142039755","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":"Renormalised energy between boundary vortices in thin-film micromagnetics with Dzyaloshinskii-Moriya interaction","authors":"Radu Ignat ,&nbsp;François L’Official","doi":"10.1016/j.na.2024.113622","DOIUrl":"10.1016/j.na.2024.113622","url":null,"abstract":"<div><p>We consider a three-dimensional micromagnetic model with Dzyaloshinskii-Moriya interaction in a thin-film regime for boundary vortices. In this regime, we prove a dimension reduction result: the nonlocal three-dimensional model reduces to a local two-dimensional Ginzburg–Landau type model in terms of the averaged magnetisation in the thickness of the film. This reduced model captures the interaction between boundary vortices (so-called renormalised energy), that we determine by a <span><math><mi>Γ</mi></math></span>-convergence result at the second order and then we analyse its minimisers. They nucleate two boundary vortices whose position depends on the Dzyaloshinskii-Moriya interaction.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X2400141X/pdfft?md5=909aaa9112d6eb58c619099b93d70d70&pid=1-s2.0-S0362546X2400141X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142041361","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":"Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems","authors":"Carolin Kreisbeck,&nbsp;Hidde Schönberger","doi":"10.1016/j.na.2024.113642","DOIUrl":"10.1016/j.na.2024.113642","url":null,"abstract":"<div><p>This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via <span><math><mi>Γ</mi></math></span>-convergence as the fractional parameter tends to 1.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001615/pdfft?md5=553f4dd248401bdbae37ffd61c633f93&pid=1-s2.0-S0362546X24001615-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142011691","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":"Partial regularity for manifold constrained quasilinear elliptic systems","authors":"Esther Cabezas-Rivas ,&nbsp;Salvador Moll ,&nbsp;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":null,"pages":null},"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}

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