Nonlinear Analysis-Theory Methods & Applications最新文献

{"title":"Blow-up and boundedness in a chemotaxis system with flux-limited diffusion and logistic source","authors":"Monica Marras , Stella Vernier-Piro , Tomomi Yokota","doi":"10.1016/j.na.2025.113868","DOIUrl":"10.1016/j.na.2025.113868","url":null,"abstract":"<div><div>In this paper we consider radially symmetric solutions of the parabolic–elliptic cross-diffusion system with flux limitation term, <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>u</mi><mo>∇</mo><mi>u</mi></mrow><mrow><msqrt><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mi>χ</mi><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>λ</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>m</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><mi>m</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>under no-flux boundary conditions, where <span><math><mrow><mi>Ω</mi><mo>=</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>R</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>N</mi><mo>≥</mo><mn>1</mn></mrow></math></span>) is a ball, <span><math><mi>χ</mi></math></span>, <span><math><mi>λ</mi></math></span>, <span><math><mi>μ</mi></math></span> are positive constants and <span><math><mrow><mi>k</mi><mo>></mo><mn>1</mn></mrow></math></span> . Under suitable conditions on the data, we prove that the solution is global in time. If <span><math><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, under conditions on the data, we prove that the solution <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> blows up in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm at finite time <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113868"},"PeriodicalIF":1.3,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306506","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":"Everywhere regularity for local minimizers of asymptotically convex non-autonomous functionals","authors":"Junjie Zhang ,&nbsp;Shenzhou Zheng","doi":"10.1016/j.na.2025.113869","DOIUrl":"10.1016/j.na.2025.113869","url":null,"abstract":"<div><div>We consider everywhere regularity of local vectorial minimizers <span><math><mrow><mi>u</mi><mo>:</mo><mi>Ω</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mrow><mo>(</mo><mi>N</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> to a class of non-autonomous functionals <span><span><span><math><mrow><mi>J</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>Ω</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>Φ</mi><mrow><mo>(</mo><mrow><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mi>x</mi><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>Ω</mi></math></span> is a bounded open subset of <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. Under assumptions that <span><math><mi>Φ</mi></math></span> is an Orlicz function and the coefficient function <span><math><mrow><mi>a</mi><mo>:</mo><mi>Ω</mi><mo>→</mo><mi>R</mi></mrow></math></span> belongs to <span><math><mrow><mi>V</mi><mi>M</mi><mi>O</mi><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, we prove that every local minimizer of such functional is an everywhere Hölder continuity with any Hölder exponent <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> by employing a perturbation method, hole filling technique and the iteration lemma. Furthermore, if the coefficient <span><math><mrow><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>κ</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>κ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> is satisfied, then a local everywhere Hölder continuity of the gradient is derived for such local minimizer. As a generalization of our main theorem, the same regularity conclusion holds for an asymptotically convex functional as follows: <span><span><span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>Ω</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mi>x</mi><mo>,</mo></mrow></math></span></span></span>where the integrand <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow></mrow></math></span> is only asymptotically regular with respect to the integrand <span><math><mrow><mi>Φ</mi><mrow><mo>(</mo><mrow><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113869"},"PeriodicalIF":1.3,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306507","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 effect of an unfavorable region on the invasion process of a species with Allee effect","authors":"Pengchao Lai ,&nbsp;Junfan Lu","doi":"10.1016/j.na.2025.113872","DOIUrl":"10.1016/j.na.2025.113872","url":null,"abstract":"<div><div>To model a propagating phenomena through the environment with an unfavorable region, we consider a reaction–diffusion equation with negative growth rate in the unfavorable region and bistable reaction outside of it. We study rigorously the influence of <span><math><mi>L</mi></math></span>, the width of the unfavorable region, on the propagation of solutions. It turns out that there exists a critical value <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> depending only on the reaction term such that, when <span><math><mrow><mi>L</mi><mo>&lt;</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>, spreading happens for any solution in the sense that it passes through the unfavorable region successfully and establish with minor defect in the region; when <span><math><mrow><mi>L</mi><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>, spreading happens only for a species with large initial population, while residue happens for a population with small initial data, in the sense that the solution converges to a small steady state; when <span><math><mrow><mi>L</mi><mo>&gt;</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> we have a trichotomy result: spreading/residue happens for a species with large/small initial population, but, for a species with medium-sized initial data, it cannot pass through the region either and converges to a transition steady state.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"260 ","pages":"Article 113872"},"PeriodicalIF":1.3,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279877","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 flow method for curvature equations","authors":"Shanwei Ding,&nbsp;Guanghan Li","doi":"10.1016/j.na.2025.113873","DOIUrl":"10.1016/j.na.2025.113873","url":null,"abstract":"<div><div>We consider a general curvature equation <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>=</mo><mi>G</mi><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>ν</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>k</mi></math></span> is the principal curvature of the hypersurface <em>M</em> with position vector <span><math><mi>X</mi></math></span>. It includes the classical prescribed curvature measures problem and area measures problem. However, Guan et al. (2015) proved that the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> estimate fails usually for general function <span><math><mi>F</mi></math></span>. Thus, in this paper, we pose some additional conditions of <span><math><mi>G</mi></math></span> to get existence results by a suitably designed parabolic flow. In particular, if <span><math><mrow><mi>F</mi><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msubsup></mrow></math></span> for <span><math><mrow><mo>∀</mo><mn>1</mn><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>, the existence result has been derived in the famous work Guan et al. (2012) with <span><math><mrow><mi>G</mi><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><mfrac><mrow><mi>X</mi></mrow><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow></mrow></mfrac><mo>)</mo></mrow><msup><mrow><mrow><mo>〈</mo><mi>X</mi><mo>,</mo><mi>ν</mi><mo>〉</mo></mrow></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup></mrow></math></span>. This result will be generalized to <span><math><mrow><mi>G</mi><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><mfrac><mrow><mi>X</mi></mrow><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow></mrow></mfrac><mo>)</mo></mrow><msup><mrow><mrow><mo>〈</mo><mi>X</mi><mo>,</mo><mi>ν</mi><mo>〉</mo></mrow></mrow><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow></mrow><mrow><mfrac><mrow><mi>q</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup></mrow></math></span> with <span><math><mrow><mi>p</mi><mo>&gt;</mo><mi>q</mi></mrow></math></span> for arbitrary <span><math><mi>k</mi></math></span> by a suitable auxiliary function. The uniqueness of the solutions in some cases is also studied.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113873"},"PeriodicalIF":1.3,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279190","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":"On nonlinear Landau damping and Gevrey regularity","authors":"Christian Zillinger","doi":"10.1016/j.na.2025.113875","DOIUrl":"10.1016/j.na.2025.113875","url":null,"abstract":"<div><div>In this article we study the problem of nonlinear Landau damping for the Vlasov–Poisson equations on the torus. We introduce <em>anisotropic Gevrey spaces</em> as a new tool to capture the time- and frequency-dependence of resonances. In particular, we show that for small initial data of size <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>ϵ</mi><mo>&lt;</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span> and time intervals <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mo>−</mo><mi>N</mi></mrow></msup><mo>)</mo></mrow></math></span> with <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi></mrow></math></span> arbitrary but fixed, nonlinear stability holds in regularity classes larger than Gevrey-3, uniformly in <span><math><mi>ϵ</mi></math></span>. As a complementary result we construct families of Sobolev regular initial data which exhibit nonlinear Landau damping. Our proof is based on the methods of Grenier, Nguyen and Rodnianski (Grenier et al., 2021).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"260 ","pages":"Article 113875"},"PeriodicalIF":1.3,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279953","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":"On the properties of rearrangement-invariant quasi-Banach function spaces","authors":"Anna Musilová ,&nbsp;Aleš Nekvinda ,&nbsp;Dalimil Peša ,&nbsp;Hana Turčinová","doi":"10.1016/j.na.2025.113854","DOIUrl":"10.1016/j.na.2025.113854","url":null,"abstract":"<div><div>This paper explores some important aspects of the theory of rearrangement-invariant quasi-Banach function spaces. We focus on two main topics. Firstly, we prove an analogue of the Luxemburg representation theorem for rearrangement-invariant quasi-Banach function spaces over resonant measure spaces. Secondly, we develop the theory of fundamental functions and endpoint spaces.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"260 ","pages":"Article 113854"},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144262332","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":"Long-time behavior of the heterogeneous SIRS epidemiological model","authors":"Romain Ducasse,&nbsp;Maxime Laborde","doi":"10.1016/j.na.2025.113867","DOIUrl":"10.1016/j.na.2025.113867","url":null,"abstract":"<div><div>We study the long-time behavior of solutions of the SIRS model, a reaction–diffusion system that appears in epidemiology to describe the spread of epidemics. We allow the system to be heterogeneous periodic. Under some hypotheses on the coefficients, we prove that the solutions converge to an equilibrium that we identify and establish some estimates on the speed of propagation.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"260 ","pages":"Article 113867"},"PeriodicalIF":1.3,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144229864","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":"De Leeuw representations of functionals on Lipschitz spaces","authors":"Ramón J. Aliaga ,&nbsp;Eva Pernecká ,&nbsp;Richard J. Smith","doi":"10.1016/j.na.2025.113851","DOIUrl":"10.1016/j.na.2025.113851","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Lip&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be the space of Lipschitz functions on a complete metric space &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; that vanish at a point &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We investigate its dual &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Lip&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; using the De Leeuw transform, which allows representing each functional on &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Lip&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; as a (non-unique) measure on &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;M&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;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; is the space of pairs &lt;span&gt;&lt;math&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;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We distinguish a set of points of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;M&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;/math&gt;&lt;/span&gt; that are “away from infinity”, which can be assigned coordinates belonging to the Lipschitz realcompactification &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;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We define a natural metric &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̄&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; on &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;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; extending &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and we show that optimal (i.e. positive and norm-minimal) De Leeuw representations of well-behaved functionals are characterised by &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̄&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-cyclical monotonicity of their support, extending known results for functionals in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, the predual of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Lip&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We also extend the Kantorovich–Rubinstein theorem to normal Hausdorff spaces, in particular to &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;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, and use this to characterise measure-induced and majorisable functionals in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Lip&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; as those admitting optimal representations ","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"260 ","pages":"Article 113851"},"PeriodicalIF":1.3,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144205303","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":"BMO estimates for Hodge–Maxwell systems with discontinuous anisotropic coefficients","authors":"Dharmendra Kumar ,&nbsp;Swarnendu Sil","doi":"10.1016/j.na.2025.113852","DOIUrl":"10.1016/j.na.2025.113852","url":null,"abstract":"<div><div>We prove up to the boundary <span><math><mi>BMO</mi></math></span> estimates for linear Maxwell–Hodge type systems for <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>-valued differential <span><math><mi>k</mi></math></span>-forms <span><math><mi>u</mi></math></span> in <span><math><mi>n</mi></math></span> dimensions <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msup><mrow><mi>d</mi></mrow><mrow><mo>∗</mo></mrow></msup><mfenced><mrow><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>u</mi></mrow></mfenced></mtd><mtd><mo>=</mo><mi>f</mi></mtd><mtd></mtd><mtd><mtext>in</mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>d</mi></mrow><mrow><mo>∗</mo></mrow></msup><mfenced><mrow><mi>B</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi></mrow></mfenced></mtd><mtd><mo>=</mo><mi>g</mi></mtd><mtd></mtd><mtd><mtext>in</mtext><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>with <span><math><mrow><mi>ν</mi><mo>∧</mo><mi>u</mi></mrow></math></span> prescribed on <span><math><mrow><mi>∂</mi><mi>Ω</mi><mo>,</mo></mrow></math></span> where the coefficient tensors <span><math><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></math></span> are only required to be bounded measurable and in a class of ‘small multipliers of BMO’. This class neither contains nor is contained in <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>.</mo></mrow></math></span> Since the coefficients are allowed to be discontinuous, the usual Korn’s freezing trick cannot be applied. As an application, we show BMO estimates hold for the time-harmonic Maxwell system in dimension three for a class of discontinuous anisotropic permeability and permittivity tensors. The regularity assumption on the coefficient is essentially sharp.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"260 ","pages":"Article 113852"},"PeriodicalIF":1.3,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194481","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":"Weighted L∞-estimates for solutions of the damped wave equation in three space dimensions and its application","authors":"Vladimir Georgiev ,&nbsp;Kosuke Kita","doi":"10.1016/j.na.2025.113850","DOIUrl":"10.1016/j.na.2025.113850","url":null,"abstract":"<div><div>In this paper, we derive a weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-estimate of the solution to the damped wave equation in three space dimensions. Our proof uses a concrete representation formula of the solution to the damped wave equation that does not rely on the Fourier transform or the energy method. Moreover, by applying our weighted estimate, we consider the global existence of solutions to nonlinear damped wave equations for small data and obtain a new pointwise decay estimate.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"260 ","pages":"Article 113850"},"PeriodicalIF":1.3,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194482","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}