Ayana Pinheiro de Castro Santana , Luís Henrique de Miranda
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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":"248 ","pages":"Article 113625"},"PeriodicalIF":1.3000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Regularizing effect for a class of Maxwell–Schrödinger systems\",\"authors\":\"Ayana Pinheiro de Castro Santana , Luís Henrique de Miranda\",\"doi\":\"10.1016/j.na.2024.113625\",\"DOIUrl\":null,\"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>></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>></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\":\"248 \",\"pages\":\"Article 113625\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001445\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001445","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文证明了以下系统弱解的存在性和正则性-div(M(x)∇u)+g(x,u,v)=finΩ-div(M(x)∇v)=h(x,u,v)inΩu=v=0on∂Ω,其中Ω是 RN 的开放有界子集,对于 N>;2,f∈Lm(Ω),M 是具有 Lipschitz 系数的矩阵,m>1 和 g, h 是两个 Carathéodory 函数。我们证明,在 g 和 h 的适当条件下,存在一些解,它们摆脱了经典的斯坦帕奇亚理论所预测的正则性,从而产生了所谓的正则化效应。
Regularizing effect for a class of Maxwell–Schrödinger systems
In this paper we prove existence and regularity of weak solutions for the following system where is an open bounded subset of , for , , is a matrix with Lipschitz coefficients, and , are two Carathéodory functions. We prove that under appropriate conditions on and , there exist solutions which escape the predicted regularity by the classical Stampacchia’s theory causing the so-called regularizing effect.
期刊介绍:
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