{"title":"具有退化扩散和赫尔德连续系数的抛物线系统的部分梯度正则性","authors":"Fabian Bäuerlein","doi":"10.1016/j.na.2024.113691","DOIUrl":null,"url":null,"abstract":"<div><div>We consider vector valued weak solutions <span><math><mrow><mi>u</mi><mo>:</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi></mrow></math></span> of degenerate or singular parabolic systems of type <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>div</mi><mspace></mspace><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>Ω</mi></math></span> denotes an open set in <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>1</mn></mrow></math></span> and <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span> a finite time. Assuming that the vector field <span><math><mi>a</mi></math></span> is not of Uhlenbeck-type structure, satisfies <span><math><mi>p</mi></math></span>-growth assumptions and <span><math><mrow><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>↦</mo><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> is Hölder continuous for every <span><math><mrow><mi>ξ</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mi>n</mi></mrow></msup></mrow></math></span>, we show that the gradient <span><math><mrow><mi>D</mi><mi>u</mi></mrow></math></span> is partially Hölder continuous, provided the vector field degenerates like that of the <span><math><mi>p</mi></math></span>-Laplacian for small gradients.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113691"},"PeriodicalIF":1.3000,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients\",\"authors\":\"Fabian Bäuerlein\",\"doi\":\"10.1016/j.na.2024.113691\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We consider vector valued weak solutions <span><math><mrow><mi>u</mi><mo>:</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi></mrow></math></span> of degenerate or singular parabolic systems of type <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>div</mi><mspace></mspace><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>Ω</mi></math></span> denotes an open set in <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>1</mn></mrow></math></span> and <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span> a finite time. Assuming that the vector field <span><math><mi>a</mi></math></span> is not of Uhlenbeck-type structure, satisfies <span><math><mi>p</mi></math></span>-growth assumptions and <span><math><mrow><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>↦</mo><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> is Hölder continuous for every <span><math><mrow><mi>ξ</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mi>n</mi></mrow></msup></mrow></math></span>, we show that the gradient <span><math><mrow><mi>D</mi><mi>u</mi></mrow></math></span> is partially Hölder continuous, provided the vector field degenerates like that of the <span><math><mi>p</mi></math></span>-Laplacian for small gradients.</div></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":\"251 \",\"pages\":\"Article 113691\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-10-31\",\"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/S0362546X24002104\",\"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/S0362546X24002104","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
我们考虑矢量值弱解 u:ΩT→RN,N∈N 的∂tu-diva(z,u,Du)=0inΩT=Ω×(0,T)类型的退化或奇异抛物线系统,其中Ω表示 Rn 中的开集,n≥1,T>0 为有限时间。假定向量场 a 不是乌伦贝克型结构,满足 p 生长假设,且 (z,u)↦a(z,u,ξ) 对于每个 ξ∈RNn 都是霍尔德连续的,我们证明梯度 Du 部分是霍尔德连续的,条件是向量场像 p-Laplacian 的梯度一样退化为小梯度。
Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients
We consider vector valued weak solutions with of degenerate or singular parabolic systems of type where denotes an open set in for and a finite time. Assuming that the vector field is not of Uhlenbeck-type structure, satisfies -growth assumptions and is Hölder continuous for every , we show that the gradient is partially Hölder continuous, provided the vector field degenerates like that of the -Laplacian for small gradients.
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