{"title":"奇异分数 p-Laplacian 的精细边界正则性","authors":"","doi":"10.1016/j.jde.2024.08.026","DOIUrl":null,"url":null,"abstract":"<div><p>We study the boundary weighted regularity of weak solutions <em>u</em> to a <em>s</em>-fractional <em>p</em>-Laplacian equation in a bounded <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We prove optimal up-to-the-boundary regularity of <em>u</em>, which is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span> for any <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and, in the singular case <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, that <span><math><mi>u</mi><mo>/</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math></span> has a Hölder continuous extension to the closure of Ω, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> meaning the distance of <em>x</em> from the complement of Ω. This last result is the singular counterpart of the one in <span><span>[30]</span></span>, where the degenerate case <span><math><mi>p</mi><mo>⩾</mo><mn>2</mn></math></span> is considered.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022039624005084/pdfft?md5=1c17576e29620614c9f4f6b0066610f0&pid=1-s2.0-S0022039624005084-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Fine boundary regularity for the singular fractional p-Laplacian\",\"authors\":\"\",\"doi\":\"10.1016/j.jde.2024.08.026\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study the boundary weighted regularity of weak solutions <em>u</em> to a <em>s</em>-fractional <em>p</em>-Laplacian equation in a bounded <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We prove optimal up-to-the-boundary regularity of <em>u</em>, which is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span> for any <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and, in the singular case <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, that <span><math><mi>u</mi><mo>/</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math></span> has a Hölder continuous extension to the closure of Ω, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> meaning the distance of <em>x</em> from the complement of Ω. This last result is the singular counterpart of the one in <span><span>[30]</span></span>, where the degenerate case <span><math><mi>p</mi><mo>⩾</mo><mn>2</mn></math></span> is considered.</p></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2024-08-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022039624005084/pdfft?md5=1c17576e29620614c9f4f6b0066610f0&pid=1-s2.0-S0022039624005084-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039624005084\",\"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":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039624005084","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
我们研究在有界 C1,1 域 Ω 中,s-分式 p-拉普拉奇方程的弱解 u 的边界加权正则性,该域具有有界反应和非局部 Dirichlet 型边界条件,s∈(0,1)。我们证明了 u 的最优达界正则性,即对于任意 p>1 均为 Cs(Ω‾);在奇异情况 p∈(1,2)下,u/dΩs 具有霍尔德连续扩展到 Ω 的闭合,dΩ(x) 指 x 与 Ω 的补集的距离。
Fine boundary regularity for the singular fractional p-Laplacian
We study the boundary weighted regularity of weak solutions u to a s-fractional p-Laplacian equation in a bounded domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with . We prove optimal up-to-the-boundary regularity of u, which is for any and, in the singular case , that has a Hölder continuous extension to the closure of Ω, meaning the distance of x from the complement of Ω. This last result is the singular counterpart of the one in [30], where the degenerate case is considered.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics