Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring
{"title":"分数阶p-泊松方程的梯度估计","authors":"Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring","doi":"10.1016/j.matpur.2025.103764","DOIUrl":null,"url":null,"abstract":"<div><div>We consider local weak solutions to the fractional <em>p</em>-Poisson equation of order <em>s</em>, i.e. <span><math><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi></math></span>. In the range <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></math></span> we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any <span><math><mi>q</mi><mo>></mo><mn>1</mn></math></span> that<span><span><span><math><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mfrac><mrow><mi>q</mi><mi>p</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msubsup><mspace></mspace><mo>⟹</mo><mspace></mspace><mi>∇</mi><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mi>q</mi><mi>p</mi></mrow></msubsup><mo>.</mo></math></span></span></span> The qualitative result is accompanied by a local quantitative estimate.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103764"},"PeriodicalIF":2.3000,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Gradient estimates for the fractional p-Poisson equation\",\"authors\":\"Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring\",\"doi\":\"10.1016/j.matpur.2025.103764\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We consider local weak solutions to the fractional <em>p</em>-Poisson equation of order <em>s</em>, i.e. <span><math><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi></math></span>. In the range <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></math></span> we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any <span><math><mi>q</mi><mo>></mo><mn>1</mn></math></span> that<span><span><span><math><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mfrac><mrow><mi>q</mi><mi>p</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msubsup><mspace></mspace><mo>⟹</mo><mspace></mspace><mi>∇</mi><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mi>q</mi><mi>p</mi></mrow></msubsup><mo>.</mo></math></span></span></span> The qualitative result is accompanied by a local quantitative estimate.</div></div>\",\"PeriodicalId\":51071,\"journal\":{\"name\":\"Journal de Mathematiques Pures et Appliquees\",\"volume\":\"204 \",\"pages\":\"Article 103764\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-06-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal de Mathematiques Pures et Appliquees\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021782425001084\",\"RegionNum\":1,\"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 de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782425001084","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Gradient estimates for the fractional p-Poisson equation
We consider local weak solutions to the fractional p-Poisson equation of order s, i.e. . In the range and we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any that The qualitative result is accompanied by a local quantitative estimate.
期刊介绍：
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.
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