{"title":"Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems","authors":"Sunghan Kim, Kaj Nyström","doi":"10.1016/j.matpur.2024.02.006","DOIUrl":null,"url":null,"abstract":"<div><p>We prove new optimal <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> regularity results for obstacle problems involving evolutionary <em>p</em>-Laplace type operators in the degenerate regime <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span>. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward <em>p</em>-paraboloids, up to the critical exponent, <span><math><mi>α</mi><mo>≤</mo><mn>2</mn><mo>/</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>, and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> intrinsic interpolative geometries.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"185 ","pages":"Pages 1-46"},"PeriodicalIF":2.1000,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000266/pdfft?md5=3fbde8e03d141f6fd4be7e7aa45c36ea&pid=1-s2.0-S0021782424000266-main.pdf","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/S0021782424000266","RegionNum":1,"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
Abstract
We prove new optimal regularity results for obstacle problems involving evolutionary p-Laplace type operators in the degenerate regime . Our main results include the optimal regularity improvement at free boundary points in intrinsic backward p-paraboloids, up to the critical exponent, , and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or intrinsic interpolative geometries.
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.