{"title":"An improved dense class in Sobolev spaces to manifolds","authors":"Antoine Detaille","doi":"10.1016/j.jfa.2025.110894","DOIUrl":null,"url":null,"abstract":"<div><div>We consider the strong density problem in the Sobolev space <span><math><msup><mrow><mi>W</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>;</mo><mi>N</mi><mo>)</mo></math></span> of maps with values into a compact Riemannian manifold <span><math><mi>N</mi></math></span>. It is known, from the seminal work of Bethuel, that such maps may always be strongly approximated by <span><math><mi>N</mi></math></span>-valued maps that are smooth outside of a finite union of <span><math><mo>(</mo><mi>m</mi><mo>−</mo><mo>⌊</mo><mi>s</mi><mi>p</mi><mo>⌋</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-planes. Our main result establishes the strong density in <span><math><msup><mrow><mi>W</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>;</mo><mi>N</mi><mo>)</mo></math></span> of an improved version of the class introduced by Bethuel, where the maps have a singular set <em>without crossings</em>. This answers a question raised by Brezis and Mironescu.</div><div>In the special case where <span><math><mi>N</mi></math></span> has a sufficiently simple topology and for some values of <em>s</em> and <em>p</em>, this result was known to follow from the <em>method of projection</em>, which takes its roots in the work of Federer and Fleming. As a first result, we implement this method in the full range of <em>s</em> and <em>p</em> in which it was expected to be applicable. In the case of a general target manifold, we devise a topological argument that allows to remove the self-intersections in the singular set of the maps obtained via Bethuel's technique.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 2","pages":"Article 110894"},"PeriodicalIF":1.7000,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S002212362500076X","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
Abstract
We consider the strong density problem in the Sobolev space of maps with values into a compact Riemannian manifold . It is known, from the seminal work of Bethuel, that such maps may always be strongly approximated by -valued maps that are smooth outside of a finite union of -planes. Our main result establishes the strong density in of an improved version of the class introduced by Bethuel, where the maps have a singular set without crossings. This answers a question raised by Brezis and Mironescu.
In the special case where has a sufficiently simple topology and for some values of s and p, this result was known to follow from the method of projection, which takes its roots in the work of Federer and Fleming. As a first result, we implement this method in the full range of s and p in which it was expected to be applicable. In the case of a general target manifold, we devise a topological argument that allows to remove the self-intersections in the singular set of the maps obtained via Bethuel's technique.
期刊介绍:
The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published.
Research Areas Include:
• Significant applications of functional analysis, including those to other areas of mathematics
• New developments in functional analysis
• Contributions to important problems in and challenges to functional analysis