{"title":"二维拉普拉斯算子奇异扰动的索波列夫空间","authors":"Vladimir Georgiev , Mario Rastrelli","doi":"10.1016/j.na.2024.113710","DOIUrl":null,"url":null,"abstract":"<div><div>We study the perturbed Sobolev space <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>α</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msubsup></math></span>, <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> associated with singular perturbation <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> of Laplace operator in Euclidean space of dimension <span><math><mrow><mn>2</mn><mo>.</mo></mrow></math></span> The main results give the possibility to extend the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> theory of perturbed Sobolev space to the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> case. When <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> we have appropriate representation of the functions in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>α</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msubsup></math></span> in regular and singular part. An application to local well-posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113710"},"PeriodicalIF":1.3000,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sobolev spaces for singular perturbation of 2D Laplace operator\",\"authors\":\"Vladimir Georgiev , Mario Rastrelli\",\"doi\":\"10.1016/j.na.2024.113710\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We study the perturbed Sobolev space <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>α</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msubsup></math></span>, <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> associated with singular perturbation <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> of Laplace operator in Euclidean space of dimension <span><math><mrow><mn>2</mn><mo>.</mo></mrow></math></span> The main results give the possibility to extend the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> theory of perturbed Sobolev space to the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> case. When <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> we have appropriate representation of the functions in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>α</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msubsup></math></span> in regular and singular part. An application to local well-posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.</div></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":\"251 \",\"pages\":\"Article 113710\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-11-15\",\"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/S0362546X24002293\",\"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/S0362546X24002293","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Sobolev spaces for singular perturbation of 2D Laplace operator
We study the perturbed Sobolev space , associated with singular perturbation of Laplace operator in Euclidean space of dimension The main results give the possibility to extend the theory of perturbed Sobolev space to the case. When we have appropriate representation of the functions in in regular and singular part. An application to local well-posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.
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Nonlinear Analysis focuses on papers that address significant problems in Nonlinear Analysis that have a sustainable and important impact on the development of new directions in the theory as well as potential applications. Review articles on important topics in Nonlinear Analysis are welcome as well. In particular, only papers within the areas of specialization of the Editorial Board Members will be considered. Authors are encouraged to check the areas of expertise of the Editorial Board in order to decide whether or not their papers are appropriate for this journal. The journal aims to apply very high standards in accepting papers for publication.
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