{"title":"Approximation, regularity and positivity preservation on Riemannian manifolds","authors":"Stefano Pigola, Daniele Valtorta, Giona Veronelli","doi":"10.1016/j.na.2024.113570","DOIUrl":null,"url":null,"abstract":"<div><p>The paper focuses on the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-Positivity Preservation property (<span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-PP for short) on a Riemannian manifold <span><math><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></math></span>. It states that any <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> function <span><math><mi>u</mi></math></span> with <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>+</mo><mi>∞</mi></mrow></math></span>, which solves <span><math><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>u</mi><mo>≥</mo><mn>0</mn></mrow></math></span> on <span><math><mi>M</mi></math></span> in the sense of distributions must be non-negative. Our main result is that the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-PP holds if (the possibly incomplete) <span><math><mi>M</mi></math></span> has a finite number of ends with respect to some compact domain, each of which is <span><math><mi>q</mi></math></span>-parabolic for some, possibly different, values <span><math><mrow><mn>2</mn><mi>p</mi><mo>/</mo><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo><</mo><mi>q</mi><mo>≤</mo><mo>+</mo><mi>∞</mi></mrow></math></span>. When <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math></span>, since <span><math><mi>∞</mi></math></span>-parabolicity coincides with geodesic completeness, our result settles in the affirmative a conjecture by M. Braverman, O. Milatovic and M. Shubin in 2002. On the other hand, we also show that the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-PP is stable by removing from a complete manifold a possibly singular set with Hausdorff co-dimension strictly larger than <span><math><mrow><mn>2</mn><mi>p</mi><mo>/</mo><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> or with a uniform Minkowski-type upper estimate of order <span><math><mrow><mn>2</mn><mi>p</mi><mo>/</mo><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. The threshold value <span><math><mrow><mn>2</mn><mi>p</mi><mo>/</mo><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> is sharp as we show that when the Hausdorff co-dimension of the removed set is strictly smaller, then the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-PP fails. This gives a rather complete picture. The tools developed to carry out our investigations include smooth monotonic approximation and consequent regularity results for subharmonic distributions, a manifold version of the Brezis–Kato inequality, Liouville-type theorems in low regularity, removable singularities results for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-subharmonic distributions and a Frostman-type lemma. Since the seminal works by T. Kato, the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-PP has been linked to the spectral theory of Schrödinger operators with singular potentials <span><math><mrow><mi>Δ</mi><mo>−</mo><mi>V</mi></mrow></math></span>. Here we present some applications of the main results of this paper to the case where <span><math><mrow><mi>V</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mrow></math></span>, addressing the essential self-adjointness of the operator when <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math></span> and whether or not <span><math><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi>∞</mi></mrow></msubsup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> is an operator core for <span><math><mrow><mi>Δ</mi><mo>−</mo><mi>V</mi></mrow></math></span> in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"245 ","pages":"Article 113570"},"PeriodicalIF":1.3000,"publicationDate":"2024-05-18","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/S0362546X24000890","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
The paper focuses on the -Positivity Preservation property (-PP for short) on a Riemannian manifold . It states that any function with , which solves on in the sense of distributions must be non-negative. Our main result is that the -PP holds if (the possibly incomplete) has a finite number of ends with respect to some compact domain, each of which is -parabolic for some, possibly different, values . When , since -parabolicity coincides with geodesic completeness, our result settles in the affirmative a conjecture by M. Braverman, O. Milatovic and M. Shubin in 2002. On the other hand, we also show that the -PP is stable by removing from a complete manifold a possibly singular set with Hausdorff co-dimension strictly larger than or with a uniform Minkowski-type upper estimate of order . The threshold value is sharp as we show that when the Hausdorff co-dimension of the removed set is strictly smaller, then the -PP fails. This gives a rather complete picture. The tools developed to carry out our investigations include smooth monotonic approximation and consequent regularity results for subharmonic distributions, a manifold version of the Brezis–Kato inequality, Liouville-type theorems in low regularity, removable singularities results for -subharmonic distributions and a Frostman-type lemma. Since the seminal works by T. Kato, the -PP has been linked to the spectral theory of Schrödinger operators with singular potentials . Here we present some applications of the main results of this paper to the case where , addressing the essential self-adjointness of the operator when and whether or not is an operator core for in .
期刊介绍:
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.