Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited
In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with...
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MDPI AG
2022-02-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/3/522 |
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| author | Ali Behzadan Michael Holst |
| author_facet | Ali Behzadan Michael Holst |
| author_sort | Ali Behzadan |
| collection | DOAJ |
| description | In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, it is not necessarily true that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>W</mi><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow><mo>↪</mo><msup><mi>W</mi><mrow><mi>t</mi><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here. |
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| spelling | doaj.art-50debb14d3144233990e36bb6cf9441c2023-11-23T17:08:55ZengMDPI AGMathematics2227-73902022-02-0110352210.3390/math10030522Sobolev-Slobodeckij Spaces on Compact Manifolds, RevisitedAli Behzadan0Michael Holst1Department of Mathematics and Statistics, California State University Sacramento, Sacramento, CA 95819, USADepartment of Mathematics, University of California San Diego, La Jolla, San Diego, CA 92093, USAIn this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, it is not necessarily true that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>W</mi><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow><mo>↪</mo><msup><mi>W</mi><mrow><mi>t</mi><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.https://www.mdpi.com/2227-7390/10/3/522Sobolev spacescompact manifoldstensor bundlesdifferential operators |
| spellingShingle | Ali Behzadan Michael Holst Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited Mathematics Sobolev spaces compact manifolds tensor bundles differential operators |
| title | Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited |
| title_full | Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited |
| title_fullStr | Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited |
| title_full_unstemmed | Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited |
| title_short | Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited |
| title_sort | sobolev slobodeckij spaces on compact manifolds revisited |
| topic | Sobolev spaces compact manifolds tensor bundles differential operators |
| url | https://www.mdpi.com/2227-7390/10/3/522 |
| work_keys_str_mv | AT alibehzadan sobolevslobodeckijspacesoncompactmanifoldsrevisited AT michaelholst sobolevslobodeckijspacesoncompactmanifoldsrevisited |