On the constants for some Sobolev imbeddings
<p/> <p>We consider the imbedding inequality <inline-formula><graphic file="1029-242X-2001-823805-i1.gif"/></inline-formula> is the Sobolev space (or Bessel potential space) of <inline-formula><graphic file="1029-242X-2001-823805-i2.gif"/>...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2001-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/6/823805 |
| _version_ | 1819059924952416256 |
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| author | Pizzocchero Livio Morosi Carlo |
| author_facet | Pizzocchero Livio Morosi Carlo |
| author_sort | Pizzocchero Livio |
| collection | DOAJ |
| description | <p/> <p>We consider the imbedding inequality <inline-formula><graphic file="1029-242X-2001-823805-i1.gif"/></inline-formula> is the Sobolev space (or Bessel potential space) of <inline-formula><graphic file="1029-242X-2001-823805-i2.gif"/></inline-formula> type and (integer or fractional) order <inline-formula><graphic file="1029-242X-2001-823805-i3.gif"/></inline-formula>. We write down upper bounds for the constants <inline-formula><graphic file="1029-242X-2001-823805-i4.gif"/></inline-formula>, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if <inline-formula><graphic file="1029-242X-2001-823805-i5.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2001-823805-i6.gif"/></inline-formula>, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on <inline-formula><graphic file="1029-242X-2001-823805-i7.gif"/></inline-formula> for <inline-formula><graphic file="1029-242X-2001-823805-i8.gif"/></inline-formula> in many cases these are close to the previous upper bounds, as illustrated by a number of examples, thus characterizing the sharp constants with little uncertainty.</p> |
| first_indexed | 2024-12-21T14:18:50Z |
| format | Article |
| id | doaj.art-302fec60347946ea9bab655a77141f2e |
| institution | Directory Open Access Journal |
| issn | 1025-5834 1029-242X |
| language | English |
| last_indexed | 2024-12-21T14:18:50Z |
| publishDate | 2001-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-302fec60347946ea9bab655a77141f2e2022-12-21T19:00:50ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2001-01-0120016823805On the constants for some Sobolev imbeddingsPizzocchero LivioMorosi Carlo<p/> <p>We consider the imbedding inequality <inline-formula><graphic file="1029-242X-2001-823805-i1.gif"/></inline-formula> is the Sobolev space (or Bessel potential space) of <inline-formula><graphic file="1029-242X-2001-823805-i2.gif"/></inline-formula> type and (integer or fractional) order <inline-formula><graphic file="1029-242X-2001-823805-i3.gif"/></inline-formula>. We write down upper bounds for the constants <inline-formula><graphic file="1029-242X-2001-823805-i4.gif"/></inline-formula>, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if <inline-formula><graphic file="1029-242X-2001-823805-i5.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2001-823805-i6.gif"/></inline-formula>, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on <inline-formula><graphic file="1029-242X-2001-823805-i7.gif"/></inline-formula> for <inline-formula><graphic file="1029-242X-2001-823805-i8.gif"/></inline-formula> in many cases these are close to the previous upper bounds, as illustrated by a number of examples, thus characterizing the sharp constants with little uncertainty.</p>http://www.journalofinequalitiesandapplications.com/content/6/823805Sobolev spacesImbedding inequalities |
| spellingShingle | Pizzocchero Livio Morosi Carlo On the constants for some Sobolev imbeddings Journal of Inequalities and Applications Sobolev spaces Imbedding inequalities |
| title | On the constants for some Sobolev imbeddings |
| title_full | On the constants for some Sobolev imbeddings |
| title_fullStr | On the constants for some Sobolev imbeddings |
| title_full_unstemmed | On the constants for some Sobolev imbeddings |
| title_short | On the constants for some Sobolev imbeddings |
| title_sort | on the constants for some sobolev imbeddings |
| topic | Sobolev spaces Imbedding inequalities |
| url | http://www.journalofinequalitiesandapplications.com/content/6/823805 |
| work_keys_str_mv | AT pizzoccherolivio ontheconstantsforsomesobolevimbeddings AT morosicarlo ontheconstantsforsomesobolevimbeddings |