Extremal functions for Morrey’s inequality in convex domains
For a bounded domain Ω ⊂ R[superscript n] and p>n , Morrey’s inequality implies that there is c>0 such that c∥u∥p[subscript ∞]≤∫[subscript Ω]|Du|p[subscript dx] for each u belonging to the Sobolev space W[superscript 1,p][subscript 0](Ω) . We show that the ratio of any two extremal functions i...
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2018
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| Online Access: | http://hdl.handle.net/1721.1/119141 |
| _version_ | 1826201221093916672 |
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| author | Lindgren, Erik Hynd, Ryan C |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Lindgren, Erik Hynd, Ryan C |
| author_sort | Lindgren, Erik |
| collection | MIT |
| description | For a bounded domain Ω ⊂ R[superscript n] and p>n , Morrey’s inequality implies that there is c>0 such that c∥u∥p[subscript ∞]≤∫[subscript Ω]|Du|p[subscript dx] for each u belonging to the Sobolev space W[superscript 1,p][subscript 0](Ω) . We show that the ratio of any two extremal functions is constant provided that Ω is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green’s function for the p-Laplacian. |
| first_indexed | 2024-09-23T11:48:28Z |
| format | Article |
| id | mit-1721.1/119141 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:48:28Z |
| publishDate | 2018 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1191412022-10-01T06:10:26Z Extremal functions for Morrey’s inequality in convex domains Lindgren, Erik Hynd, Ryan C Massachusetts Institute of Technology. Department of Mathematics Hynd, Ryan C For a bounded domain Ω ⊂ R[superscript n] and p>n , Morrey’s inequality implies that there is c>0 such that c∥u∥p[subscript ∞]≤∫[subscript Ω]|Du|p[subscript dx] for each u belonging to the Sobolev space W[superscript 1,p][subscript 0](Ω) . We show that the ratio of any two extremal functions is constant provided that Ω is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green’s function for the p-Laplacian. 2018-11-16T15:10:11Z 2018-11-16T15:10:11Z 2018-11 2018-11-14T06:43:04Z Article http://purl.org/eprint/type/JournalArticle 0025-5831 1432-1807 http://hdl.handle.net/1721.1/119141 Hynd, Ryan, and Erik Lindgren. “Extremal Functions for Morrey’s Inequality in Convex Domains.” Mathematische Annalen, Nov. 2018. © 2018 The Authors en https://doi.org/10.1007/s00208-018-1775-8 Mathematische Annalen Creative Commons Attribution http://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Lindgren, Erik Hynd, Ryan C Extremal functions for Morrey’s inequality in convex domains |
| title | Extremal functions for Morrey’s inequality in convex domains |
| title_full | Extremal functions for Morrey’s inequality in convex domains |
| title_fullStr | Extremal functions for Morrey’s inequality in convex domains |
| title_full_unstemmed | Extremal functions for Morrey’s inequality in convex domains |
| title_short | Extremal functions for Morrey’s inequality in convex domains |
| title_sort | extremal functions for morrey s inequality in convex domains |
| url | http://hdl.handle.net/1721.1/119141 |
| work_keys_str_mv | AT lindgrenerik extremalfunctionsformorreysinequalityinconvexdomains AT hyndryanc extremalfunctionsformorreysinequalityinconvexdomains |