On the logarithmic epiperimetric inequality for the obstacle problem
We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2021-10-01
|
| Series: | Mathematics in Engineering |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/10.3934/mine.2021004/fulltext.html |
| _version_ | 1818314184880816128 |
|---|---|
| author | Luca Spolaor Bozhidar Velichkov |
| author_facet | Luca Spolaor Bozhidar Velichkov |
| author_sort | Luca Spolaor |
| collection | DOAJ |
| description | We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities. |
| first_indexed | 2024-12-13T08:45:37Z |
| format | Article |
| id | doaj.art-905543fcf7da4980aa2885a4b0fd2a92 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-12-13T08:45:37Z |
| publishDate | 2021-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-905543fcf7da4980aa2885a4b0fd2a922022-12-21T23:53:28ZengAIMS PressMathematics in Engineering2640-35012021-10-013114210.3934/mine.2021004On the logarithmic epiperimetric inequality for the obstacle problemLuca Spolaor0Bozhidar Velichkov11 Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093, USA2 Dipartimento di Matematica, University of Pisa, Largo B. Pontecorvo 5, 56127 Pisa, ItalyWe give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities.https://www.aimspress.com/article/10.3934/mine.2021004/fulltext.htmlepiperimetric inequalityobstacle problemfree boundarysingular points |
| spellingShingle | Luca Spolaor Bozhidar Velichkov On the logarithmic epiperimetric inequality for the obstacle problem Mathematics in Engineering epiperimetric inequality obstacle problem free boundary singular points |
| title | On the logarithmic epiperimetric inequality for the obstacle problem |
| title_full | On the logarithmic epiperimetric inequality for the obstacle problem |
| title_fullStr | On the logarithmic epiperimetric inequality for the obstacle problem |
| title_full_unstemmed | On the logarithmic epiperimetric inequality for the obstacle problem |
| title_short | On the logarithmic epiperimetric inequality for the obstacle problem |
| title_sort | on the logarithmic epiperimetric inequality for the obstacle problem |
| topic | epiperimetric inequality obstacle problem free boundary singular points |
| url | https://www.aimspress.com/article/10.3934/mine.2021004/fulltext.html |
| work_keys_str_mv | AT lucaspolaor onthelogarithmicepiperimetricinequalityfortheobstacleproblem AT bozhidarvelichkov onthelogarithmicepiperimetricinequalityfortheobstacleproblem |