An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions
We present an elementary proof of the Harnack inequality for non-negative viscosity supersolutions of $Delta_{infty}u=0$. This was originally proven by Lindqvist and Manfredi using sequences of solutions of the $p$-Laplacian. We work directly with the $Delta_{infty}$ operator using the distance func...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2001-06-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2001/44/abstr.html |
| _version_ | 1819037943762780160 |
|---|---|
| author | Tilak Bhattacharya |
| author_facet | Tilak Bhattacharya |
| author_sort | Tilak Bhattacharya |
| collection | DOAJ |
| description | We present an elementary proof of the Harnack inequality for non-negative viscosity supersolutions of $Delta_{infty}u=0$. This was originally proven by Lindqvist and Manfredi using sequences of solutions of the $p$-Laplacian. We work directly with the $Delta_{infty}$ operator using the distance function as a test function. We also provide simple proofs of the Liouville property, Hopf boundary point lemma and Lipschitz continuity. |
| first_indexed | 2024-12-21T08:29:27Z |
| format | Article |
| id | doaj.art-178014ba26c047b28817d5b7c30f375f |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-21T08:29:27Z |
| publishDate | 2001-06-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-178014ba26c047b28817d5b7c30f375f2022-12-21T19:10:15ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912001-06-0120014418An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functionsTilak BhattacharyaWe present an elementary proof of the Harnack inequality for non-negative viscosity supersolutions of $Delta_{infty}u=0$. This was originally proven by Lindqvist and Manfredi using sequences of solutions of the $p$-Laplacian. We work directly with the $Delta_{infty}$ operator using the distance function as a test function. We also provide simple proofs of the Liouville property, Hopf boundary point lemma and Lipschitz continuity.http://ejde.math.txstate.edu/Volumes/2001/44/abstr.htmlViscosity solutionsHarnack inequalityinfinite harmonic operatordistance function. |
| spellingShingle | Tilak Bhattacharya An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions Electronic Journal of Differential Equations Viscosity solutions Harnack inequality infinite harmonic operator distance function. |
| title | An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions |
| title_full | An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions |
| title_fullStr | An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions |
| title_full_unstemmed | An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions |
| title_short | An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions |
| title_sort | elementary proof of the harnack inequality for non negative infinity superharmonic functions |
| topic | Viscosity solutions Harnack inequality infinite harmonic operator distance function. |
| url | http://ejde.math.txstate.edu/Volumes/2001/44/abstr.html |
| work_keys_str_mv | AT tilakbhattacharya anelementaryproofoftheharnackinequalityfornonnegativeinfinitysuperharmonicfunctions AT tilakbhattacharya elementaryproofoftheharnackinequalityfornonnegativeinfinitysuperharmonicfunctions |