A remark on $C^2$ infinity-harmonic functions
In this paper, we prove that any nonconstant, $C^2$ solution of the infinity Laplacian equation $u_{x_i}u_{x_j}u_{x_ix_j}=0$ can not have interior critical points. This result was first proved by Aronsson [2] in two dimensions. When the solution is $C^4$, Evans [6] established a Harnack in...
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| Format: | Article |
| Language: | English |
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Texas State University
2006-10-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2006/122/abstr.thml |
| _version_ | 1819030156471173120 |
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| author | Yifeng Yu |
| author_facet | Yifeng Yu |
| author_sort | Yifeng Yu |
| collection | DOAJ |
| description | In this paper, we prove that any nonconstant, $C^2$ solution of the infinity Laplacian equation $u_{x_i}u_{x_j}u_{x_ix_j}=0$ can not have interior critical points. This result was first proved by Aronsson [2] in two dimensions. When the solution is $C^4$, Evans [6] established a Harnack inequality for $|Du|$, which implies that non-constant $C^4$ solutions have no interior critical points for any dimension. Our method is strongly motivated by the work in [6]. |
| first_indexed | 2024-12-21T06:25:40Z |
| format | Article |
| id | doaj.art-5cb3a6a622f94b93808486e692e2ff97 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-21T06:25:40Z |
| publishDate | 2006-10-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-5cb3a6a622f94b93808486e692e2ff972022-12-21T19:13:07ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912006-10-01200612214A remark on $C^2$ infinity-harmonic functionsYifeng YuIn this paper, we prove that any nonconstant, $C^2$ solution of the infinity Laplacian equation $u_{x_i}u_{x_j}u_{x_ix_j}=0$ can not have interior critical points. This result was first proved by Aronsson [2] in two dimensions. When the solution is $C^4$, Evans [6] established a Harnack inequality for $|Du|$, which implies that non-constant $C^4$ solutions have no interior critical points for any dimension. Our method is strongly motivated by the work in [6].http://ejde.math.txstate.edu/Volumes/2006/122/abstr.thmlInfinity Laplacian equationinfinity harmonic functionviscosity solutions. |
| spellingShingle | Yifeng Yu A remark on $C^2$ infinity-harmonic functions Electronic Journal of Differential Equations Infinity Laplacian equation infinity harmonic function viscosity solutions. |
| title | A remark on $C^2$ infinity-harmonic functions |
| title_full | A remark on $C^2$ infinity-harmonic functions |
| title_fullStr | A remark on $C^2$ infinity-harmonic functions |
| title_full_unstemmed | A remark on $C^2$ infinity-harmonic functions |
| title_short | A remark on $C^2$ infinity-harmonic functions |
| title_sort | remark on c 2 infinity harmonic functions |
| topic | Infinity Laplacian equation infinity harmonic function viscosity solutions. |
| url | http://ejde.math.txstate.edu/Volumes/2006/122/abstr.thml |
| work_keys_str_mv | AT yifengyu aremarkonc2infinityharmonicfunctions AT yifengyu remarkonc2infinityharmonicfunctions |