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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Texas State University
2006-10-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2006/122/abstr.thml |
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doaj-5cb3a6a622f94b93808486e692e2ff972020-11-24T22:59:39ZengTexas 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. |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Yifeng Yu |
| spellingShingle |
Yifeng Yu A remark on $C^2$ infinity-harmonic functions Electronic Journal of Differential Equations Infinity Laplacian equation infinity harmonic function viscosity solutions. |
| author_facet |
Yifeng Yu |
| author_sort |
Yifeng Yu |
| title |
A remark on $C^2$ infinity-harmonic functions |
| title_short |
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_sort |
remark on $c^2$ infinity-harmonic functions |
| publisher |
Texas State University |
| series |
Electronic Journal of Differential Equations |
| issn |
1072-6691 |
| publishDate |
2006-10-01 |
| 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]. |
| 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 |
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1725644466346262528 |