Geodesic
A remark on \(C^2\) infinity-harmonic functions
Electronic journal of differential equations, Tome 2006 (2006)
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].
Classification : 35B38
Keywords: infinity Laplacian equation, infinity harmonic function, viscosity solutions 
@article{EJDE_2006__2006__a46,
     author = {Yu,  Yifeng},
     title = {A remark on {\(C^2\)} infinity-harmonic functions},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1113.35026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a46/}
}
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Yu,  Yifeng. A remark on \(C^2\) infinity-harmonic functions. Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a46/