
A natural approach to the asymptotic mean value property for the pLaplacianA natural approach to the asymptotic mean value property for the pLaplacianAA10848617 
"/Ishiwata, Michinori/"Ishiwata, Michinori ,
"/Magnanini, Rolando/"Magnanini, Rolando ,
"/Wadade, Hidemitsu(1000000466525)/"Wadade, Hidemitsu
56
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4
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, p.97 , 20170801 , Springer New York LLC
ISSN:09442669
NCID:AA10848617
Description
Let 1 ≤ p≤ ∞. We show that a function u∈ C(RN) is a viscosity solution to the normalized pLaplace equation Δpnu(x)=0 if and only if the asymptotic formula (Formula Presented.) holds as ε→ 0 in the viscosity sense. Here, μp(ε, u) (x) is the pmean value of u on Bε(x) characterized as a unique minimizer of (Formula Presented.) with respect to λ∈ R. This kind of asymptotic mean value property (AMVP) extends to the case p= 1 previous (AMVP)’s obtained when μp(ε, u) (x) is replaced by other kinds of mean values. The natural definition of μp(ε, u) (x) makes sure that this is a monotonic and continuous (in the appropriate topology) functional of u. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic pLaplace equation. © 2017, SpringerVerlag Berlin Heidelberg.
FullText
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