||A natural approach to the asymptotic mean value property for the p-Laplacian
Ishiwata, Michinori ,
Magnanini, RolandoWadade, Hidemitsu
Calculus of Variations and Partial Differential Equations
, p.97 , 2017-08-01 , Springer New York LLC
Let 1 ≤ p≤ ∞. We show that a function u∈ C(RN) is a viscosity solution to the normalized p-Laplace 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 p-mean 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 p-Laplace equation. © 2017, Springer-Verlag Berlin Heidelberg.