A natural approach to the asymptotic mean value property for the p-Laplacian

作者：Ishiwata Michinori; Magnanini Rolando; Wadade Hidemitsu^*

来源：Calculus of Variations and Partial Differential Equations, 2017, 56(4): 97.

DOI：10.1007/s00526-017-1188-7

Let 1 <= p <= infinity. We show that a function u is an element of C(R-N) is a viscosity solution to the normalized p-Laplace equation Delta(n)(p)u(x) = 0 if and only if the asymptotic formula
u(x) = mu(p)(epsilon, u)(x) + o(epsilon(2))
holds as epsilon -> 0 in the viscosity sense. Here, mu(p)(epsilon,u)(x) is the p-mean value of u on B epsilon(x) characterized as a unique minimizer or
parallel to u - lambda parallel to L-p (B-epsilon(x))
with respect to lambda is an element of R. This kind of asymptotic mean value property (AMVP) extends to the case p = 1 previous (AMVP)' s obtained when mu(p)(epsilon, u) (x) is replaced by other kinds of mean values. The natural definition of mu(p)(epsilon, 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.

• 出版日期2017-8