Models for growth of heterogeneous sandpiles via Mosco convergence

Abstract

In this paper we study the asymptotic behavior of several classes of power-law functionals involving variable exponents p_n(·)→∞, via Mosco convergence. In the particular case p_n(·)=np(·), we show that the sequence {H_n} of functionals H_n:L²(R^N)→[0,+∞] given by

H_n(u)=∫_R^Nλ(x)ⁿ/np(x)|∇u(x)|^np(x) dx   if u∈L²(R^N)∩W^1,np(·)(R^N),

+∞  otherwise,

converges in the sense of Mosco to a functional which vanishes on the set

u∈L²(R^N): λ(x)|∇u|^p(x)≤ 1 a.e. x∈R^N

and is infinite in its complement. We also provide an example of a sequence of functionals whose Mosco limit cannot be described in terms of the characteristic function of a subset of L²(R^N).

As an application of our results we obtain a model for the growth of a sandpile in which the allowed slope of the sand depends explicitly on the position in the sample.