Models for growth of heterogeneous sandpiles via Mosco convergence
Marian F. Bocea, Mihai Mihăilescu, Mayte Pérez‐Llanos, Julio Daniel Rossi · Asymptotic Analysis · 2012
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 :L 2 (R N )→[0,+∞] given by H n (u)=∫ R N λ(x) n /np(x)|∇u(x)| np(x) dx if u∈L 2 (R N )∩W 1,np(·) (R N ), +∞ otherwise, converges in the sense of Mosco to a functional which vanishes on the set u∈L 2 (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 L 2 (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.