Affiliations: Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, USA. E-mail: [email protected] | Department of Mathematics, University of Craiova, Craiova, Romania. E-mail: [email protected] | Department of Mathematics, University of Texas at Austin, Austin, TX, USA. E-mail: [email protected] | Departamento de Análisis Matemático, Universidad de Alicante, Alicante, Spain. E-mail: [email protected]
Note: [] Corresponding author: M. Bocea, Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, IL 60660, USA. E-mail: [email protected].
Note: [] On leave from Departamento de Matemática, FCEyN UBA, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina.
Abstract: In this paper we study the asymptotic behavior of several classes of power-law functionals involving variable exponents pn(·)→∞, via Mosco convergence. In the particular case pn(·)=np(·), we show that the sequence {Hn} of functionals Hn:L2(RN)→[0,+∞] given by Hn(u)=∫RNλ(x)n/np(x)|∇u(x)|np(x) dx if u∈L2(RN)∩W1,np(·)(RN), +∞ otherwise, converges in the sense of Mosco to a functional which vanishes on the set u∈L2(RN): λ(x)|∇u|p(x)≤ 1 a.e. x∈RN 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(RN). 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.
