Asymptotic Analysis - Volume 78, issue 1-2

Authors: Croce, Gisella

Article Type: Research Article

Abstract: We study a Dirichlet problem for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side. We will show that the right-hand side has some regularizing effects on the solutions, even if it is singular.

Keywords: degenerate elliptic problem, boundary value problem, weak solutions, singular lower order term

DOI: 10.3233/ASY-2011-1078

Citation: Asymptotic Analysis, vol. 78, no. 1-2, pp. 1-10, 2012

Authors: Bocea, M. | Mihăilescu, M. | Pérez-Llanos, M. | Rossi, J.D.

Article Type: Research Article

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)=∫RN λ(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. Show more

Keywords: Mosco convergence, power-law functionals, variable exponent spaces, sandpile models

DOI: 10.3233/ASY-2011-1083

Citation: Asymptotic Analysis, vol. 78, no. 1-2, pp. 11-36, 2012

Authors: Akian, Jean-Luc

Article Type: Research Article

Abstract: This paper is devoted to the linear elastodynamics equations in an open bounded set with a smooth boundary in the high-frequency limit. The boundary conditions are of Dirichlet or Neumann type. Semiclassical (or Wigner) measures enable to estimate the energy density related to these equations for high frequency phenomena. We determine here the boundary conditions verified by the space–time semiclassical measures related to the elastodynamics equations for the doubly hyperbolic set (under a non-interference hypothesis on the incident semiclassical measures) and for the hyperbolic–elliptic set. The non-interference hypothesis is similar to that of L. Miller in the case of transmission …problems for the wave equation. Show more

Keywords: high frequency, semiclassical measure, elastodynamics, Dirichlet, Neumann

DOI: 10.3233/ASY-2011-1084

Citation: Asymptotic Analysis, vol. 78, no. 1-2, pp. 37-83, 2012

Authors: Chupin, Laurent

Article Type: Research Article

Abstract: We study the effect of a periodic roughness on a Neumann boundary condition. We show that, as in the case of a Dirichlet boundary condition, it is possible to approach this condition by a more complex law on a domain without rugosity, called wall law. This approach is however different from that usually used in Dirichlet case. In particular, we show that this wall law can be explicitly written using an energy developed in the roughness boundary layer. The first part deals with the case of a Laplace operator in a simple domain but many more general results are next …given: when the domain or the operator are more complex or with Robin–Fourier boundary conditions. Some numerical illustrations are used to obtain magnitudes for the coefficients appearing in the new wall laws. Finally, these wall laws can be interpreted using a fictive boundary without rugosity. That allows to give an application to the water waves equation. Show more

Keywords: Neumann or Robin–Fourier boundary conditions, asymptotic development, rough boundaries, hight order approximation, wall laws, Laplace equation, water wave, Dirichlet–Neumann operator

DOI: 10.3233/ASY-2011-1086

Citation: Asymptotic Analysis, vol. 78, no. 1-2, pp. 85-121, 2012