Asymptotic Analysis - Volume 17, issue 3

Authors: Andrews, Kevin T. | Wright, Steve

Article Type: Research Article

Abstract: Using stochastic two‐scale mean convergence, homogenization results are obtained for the Dirichlet problem for elliptic equations and systems in divergence form. The coefficients in the equations are assumed to have the bivariate form a(x, T(\varepsilon^{-1}x)\omega) , where a is only assumed to be in L^\infty(Q \times {\varOmega}) and T is an n ‐parameter dynamical system defined on a separable probability space {\varOmega} which induces random oscillations. The nonhomogeneous terms in the equations are assumed to be in L^p and the boundary data is taken to be in W^{1-(1/p),p} …. Results are obtained for both unperforated and randomly perforated domains. Show more

Keywords: Stochastic homogenization , random coefficients , elliptic equation , boundary‐value problem , randomly perforated domain

Citation: Asymptotic Analysis, vol. 17, no. 3, pp. 165-184, 1998

Authors: Budd, C.J. | Humphries, A.R.

Article Type: Research Article

Abstract: We make a formal study of the differential equation \begin{equation} u_{rr} + \frac{2}{r} u_r + \lambda u + u^{5+\varepsilon} = 0,\quad u_r(0) = u(1) = 0,\quad u > 0\quad \hbox{if}\ 0\mathrel{\hbox{{\char"36}}}r<1, \end{equation} when posed as a variational problem over a finite‐dimensional subset S_h of H^1_0 comprising piecewise‐linear functions defined on a mesh of size h . We determine critical points U_h \in S_h of the variational form of (1). Such functions are perturbations of u when a solution of (1) exists, but we show that U_h …can also exist when (1) has no solution and we determine an asymptotic expression for the solution branch (\lambda,U_h) when \Vert U_h\Vert _{\infty} is large and h \Vert U_h\Vert _{\infty}^2 is small. If \varepsilon = 0 , then u exists if \lambda > \mbox{{\char"19}}^2/4 , and we give a formula expressing U_h as a perturbation of u . If \lambda \mathrel{\hbox{{\char"36}}}\mbox{{\char"19}}^2/4 , then a solution of the differential equation does not exist, and U_h grows as h \rightarrow 0. We show that the rate of growth is proportional to h^{-1/4} if \lambda = \mbox{{\char"19}}^2/4 , and h^{-1/3} if \lambda = 0 . We compare these results with estimates for the solutions of (1) when \varepsilon \rightarrow 0^- . Our results are obtained by using formal asymptotic methods – particulary the method of matched asymptotic expansions – and are supported by some numerical calculations. Show more

Citation: Asymptotic Analysis, vol. 17, no. 3, pp. 185-220, 1998

Authors: Junca, Stéphane

Article Type: Research Article

Abstract: We prove a general result of two‐scale convergence for BV‐functions. We then use this result to show the validity of nonlinear geometric optics for weak solutions to scalar conservation laws when the small perturbation of the initial datum is only BV and periodic with respect to the slow and the fast variable.

Citation: Asymptotic Analysis, vol. 17, no. 3, pp. 221-238, 1998