Asymptotic Analysis - Volume 77, issue 3-4

Authors: Bal, Guillaume | Garnier, Josselin | Gu, Yu | Jing, Wenjia

Article Type: Research Article

Abstract: We consider an elliptic pseudo-differential equation with a highly oscillating linear potential modeled as a stationary ergodic random field. The random field is a function composed with a centered long-range correlated Gaussian process. In the limiting of vanishing correlation length, the heterogeneous solution converges to a deterministic solution obtained by averaging the random potential. We characterize the deterministic and stochastic correctors. With proper rescaling, the mean-zero stochastic corrector converges to a Gaussian random process in probability and weakly in the spatial variables. In addition, for two prototype equations involving the Laplacian and the fractional Laplacian operators, we prove that the …limit holds in distribution in some Hilbert spaces. We also determine the size of the deterministic corrector when it is larger than the stochastic corrector. Depending on the correlation structure of the random field and on the singularities of the Green's function, we show that either the deterministic or the random part of the corrector dominates. Show more

Keywords: corrector theory, random homogenization, long-range correlations, Gaussian random field, weak convergence of probability measures

DOI: 10.3233/ASY-2011-1072

Citation: Asymptotic Analysis, vol. 77, no. 3-4, pp. 123-145, 2012

Authors: von Below, Joachim | Lubary, José A.

Article Type: Research Article

Abstract: We determine the asymptotic behavior of the eigenvalues of second-order elliptic operators on finite networks under continuity and weighted Kirchhoff flow conditions at the vertices. It turns out that the quadratic growth formula in the symmetrizable case [Linear Algebra Appl. 121 (1989), 692–697; Math. Meth. Appl. Sci. 10 (1988), 383–395; Parabolic Network Equations, 2nd edn, Universitätsverlag, Tübingen, 1994] holds also in the general case when the problem cannot be reduced to a self-adjoint problem and when non-real eigenvalues occur.

Keywords: elliptic operators on networks, eigenvalue asymptotics, adjacency and transition operators

DOI: 10.3233/ASY-2011-1073

Citation: Asymptotic Analysis, vol. 77, no. 3-4, pp. 147-167, 2012

Authors: Kałamajska, Agnieszka | Peszek, Jan

Article Type: Research Article

Abstract: We derive the inequality ∫R |f′(x)|p h(f(x)) dx≤(\sqrt{p−1})p ∫R (\sqrt{|f″(x)𝒯h (f(x))|})p h(f(x)) dx, where f belongs locally to the Sobolev space W2,1 and f′ has bounded support. Here h(·) is a given function and 𝒯h (·) is its given transform, it is independent of p. In case when h≡1 we retrieve the well-known inequality: ∫R |f′(x)|p  dx≤(\sqrt{p−1})p ∫R (\sqrt{|f″(x)f(x)|})p  dx. Our inequalities have a form similar to the classical second-order Opial inequalities. They also extend certain class of inequalities due to Mazya, used to obtain second-order isoperimetric inequalities and capacitary estimates. We apply them to obtain new a priori …estimates for nonlinear eigenvalue problems. Show more

Keywords: Gagliardo–Nirenberg inequalities, interpolation inequalities, nonlinear eigenvalue problems

DOI: 10.3233/ASY-2011-1079

Citation: Asymptotic Analysis, vol. 77, no. 3-4, pp. 169-196, 2012

Authors: Capdeboscq, Yves

Article Type: Research Article

Abstract: We consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk of radius ε and another one outside. We derive sharp estimates of the size of the scattered field caused by this disk inhomogeneity, for any frequencies and any contrast. We also provide a broadband estimate, that is, a uniform bound for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with ε.

Keywords: uniform scattering estimates, Helmholtz equation, Bessel function

DOI: 10.3233/ASY-2011-1080

Citation: Asymptotic Analysis, vol. 77, no. 3-4, pp. 197-246, 2012

Article Type: Other

Citation: Asymptotic Analysis, vol. 77, no. 3-4, pp. 247-247, 2012