Asymptotic Analysis - Volume 2, issue 4

Authors: Ciarlet, Philippe G. | Le Dret, Hervé

Article Type: Research Article

Abstract: On considère un problème d'élasticité linéarisée tridimensionnelle, posé sur un domaine constitué d'une plaque d'épaisseur 2ε, insérée dans un support tridimensionnel élastique. En supposant que les constantes de Lamé du matériau constituant la plaque varient comme ε−3 et que celles du support varient comme ε−2−s , s>0, on établit la convergence en norme H1 , lorsque ε tend vers zéro, des composantes (convenablement mises à l'échelle) du champ de déplacements vers la solution du problème aux limites bien connu pour une plaque encastrée. De la sorte, on montre que les conditions aux limites classiques pour l'encastrement d'une plaque peuvent …être obtenus de façon rigoureuse par une “analyse limite” qui prend également en compte le caractère élastique du support. Show more

DOI: 10.3233/ASY-1989-2401

Citation: Asymptotic Analysis, vol. 2, no. 4, pp. 257-277, 1989

Authors: Kinderlehrer, David | Vergara-Caffarelli, Giorgio

Article Type: Research Article

Abstract: The relaxation of a variational principle is determined when a superficial term is present. Lower semicontinuity of the functional may fail even when both integrands are convex. Generalized solutions, in terms of parametrized measures or Young measures, are introduced and analyzed. Several examples are given.

DOI: 10.3233/ASY-1989-2402

Citation: Asymptotic Analysis, vol. 2, no. 4, pp. 279-298, 1989

Authors: Duzaar, Frank | Steffen, Klaus

Article Type: Research Article

Abstract: For a Riemannian manifold M with boundary, a Riemannian manifold N, an open subset Σ≠∅ of ∂M and a submanifold S of N, we prove that every map u:M→N of class H1,2 which is energy minimizing with respect to the free boundary condition u(Σ)⊂S is regular on an open subset Reg(u) of M∪Σ whose complement Sing(u)=(M∪Σ)\Reg(u) has vanishing m−2 dimensional measure. The result is new for the singular set Σ∩Sing(u) contained in the free boundary Σ. There are no restrictions on the image of u in N and no dimensional restrictions on M, N and S. The proof …is based on a method introduced by R. Schoen and K. Uhlenbeck in the interior partial regularity theory for harmonic mappings, a novel partial reflection construction ũ extending u and a precise control of the local averages ur,a , ũr,a on balls Br,a with radius r>0 and center a∈Σ in terms of the small normalized energy ε of u on Br,a . The crucial estim ate is dist(ur,a ,S)$\lesssim$ ε1/2 and dist(ũr,a ,S)$\lesssim$ ε2/3 as ε↓0. Show more

DOI: 10.3233/ASY-1989-2403

Citation: Asymptotic Analysis, vol. 2, no. 4, pp. 299-343, 1989

Authors: Stengle, Gilbert

Article Type: Research Article

Abstract: We consider systems of linear differential equations of the form εP dy/dx=A(k,ε)y where A is an n×n matrix-valued function holomorphic in x possessing an asymptotic power series expansion in ε, A(k,ε)≈Σk Ak (x)εk uniformly valid on a neighborhood |x|<r of x=0 as ε→0+ . We show that any such system can be reduced by a change of variable y→Ty to a certain generic canonical form depending on the Jordan canonical form of A(0, 0). Here T(x, ε) is an invertible matrix-valued function similarly possessing an asymptotic expansion T≈Σk Tk (x)εk uniformly valid on an ε-dependent domain |x|<rδ(ε) the …size of which is determined by certain finer properties of the asymptotic expansion of A. We also obtain new results of classical type in which this domain is independent of ε. The method of proof seems both new and of general purpose. It is based on an exacting analysis of finer termwise properties of divergent infinite formal series transformations which shows certain near-optimal finite sections to be powerful first approximations to actual transformations. Show more

DOI: 10.3233/ASY-1989-2404

Citation: Asymptotic Analysis, vol. 2, no. 4, pp. 345-355, 1989

Article Type: Other

Citation: Asymptotic Analysis, vol. 2, no. 4, pp. 357-357, 1989