By S. L. Sobolev
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Extra resources for Applications of Functional Analysis in Mathematical Physics (Translations of Mathematical Monographs, Vol 7)
Hn−1 )), n=1 7 Note that T∗iL = AiL (H) = C(1)iLjM HjM + C(2)iLjM hN HjM HhN + · · · . 16. Signorini’s Method for Traction Problems 31 with En = (Hn + HTn )/2. 6). 142) where we have used the notations: ρ∗ bn ≡ ρ∗ bn + ∇X · Bn (H1 , . . , Hn−1 ), t∗n ≡ t∗n − Bn (H1 , . . , Hn−1 )N∗ . 144) For n = 1, the above system describes a mixed boundary value problem of linear elasticity with loads of b1 = b1 and t∗1 = t1 . 144) has been solved for n = 1, . . , m − 1. Then, for n = m, we have a new mixed boundary value problem for the same material and the same domain with external loads that depend in a known way on the displacements u1 , .
To prove this, we make use of the inverse mapping theorem. 209) where H = ∇X u(X) and u(X) is an arbitrary displacement ﬁeld such that u(X) = 0 on ∂C∗ − ∂C∗ . We denote by F the vector space of the functions u(X), which are suitably regular in X ∈ C∗ and vanish on ∂C∗ − ∂C∗ . We also suppose that it is possible to introduce a norm u(X) into F such that F becomes a Banach space. Similarly, we consider the Banach space F of the pairs (h(X), g(X)), as equipped with a convenient norm (h(X), g(X)) .
269) Singular Waves in Nonlinear Elasticity Let S be an elastic continuous system that is homogeneous, compressible, and isotropic in the reference conﬁguration C∗ . If we denote the mass density in the actual conﬁguration C(t) by ρ, the displacement ﬁeld from C∗ to C(t) by u(X, t), the Cauchy stress tensor by T, and the speciﬁc body force by b, the Eulerian local momentum balance is expressed by the following equation: ρ¨ u = ∇ · T + ρb. 270) If H = ∇u is the displacement gradient, let T(H) be the constitutive equation of the elastic material S.