Iteration method for the Dirichlet problem of the elasticity theory

A A rough estimate of λ 1

It is evident that the operators (2.20) and (3.15) have the same sequence of eigenvalues, in particular, they have the same greatest eigenvalue λ 1 (see Remark 2.9). Since λ 1 equals the norm of the corresponding operator (see, e.g., [10, Chapter 6]), it follows that the norms of the operators (2.20) and (3.15) equal to each other. To choose an explicit bound for the parameter τ in the iteration relation (3.2), one needs at least a rough estimate of the eigenvalue λ 1 . To obtain such a rough estimate in the case of a Lipschitz surface S, we consider operator (3.15) with

and estimate its L 2 -norm 𝐇 = ∥ - ℋ ∥ [ L 2 ⁢ ( S ) ] 3 → [ L 2 ⁢ ( S ) ] 3 .

Let

Evidently, there is a positive constant C 1 , depending only on the material parameters of the elastic body under consideration, such that the entries of the fundamental solution matrix satisfy the inequalities

For example, one such constant is (see (2.3))

In the case of the fundamental solution matrix of the Lamé system, in view of the inequality μ > 0 and formulas (2.10) and (2.11), we find

From (A.1) and (A.2) we have

Using the Cauchy–Schwartz inequality, we derive

where

By Fubini’s theorem, from (A.3) and (A.4) we get

i.e.,

Consequently,

It is evident that for particular elastic material parameters and particular Lipschitz surfaces the constants C 1 and C 2 can be explicitly calculated, which gives possibility to choose the parameter τ efficiently satisfying inequality (3.3).