Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain

1.1 Statement of the problem

The model thin aneurysm-type domain Ω_ε consists of three thin curvilinear cylinders

that are joined through a domain Ωε(0) (referred in the sequel "aneurysm"). Here ε is a small parameter; ℓ∈(0,13); the positive functions {hi}i=13 belong to the space C¹([0,1]) and they are equal to some constants in neighborhoods at the points x = 0 and x_i = 1, i = 1, 2, 3; the symbol δ_ij is the Kroneker delta, i.e., δ_ii = 1 and δ_ij = 0 if i ≠ j.

The aneurysm Ωε(0) (see Fig. 1) is formed by the homothetic transformation with coefficient ε from a bounded domain Ξ⁽⁰⁾ ∈ ℝ³, i.e., Ωε(0) = εΞ⁽⁰⁾. In addition, we assume that its boundary contains the disks

and denote Γε(0):=∂Ωε(0)∖{Υε(1)(εℓ)¯∪Υε(2)(εℓ)¯∪Υε(3)(εℓ)¯}.

Fig. 1

The aneurysm Ωε(0)

Thus the model thin aneurysm-type domain Ω_ε (see Fig. 2) is the interior of the union ⋃k=03Ωε(k)¯ and we assume that it has the Lipschitz boundary.

Fig. 2

The model thin aneurysm-type domain Ω_ε

1.2 Existence and uniqueness of the weak solution

In order to obtain operator statement for the problem (1) we introduce the new norm ∥⋅∥_ε in 𝓗_ε, which is generated by the scalar product

Due to the uniform Dirichlet condition on Υε(i) (1), i = 1, 2, 3, the norm ∥⋅∥_ε and the ordinary norm ∥⋅∥_H¹(Ωε) are uniformly equivalent, i. e., there exist constants C₁ > 0 and ε₀ > 0 such that for all ε ∈ (0, ε₀) and for all u ∈ 𝓗_ε the following estimates hold:

2.1 Regular part

Substituting the representation (8) for each fixed index i ∈{1, 2, 3 } into the differential equation of the problem (1), using Taylor’s formula for the function κ₀ at s=ω0(i)(xi) and the function f at the point x_i = (0,0), and collecting coefficients at ε⁰, we obtain

where ξ¯i=x¯iε and f0(i)(xi):=f(x)|x¯i=(0,0).

It is easy to calculate the outer unit normal to Γε(i) :

where ν¯i(x¯iε) is the outward normal for the disk Υε(i)(xi):={ξ¯i∈R2:|ξ¯i|<hi(xi)}.

Taking the view of the outer unit normal into account and putting the sum (8) into the third relation of the problem (1), we get with the help of Taylor’s formula for the function κ_i at s=ω0(i)(xi) the following relation:

Relations (10) and (11) form the linear inhomogeneous Neumann boundary-value problem

to define u2(i). Here 〈u(xi,⋅)〉Υi(xi):=∫Υi(xi)u(xi,ξ¯i)dξ¯i, the variable x_i is regarded as a parameter from the interval Iε(i) : = {x : x_i ∈(εℓ, 1), x_i = (0,0 We add the third relation in (12) for the uniqueness of a solution.

Writing down the necessary and sufficient conditions for the solvability of the problem (12), we derive the differential equation

to define ω0(i) (i ∈{1,2,3}).

Let ω0(i) be a solution of the differential equation (13) (the existence will be proved in the subsection 2.2.1). Thus, there exists a unique solution to the problem (12) for each i ∈{1, 2, 3 }.

For determination of the coefficients u3(i) , i = 1, 2, 3, we similarly obtain the following problems:

for each i ∈{1, 2, 3 }. Here

Repeating the previous reasoning, we find that the coefficients {ω1(i)}i=13 have to be solutions to the respective linear ordinary differential equation

2.2 Inner part

To obtain conditions for the functions {ωk(i)}, i = 1, 2, 3, k ∈{0, 1 } at the point (0,0,0), we introduce the inner part of the asymptotic approximation (9) in a neighborhood of the aneurysm Ωε(0) . If we pass to the “fast variables” ξ=xε and tend ε to 0, the domain Ω_ε is transformed into the unbounded domain Ξ that is the union of the domain Ξ⁽⁰⁾ and three semibounded cylinders

i.e., Ξ is the interior of ⋃i=03 Ξ⁽ⁱ⁾ (see Fig. 3).

Fig. 3

Domain Ξ

Let us introduce the following notation for parts of the boundary of the domain Ξ:

1. Γ_i = {ξ ∈ ℝ³: ℓ < ξ_i < + ∞, |ξ_i| = h_i(0)}, i = 1, 2, 3,

2. Γ₀ = ∂Ξ∖ ( ⋃i=13 Γ_i).

Substituting (9) into the problem (1) and equating coefficients at the same powers of ε, we derive the following relations for N_k, (k ∈{0,1,2}) :

Here

The right hand sides in the differential equation and boundary conditions on {Γ_i} of the problem (17) are obtained with the help of the Taylor decomposition of the functions f and φ⁽ⁱ⁾ at the points x = 0 and x_i = 0, i = 1, 2, 3, respectively.

The fourth condition in (17) appears by matching the regular and inner asymptotics in a neighborhood of the aneurysm, namely the asymptotics of the terms {N_k} as ξ_i →+ ∞ have to coincide with the corresponding asymptotics of the terms {ωk(i)} as x_i = εξ_i →+0, i = 1, 2, 3, respectively. Expanding formally each term of the regular asymptotics in the Taylor series at the points x_i = 0 and collecting the coefficients of the same powers of ε, we get

A solution of the problem (17) at k = 1, 2 is sought in the form

where χ_i ∈ C^∞(ℝ₊), 0 ≤ χ_i ≤ 1 and

Then N͠_k has to be a solution of the problem

where

and

In addition, we demand that N͠_k satisfies the following stabilization conditions:

The existence of a solution to the problem (20) in the corresponding energetic space can be obtained from general results about the asymptotic behavior of solutions to elliptic problems in domains with different exits to infinity (see e.g. [8, 24]). We will use approach proposed in [6, 8].

Let C0,ξ∞(Ξ¯) be a space of functions infinitely differentiable in -Ξ and finite with respect to ξ, i.e.,

We now define a space H:=(C0,ξ∞(Ξ¯),∥⋅∥H)¯, where

and the weight function ρ ∈ C^∞(ℝ³), 0 ≤ ρ ≤ 1 and