The Well-Posedness of the Dirichlet Problem in the Cylindric Domain for the Multidimensional Wave Equation

Mathematical Problems in Engineering Volume 2010, Article ID 653215,7pages doi:10.1155/2010/653215

Research Article

The Well-Posedness of the Dirichlet Problem in the Cylindric Domain for the Multidimensional Wave Equation

Serik A. Aldashev

Aktobe State University, AGU, Br. Zhubanov Str 263, Aktobe 030000, Kazakhstan

Correspondence should be addressed to Serik A. Aldashev,[email protected] Received 26 October 2009; Accepted 29 April 2010

Academic Editor: Carlo Cattani

Copyrightq2010 Serik A. Aldashev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the theory of hyperbolic PDEs, the boundary-value problems with conditions on the entire boundary of the domain serve typically as the examples of the ill-posedness. The paper shows the unique solvability of the Dirichlet problem in the cylindric domain for the multidimensional wave equation. We also establish the criterion for the unique solvability of the equation.

One of the fundamental problems of mathematical physics—the analysis of the behavior of the vibrating string—has been shown to be ill-posed when the boundary-value conditions are defined on the entire boundary 1. Furthermore, this problem known as Dirichlet problemhas been shown to be ill-posed not only for the wave equation but for hyperbolic PDEs more generallysee2,3. Some progress was done in4which showed that for some rectangles the solution of this problem existed under sufficient differentiability conditions.

Further analyses of this problem reverted to functional analysis methods see, e.g., 5, which has the serious shortcoming of making the applications of such results in physics and engineering highly difficult. Moreover, most studies have concentrated so far on the 2D wave equation.

This paper studies the Dirichlet problem, using the classical methods, in the cylindric domain for the multidimensional wave equation. We show that the problem is well-posed.

We also establish the criterion for the unique solvability of the problem.

LetΩα be the cylindric domain of the Euclidean space E_{m 1} of pointsx1, . . . , x_m, t, bounded by the cylinderΓ {x, t:|x|1}, the planestα >0 andt0, where|x|is the length of the vectorx x1, . . . , xm.

Let us denote, respectively, withΓα, Sα, andS0the parts of these surfaces that form the boundary∂Ωαof the domainΩα.

We study, in the domainΩα, the multidimensional wave equation

Δxu−utt0, 1

whereΔxis the Laplace operator on the variablesx1, . . . , xm, m≥2.

Hereafter, it is useful to move from the Cartesian coordinates x₁, . . . , x_m, t to the spherical onesr, θ₁, . . . , θ_m, t, r≥0, 0≤θ₁<2π, 0≤θ_i≤π, i2,3, . . . , m−1.

Problem 1Dirichlet. Find the solution of1in the domainΩα, in the classCΩα∩C²Ωα, that satisfies the following boundary-value conditions:

u|_Sα ϕr, θ, u|_Γα ψt, θ, u|_S0 τr, θ. 2

Let{Y_n,m^k θ}be a system of linearly independent spherical functions of ordern, 1 ≤ k≤k_n,m−2!n!kn n m−3!2n m−2,and letW₂^lS0, l0,1, . . .be Sobolev spaces.

The following lemmata hold6.

Lemma 1. Letfr, θ∈W₂^lS0. Ifl≥m−1, then the series

fr, θ ^∞

n0 kn

k1

f_n^krY_n,m^k θ, 3

as well as the series obtained through its differentiation of orderp≤l−m 1, converge absolutely and uniformly.

Lemma 2. Forfr, θ∈W₂^lS0, it is necessary and sufficient that the coefficients of the series3 satisfy the inequalities

f₀¹r≤c1, ∞ n0

kn

k1

n^2lf_n^kr²≤c2, c1, c2 const. 4 Let’s denote asϕ^k_nr, ψ_n^kt,andτ^k_nrthe coefficients of the series3, respectively, of the functionsϕr, θ, ψt, θ,andτr, θ.

Theorem 3. Ifϕr, θ∈W₂^lSα, ψt, θ∈W₂^lΓα, τr, θ∈W₂^lS0, l >3m/2, and

sinμ_sα /0, s1,2, . . . , 5

then Problem1is uniquely solvable, whereμ_sare the positive nulls of the Bessel function of first type J_{n m−2/2}z.

Theorem 4. The solution of Problem1is unique if and only if condition5is satisfied.

Proof ofTheorem 3. In the spherical coordinates,1takes the form

urr

m−1 r ur− 1

r²δu−utt0, δ≡ −^m−1

j1

1 g_jsin^m−j−1θ_j

∂

∂θj

sin^m−j−1θ_j ∂

∂θj

, g₁1, g_j

sinθ₁· · ·sinθ_j−12 , j >1.

6

It is knownsee6that the spectrum of the operatorδconsists of eigenvaluesλ_n nn m−2, n 0,1, . . . ,to each of which correspondk_n orthonormalized eigenfunctions Y_n,m^k θ.

Given that solution of the problem that we are looking for belongs to the classCΩα∩ C²Ωα, we can look for it in the form of the series

ur, θ, t ^∞

n0 kn

k1

u^k_nr, tY_n,m^k θ, 7

whereu^k_nr, tare the functions to be determined.

Substituting7into6and using the orthogonality of the spherical functionsY_n,m^k θ 6, we get

u^k_nrr m−1

r u^k_nr−u^k_ntt−λ_n

r²u^k_n0, k1, k_n, n0,1, . . . , 8 and given this, the boundary-value conditions 2, taking into accountLemma 1, will take the form

u^k_nr,0 τ^k_nr, u^k_nr, α ϕ^k_nr, u^k_n1, t ψ_n^kt, k1, kn, n0,1, . . . . 9

In8and9, making the substitution of variables

ϑ^k_nr, t u^k_nr, t−ψ^k_nt, 10 we get

ϑ^k_nrr m−1

r ϑ^k_nr−ϑ^k_ntt−λ_n

r²ϑ^k_nf^k_nr, t,

ϑ^k_nr,0 τ^k_nr, ϑ^k_nr, α ϕ^k_nr, ϑ^k_n1, t 0, k1, kn, n0,1, . . . , f^k_nr, t ψ_ntt^k λ_n

r²ψ^k_n, τ_n^kr τ^k_nr−ψ_n^k0, ϕ^k_nr ϕ^k_nr−ψ_n^kα.

11

Making the substitution of the variableϑ^k_nr, t r^1−m/2ϑ^k_nr, t, we can reduce the problem11to the following problem

Lϑ^k_n≡ϑ_nrr^k −ϑ^k_ntt λ_n

r²ϑ^k_nf_n^kr, t,

ϑ^k_nr,0 τ_n^kr, ϑ^k_nr, α ϕ^k_nr, ϑ^k_n1, t 0, λ_n m−13−m−4λ_n

4 , f_n^kr, t r^1−m/2f^k_nr, t,

τ_n^kr r^1−m/2τ_n^kr, ϕ^k_nr r^1−m/2ϕ^k_nr.

12

We look for the solution of the problem12in the formϑ^k_nr, t ϑ_1n^k r, t ϑ^k_2nr, t, whereϑ_1n^kr, tis the solution of the problem

Lϑ^k_1nf_n^kr, t,

ϑ^k_1nr,0 0, ϑ_1n^k r, α 0, ϑ_1n^k1, t 0

13

whereasϑ^k_2nr, tis the solution of the problem Lϑ^k_2n0,

ϑ^k_2nr,0 τ_n^kr, ϑ^k_2nr, α ϕ^k_nr, ϑ^k_2n1, t 0. 14 We analyze the solutions of the above problems, analogously to7, in the form

ϑ^k_nr, t ^∞

s1

RsrTst; 15

moreover, let

f_n^kr, t ^∞

s1

astRsr, τ_n^kr ^∞

s1

bsRsr, ϕ^k_nr ^∞

s1

dsRsr. 16

Substituting15into13and taking into account16, we get

Rsrr λn

r²Rs μRs0, 0< r <1, 17

R_s1 0, |Rs0|<∞, 18

Tstt μTs−ast, 0< t < α, 19

Ts0 Tsα 0. 20

The bounded solution of the problems17and18issee8 Rsr √

rJυ

μsr

, 21

whereυn m−2/2, μμ²_s.

The general solution of19can be represented in the formsee8

Tst c1scosμst c2ssinμst cosμst μ_s

t 0

asξsinμsξ dξ−sinμst μ_s

t 0

asξcosμsξ dξ, 22

wherec_1sandc_2sare arbitrary constants; satisfying the condition20, we will get

c1s0, c_2sμ_ssinμα−cosμ_sα

α 0

a_sξsinμ_sξ dξ−sinμ_sα

α 0

a_sξcosμ_sξ dξ. 23

Substituting21into16, we get

r^−1/2f_n^kr, t ^∞

s1

a_stJυ

μ_sr

, r^−1/2τ_n^kr ^∞

s1

b_sJ_υ μ_sr

,

r^−1/2ϕ^k_nr ^∞

s1

d_sJ_υ μ_sr

, 0< r <1.

24

Series24are the decompositions into the Fourier-Bessel seriessee9, if

a_st 2

J_{υ 1}μs₂

1 0

ξf_n^kξ, tJυ

μ_sξ

dξ, 25

b_s 2

J_{υ 1}μs₂

1 0

ξτ_n^kξJυ

μ_sξ

dξ, d_s 2

J_{υ 1}μs₂

1 0

ξϕ^k_nξJυ

μ_sξ

dξ, 26

μ_s, s1,2, . . .are positive nulls of the Bessel functions, set in the increasing order.

From21–23we get the solution of the problem13:

ϑ^k_1nr, t ^∞

s1

√r μs

α 0

a_sξcosμ_sξdξ−cotμ_sα

α 0

a_sξsinμ_sdξ

sinμ_st

cosμst

t 0

asξsinμsξdξ−sinμst

t 0

asξcosμsdξ

Jυ

μsr ,

27

wherea_stis determined from25.

Next, substituting15into14and taking into account16, we will get

T_stt μ²_sT_s0, 0< t < α, 28

T_s0 b_s, T_sα d_s. 29

The general solution of28will become

T_st c_1scosμ_st c_2ssinμ_st; 30

satisfying the condition29, we will get

c_1sbs, c_2s ds

sinμ_sα−bscotμsα. 31

From21,30, and31we find the solution of the problem 14:

ϑ_2n^kr, t ^∞

s1

√r

bscosμst− d_s

sinμsα−bscotμsα

sinμst

Jυ

μsr

, 32

wherebsanddsare found from26.

Thus, the unique solution of Problem1is the function

ur, θ, t ^∞

n0 kn

k1

ψ_n^kt r^1−m/2

ϑ^k_1nr, t ϑ^k_2nr, t

Y_n,m^k θ, t >0, 33

whereϑ_1n^kr, tandϑ_2n^k r, tare determined from27and32.

Taking into account the formulasee9J_υz J_υ−1z J_{υ 1}z,the estimatessee 6,9

|Jυz| ≤ 1 Γ1 υ

z 2

υ

, kn≤c1n^m−2,

∂^q

∂θ_j^qY_n,m^k θ

≤c2n^{m/2−1 q}, j 1, m−1, q0,1, . . . ,

34

where Γz is the gamma-function, the lemmata, and the bounds on the given functions ϕr, θ, ψt, θ,andτr, θ,we can show that the obtained solution33belongs to the class CΩα∩C²Ωα.

Theorem 3is proven.

Proof ofTheorem 4. If condition 5 is satisfied, then from Theorem 3, it follows that the solution of Problem1is unique.

Now, suppose condition5does not hold, at least for ones1.

Then, if we look for the solution of the homogeneous problem, corresponding to Problem1, in the form7, then we get to the problem

Lϑ^k_n0,

ϑ^k_nr,0 0, ϑ_n^kr, α 0, ϑ^k_n1, t 0, k1, k_n, n0,1, . . . ,

35

the solution of which is the function ϑ_n^kr, t √

rsinμltJ_{n m−2/2} μlr

. 36

Therefore, the nontrivial solution of homogeneous Problem1is written as

ur, θ, t ^∞

n2 kn

k1

n^−lr^2−m/2sinμ_ltJ_{n m−2/2} μ_lr

Y_n,m^k θ. 37

From estimates34it follows thatu∈CΩα∩C²Ωα,ifl >3m/2.

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