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

Abstract

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.

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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 problem) has been shown to be ill-posed not only for the wave equation but for hyperbolic PDEs more generally (see [2, 3]). Some progress was done in [4] which 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.

Let Ω_𝛼 be the cylindric domain of the Euclidean space 𝐸_𝑚+1 of points (𝑥₁,…,𝑥_𝑚,𝑡), bounded by the cylinder Γ={(𝑥,𝑡)∶|𝑥|=1}, the planes 𝑡=𝛼>0 and 𝑡=0, where |𝑥| is the length of the vector 𝑥=(𝑥₁,…,𝑥_𝑚).

Let us denote, respectively, with Γ_𝛼,𝑆_𝛼, and 𝑆₀ the parts of these surfaces that form the boundary 𝜕Ω_𝛼 of the domain Ω_𝛼.

We study, in the domain Ω_𝛼, the multidimensional wave equationΔ_𝑥𝑢−𝑢_𝑡𝑡=0, (1) where Δ_𝑥 is the Laplace operator on the variables 𝑥₁,…,𝑥_𝑚,𝑚≥2.

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

Problem 1 (Dirichlet). Find the solution of (1) in the domain Ω_𝛼, in the class 𝐶(Ω_𝛼)∩𝐶²(Ω_𝛼), that satisfies the following boundary-value conditions: 𝑢|_𝑆𝛼=𝜑(𝑟,𝜃),𝑢|_Γ𝛼=𝜓(𝑡,𝜃),𝑢|_𝑆0=𝜏(𝑟,𝜃). (2)

Let {𝑌^𝑘_𝑛,𝑚(𝜃)} be a system of linearly independent spherical functions of order 𝑛,1≤𝑘≤𝑘_𝑛,(𝑚−2)!𝑛!𝑘_𝑛=(𝑛+𝑚−3)!(2𝑛+𝑚−2),and let 𝑊^𝑙₂(𝑆₀),𝑙=0,1,… be Sobolev spaces.

The following lemmata hold ([6]).

Lemma 1. Let 𝑓(𝑟,𝜃)∈𝑊^𝑙₂(𝑆₀). If 𝑙≥𝑚−1, then the series𝑓(𝑟,𝜃)=^∞𝑘_𝑛=0^𝑛_𝑘=1𝑓^𝑘_𝑛(𝑟)𝑌^𝑘_𝑛,𝑚(𝜃), (3) as well as the series obtained through its differentiation of order 𝑝≤𝑙−𝑚+1, converge absolutely and uniformly.

Lemma 2. For 𝑓(𝑟,𝜃)∈𝑊^𝑙₂(𝑆₀), it is necessary and sufficient that the coefficients of the series (3) satisfy the inequalities |𝑓¹₀(𝑟)|≤𝑐₁,^∞𝑘_𝑛=0^𝑛_𝑘=1𝑛^2𝑙|𝑓^𝑘_𝑛(𝑟)|²≤𝑐₂,𝑐₁,𝑐₂=const. (4)

Let's denote as 𝜑^𝑘_𝑛(𝑟),𝜓^𝑘_𝑛(𝑡), and 𝜏^𝑘_𝑛(𝑟) the coefficients of the series (3), respectively, of the functions 𝜑(𝑟,𝜃),𝜓(𝑡,𝜃), and 𝜏(𝑟,𝜃).

Theorem 3. If 𝜑(𝑟,𝜃)∈𝑊^𝑙₂(𝑆_𝛼),𝜓(𝑡,𝜃)∈𝑊^𝑙₂(Γ_𝛼),𝜏(𝑟,𝜃)∈𝑊^𝑙₂(𝑆₀),𝑙>3𝑚/2, andsin𝜇_𝑠𝛼≠0,𝑠=1,2,…, (5) then Problem 1 is uniquely solvable, where 𝜇_𝑠 are the positive nulls of the Bessel function of first type 𝐽_{𝑛+(𝑚−2)/2}(𝑧).

Theorem 4. The solution of Problem 1 is unique if and only if condition (5) is satisfied.

Proof of Theorem 3. In the spherical coordinates, (1) takes the form 𝑢_𝑟𝑟+𝑚−1𝑟𝑢_𝑟−1𝑟²𝛿𝑢−𝑢_𝑡𝑡=0,𝛿≡−^𝑚−1_𝑗=11𝑔_𝑗sin^{𝑚−𝑗−1}𝜃_𝑗𝜕𝜕𝜃_𝑗(sin^{𝑚−𝑗−1}𝜃_𝑗𝜕𝜕𝜃_𝑗),𝑔₁=1,𝑔_𝑗=(sin𝜃₁⋯sin𝜃_𝑗−1)²,𝑗>1. (6)
It is known (see [6]) that the spectrum of the operator 𝛿 consists of eigenvalues 𝜆_𝑛=𝑛(𝑛+𝑚−2),𝑛=0,1,…, to each of which correspond 𝑘_𝑛 orthonormalized eigenfunctions 𝑌^𝑘_𝑛,𝑚(𝜃).
Given that solution of the problem that we are looking for belongs to the class 𝐶(Ω_𝛼)∩𝐶²(Ω_𝛼), we can look for it in the form of the series 𝑢(𝑟,𝜃,𝑡)=^∞𝑘_𝑛=0^𝑛_𝑘=1𝑢^𝑘_𝑛(𝑟,𝑡)𝑌^𝑘_𝑛,𝑚(𝜃), (7) where 𝑢^𝑘_𝑛(𝑟,𝑡) are the functions to be determined.
Substituting (7) into (6) and using the orthogonality of the spherical functions 𝑌^𝑘_𝑛,𝑚(𝜃) ([6]), we get 𝑢^𝑘_𝑛𝑟𝑟+𝑚−1𝑟𝑢^𝑘_𝑛𝑟−𝑢^𝑘_𝑛𝑡𝑡−𝜆_𝑛𝑟²𝑢^𝑘_𝑛=0,𝑘=1,𝑘_𝑛,𝑛=0,1,…, (8) and given this, the boundary-value conditions (2), taking into account Lemma 1, will take the form 𝑢^𝑘_𝑛(𝑟,0)=𝜏^𝑘_𝑛(𝑟),𝑢^𝑘_𝑛(𝑟,𝛼)=𝜑^𝑘_𝑛(𝑟),𝑢^𝑘_𝑛(1,𝑡)=𝜓^𝑘_𝑛(𝑡),𝑘=1,𝑘_𝑛,𝑛=0,1,…. (9) In (8) and (9), making the substitution of variables 𝜗^𝑘_𝑛(𝑟,𝑡)=𝑢^𝑘_𝑛(𝑟,𝑡)−𝜓^𝑘_𝑛(𝑡), (10) we get 𝜗^𝑘_𝑛𝑟𝑟+𝑚−1𝑟𝜗^𝑘_𝑛𝑟−𝜗^𝑘_𝑛𝑡𝑡−𝜆_𝑛𝑟²𝜗^𝑘_𝑛=𝑓^𝑘_𝑛(𝑟,𝑡),𝜗^𝑘_𝑛(𝑟,0)=𝜏^𝑘_𝑛(𝑟),𝜗^𝑘_𝑛(𝑟,𝛼)=𝜑^𝑘_𝑛(𝑟),𝜗^𝑘_𝑛(1,𝑡)=0,𝑘=1,𝑘_𝑛,𝑛=0,1,…,𝑓^𝑘_𝑛(𝑟,𝑡)=𝜓^𝑘_𝑛𝑡𝑡+𝜆_𝑛𝑟²𝜓^𝑘_𝑛,𝜏^𝑘_𝑛(𝑟)=𝜏^𝑘_𝑛(𝑟)−𝜓^𝑘_𝑛(0),𝜑^𝑘_𝑛(𝑟)=𝜑^𝑘_𝑛(𝑟)−𝜓^𝑘_𝑛(𝛼). (11)
Making the substitution of the variable 𝜗^𝑘_𝑛(𝑟,𝑡)=𝑟^(1−𝑚)/2𝜗^𝑘_𝑛(𝑟,𝑡), we can reduce the problem (11) to the following problem 𝐿𝜗^𝑘_𝑛≡𝜗^𝑘_𝑛𝑟𝑟−𝜗^𝑘_𝑛𝑡𝑡+𝜆_𝑛𝑟²𝜗^𝑘_𝑛=𝑓^𝑘_𝑛𝜗(𝑟,𝑡),_𝑘𝑛(𝑟,0)=𝜏^𝑘_𝑛(𝑟),𝜗^𝑘_𝑛(𝑟,𝛼)=𝜑^𝑘_𝑛(𝑟),𝜗^𝑘_𝑛(1,𝑡)=0,𝜆_𝑛=(𝑚−1)(3−𝑚)−4𝜆_𝑛4,𝑓^𝑘_𝑛(𝑟,𝑡)=𝑟^(1−𝑚)/2𝑓^𝑘_𝑛𝜏(𝑟,𝑡),_𝑘𝑛(𝑟)=𝑟^(1−𝑚)/2𝜏^𝑘_𝑛(𝑟),𝜑^𝑘_𝑛(𝑟)=𝑟^(1−𝑚)/2𝜑^𝑘_𝑛(𝑟). (12)
We look for the solution of the problem (12) in the form 𝜗^𝑘_𝑛(𝑟,𝑡)=𝜗^𝑘_1𝑛(𝑟,𝑡)+𝜗^𝑘_2𝑛(𝑟,𝑡), where 𝜗^𝑘_1𝑛(𝑟,𝑡) is the solution of the problem 𝐿𝜗^𝑘_1𝑛=𝑓^𝑘_𝑛𝜗(𝑟,𝑡),_𝑘1𝑛(𝑟,0)=0,𝜗^𝑘_1𝑛(𝑟,𝛼)=0,𝜗^𝑘_1𝑛(1,𝑡)=0 (13) whereas 𝜗^𝑘_2𝑛(𝑟,𝑡) is the solution of the problem 𝐿𝜗^𝑘_2𝑛𝜗=0,_𝑘2𝑛(𝑟,0)=𝜏^𝑘_𝑛(𝑟),𝜗^𝑘_2𝑛(𝑟,𝛼)=𝜑^𝑘_𝑛(𝑟),𝜗^𝑘_2𝑛(1,𝑡)=0. (14)
We analyze the solutions of the above problems, analogously to [7], in the form 𝜗^𝑘_𝑛(𝑟,𝑡)=^∞_𝑠=1𝑅_𝑠(𝑟)𝑇_𝑠(𝑡); (15) moreover, let 𝑓^𝑘_𝑛(𝑟,𝑡)=^∞_𝑠=1𝑎_𝑠(𝑡)𝑅_𝑠(𝑟),𝜏^𝑘_𝑛(𝑟)=^∞_𝑠=1𝑏_𝑠𝑅_𝑠(𝑟),𝜑^𝑘_𝑛(𝑟)=^∞_𝑠=1𝑑_𝑠𝑅_𝑠(𝑟). (16)
Substituting (15) into (13) and taking into account (16), we get 𝑅_𝑠𝑟𝑟+𝜆_𝑛𝑟²𝑅_𝑠+𝜇𝑅_𝑠𝑅=0,0<𝑟<1,_𝑠(1)=0,|𝑅_𝑠𝑇(0)|<∞,_𝑠𝑡𝑡+𝜇𝑇_𝑠=−𝑎_𝑠𝑇(𝑡),0<𝑡<𝛼,_𝑠(0)=𝑇_𝑠(𝛼)=0. (17)(18)(19)(20)
The bounded solution of the problems (17) and (18) is (see [8]) 𝑅_𝑠(𝑟)=√𝑟𝐽_𝜐(𝜇_𝑠𝑟), (21) where 𝜐=𝑛+(𝑚−2)/2,𝜇=𝜇²_𝑠.
The general solution of (19) can be represented in the form (see [8]) 𝑇_𝑠(𝑡)=𝑐_1𝑠cos𝜇_𝑠𝑡+𝑐_2𝑠sin𝜇_𝑠𝑡+cos𝜇_𝑠𝑡𝜇_𝑠^𝑡₀𝑎_𝑠(𝜉)sin𝜇_𝑠𝜉𝑑𝜉−sin𝜇_𝑠𝑡𝜇_𝑠^𝑡₀𝑎_𝑠(𝜉)cos𝜇_𝑠𝜉𝑑𝜉, (22) where 𝑐_1𝑠 and 𝑐_2𝑠 are arbitrary constants; satisfying the condition (20), we will get 𝑐_1𝑠𝑐=0,_2𝑠𝜇_𝑠sin𝜇𝛼=−cos𝜇_𝑠𝛼^𝛼₀𝑎_𝑠(𝜉)sin𝜇_𝑠𝜉𝑑𝜉−sin𝜇_𝑠𝛼^𝛼₀𝑎_𝑠(𝜉)cos𝜇_𝑠𝜉𝑑𝜉. (23)
Substituting (21) into (16), we get 𝑟^−1/2𝑓^𝑘_𝑛(𝑟,𝑡)=^∞_𝑠=1𝑎_𝑠(𝑡)𝐽_𝜐(𝜇_𝑠𝑟),𝑟^−1/2𝜏^𝑘_𝑛(𝑟)=^∞_𝑠=1𝑏_𝑠𝐽_𝜐(𝜇_𝑠𝑟𝑟),_−1/2𝜑^𝑘_𝑛(𝑟)=^∞_𝑠=1𝑑_𝑠𝐽_𝜐(𝜇_𝑠𝑟),0<𝑟<1. (24)
Series (24) are the decompositions into the Fourier-Bessel series (see [9]), if 𝑎_𝑠2(𝑡)=[𝐽_𝜐+1(𝜇_𝑠)]²¹⁰𝜉𝑓^𝑘_𝑛(𝜉,𝑡)𝐽_𝜐(𝜇_𝑠𝑏𝜉)𝑑𝜉,_𝑠=2[𝐽_𝜐+1(𝜇_𝑠)]²¹⁰𝜉𝜏^𝑘_𝑛(𝜉)𝐽_𝜐(𝜇_𝑠𝜉)𝑑𝜉,𝑑_𝑠=2[𝐽_𝜐+1(𝜇_𝑠)]²¹⁰𝜉𝜑^𝑘_𝑛(𝜉)𝐽_𝜐(𝜇_𝑠𝜉)𝑑𝜉, (25)(26)𝜇_𝑠,𝑠=1,2,… are positive nulls of the Bessel functions, set in the increasing order.
From (21)–(23) we get the solution of the problem (13): 𝜗^𝑘_1𝑛(𝑟,𝑡)=^∞_𝑠=1√𝑟𝜇_𝑠{[^𝛼₀𝑎_𝑠(𝜉)cos𝜇_𝑠𝜉𝑑𝜉−cot𝜇_𝑠𝛼^𝛼₀𝑎_𝑠(𝜉)sin𝜇_𝑠𝑑𝜉]sin𝜇_𝑠𝑡+cos𝜇_𝑠𝑡^𝑡₀𝑎_𝑠(𝜉)sin𝜇_𝑠𝜉𝑑𝜉−sin𝜇_𝑠𝑡^𝑡₀𝑎_𝑠(𝜉)cos𝜇_𝑠𝑑𝜉}𝐽_𝜐(𝜇_𝑠𝑟), (27) where 𝑎_𝑠(𝑡) is determined from (25).
Next, substituting (15) into (14) and taking into account (16), we will get 𝑇_𝑠𝑡𝑡+𝜇²_𝑠𝑇_𝑠𝑇=0,0<𝑡<𝛼,_𝑠(0)=𝑏_𝑠,𝑇_𝑠(𝛼)=𝑑_𝑠. (28)(29)
The general solution of (28) will become 𝑇_𝑠(𝑡)=𝑐^′_1𝑠cos𝜇_𝑠𝑡+𝑐^′_2𝑠sin𝜇_𝑠𝑡; (30) satisfying the condition (29), we will get 𝑐^′_1𝑠=𝑏_𝑠,𝑐^′_2𝑠=𝑑_𝑠sin𝜇_𝑠𝛼−𝑏_𝑠cot𝜇_𝑠𝛼. (31)
From (21), (30), and (31) we find the solution of the problem (14): 𝜗^𝑘_2𝑛(𝑟,𝑡)=^∞_𝑠=1√𝑟[𝑏_𝑠cos𝜇_𝑠𝑑𝑡−(^𝑠sin𝜇_𝑠𝛼−𝑏_𝑠cot𝜇_𝑠𝛼)sin𝜇_𝑠𝑡]𝐽_𝜐(𝜇_𝑠𝑟), (32) where 𝑏_𝑠 and 𝑑_𝑠 are found from (26).
Thus, the unique solution of Problem 1 is the function 𝑢(𝑟,𝜃,𝑡)=^∞𝑘_𝑛=0^𝑛_𝑘=1{𝜓^𝑘_𝑛(𝑡)+𝑟^(1−𝑚)/2[𝜗^𝑘_1𝑛(𝑟,𝑡)+𝜗^𝑘_2𝑛(𝑟,𝑡)]}𝑌^𝑘_𝑛,𝑚(𝜃),𝑡>0, (33) where 𝜗^𝑘_1𝑛(𝑟,𝑡) and 𝜗^𝑘_2𝑛(𝑟,𝑡) are determined from (27) and (32).
Taking into account the formula (see [9]) 𝐽^′_𝜐(𝑧)=𝐽_𝜐−1(𝑧)+𝐽_𝜐+1(𝑧), the estimates (see [6, 9]) |𝐽_𝜐1(𝑧)|≤(𝑧Γ(1+𝜐)2)^𝜐,𝑘_𝑛≤𝑐₁𝑛^𝑚−2,|𝜕^𝑞𝜕𝜃^𝑞_𝑗𝑌^𝑘_𝑛,𝑚(𝜃)|≤𝑐₂𝑛^{𝑚/2−1+𝑞},𝑗=1,𝑚−1,𝑞=0,1,…, (34) where Γ(𝑧) is the gamma-function, the lemmata, and the bounds on the given functions 𝜑(𝑟,𝜃),𝜓(𝑡,𝜃), and 𝜏(𝑟,𝜃), we can show that the obtained solution (33) belongs to the class 𝐶(Ω_𝛼)∩𝐶²(Ω_𝛼).
Theorem 3 is proven.

Proof of Theorem 4. If condition (5) is satisfied, then from Theorem 3, it follows that the solution of Problem 1 is unique.
Now, suppose condition (5) does not hold, at least for one 𝑠=1.
Then, if we look for the solution of the homogeneous problem, corresponding to Problem 1, in the form (7), then we get to the problem 𝐿𝜗^𝑘_𝑛𝜗=0,_𝑘𝑛(𝑟,0)=0,𝜗^𝑘_𝑛(𝑟,𝛼)=0,𝜗^𝑘_𝑛(1,𝑡)=0,𝑘=1,𝑘_𝑛,𝑛=0,1,…, (35) the solution of which is the function 𝜗^𝑘_𝑛(𝑟,𝑡)=√𝑟sin𝜇_𝑙𝑡𝐽_{𝑛+(𝑚−2)/2}(𝜇_𝑙𝑟). (36)
Therefore, the nontrivial solution of homogeneous Problem 1 is written as 𝑢(𝑟,𝜃,𝑡)=^∞𝑘_𝑛=2^𝑛_𝑘=1𝑛^−𝑙𝑟^(2−𝑚)/2sin𝜇_𝑙𝑡𝐽_{𝑛+(𝑚−2)/2}(𝜇_𝑙𝑟)𝑌^𝑘_𝑛,𝑚(𝜃). (37)
From estimates (34) it follows that 𝑢∈𝐶(Ω_𝛼)∩𝐶²(Ω_𝛼), if 𝑙>3𝑚/2.