The Dirichlet Problem for the Equation Δ𝑢−𝑘²𝑢=0 in the Exterior of Nonclosed Lipschitz Surfaces

Abstract

We study the Dirichlet problem for the equation Δ𝑢−𝑘²𝑢=0 in the exterior of nonclosed Lipschitz surfaces in 𝑅³. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.

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Weak solvability of elliptic boundary value problems with Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary conditions in Lipschitz domains has been studied in [1–6]. It is pointed out in the book [1, page 91] that domains with cracks (cuts) are not Lipschitz domains. So, solvability of elliptic boundary value problems in domains with cracks does not follow from general results on solvability of elliptic boundary value problems in Lipschitz domains. In the present paper, the weak solvability of the Dirichlet problem for the equation Δ𝑢−𝑘²𝑢=0 in the exterior of nonclosed Lipschitz surfaces (cracks) in 𝑅³ is studied. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution are proved, integral representation for a solution in the form of single-layer potential is obtained, the problem is reduced to the uniquely solvable operator equation.

The weak solvability of the Neumann problem for the Laplace equation in the exterior of several smooth nonclosed surfaces in 𝑅³ has been studied in [7]. Boundary value problems for the Helmholtz equation in the exterior of smooth nonclosed screens have been studied in [8, 9].

In Cartesian coordinates 𝑥=(𝑥₁,𝑥₂,𝑥₃) in 𝑅³ consider bounded Lipschitz domain 𝐺 with the boundary 𝑆, that is, 𝑆 is closed Lipschitz surface. Let 𝛾 be nonempty subset of the boundary 𝑆 and 𝛾≠𝑆. Assume that 𝛾 is a nonclosed Lipschitz surface with Lipschitz boundary 𝜕𝛾 in the space 𝑅³, and assume that 𝛾 includes its limiting points, or, alternatively, assume that 𝛾 is a union of finite number of such nonclosed surfaces, which do not have common points, in particular, they do not have common boundary points. In the latter case, 𝛾 is not a connected set. Notice that 𝛾 is a closed set. Let us introduce Sobolev spaces on 𝛾 as follows: 𝐻^1/2𝛾=𝑣∶𝑣=𝑉∣_{𝛾⧵𝜕𝛾},𝑉∈𝐻^1/2(𝑆,𝐻_−1/2)𝛾=𝑣∶𝑣∈𝐻^−1/2(𝑆),supp𝑣⊂𝛾. (1) Spaces 𝐻^1/2(𝛾) and 𝐻^−1/2(𝛾) are dual spaces in the sense of scalar product in 𝐿₂(𝛾) [1, pages 91-92]. Furthermore, one can set 𝐻‖𝑣‖^−1/2_(𝛾)=‖𝑣‖_𝐻^−1/2_(𝑆) for 𝑣∈𝐻^−1/2(𝛾) (see [1, page 79]), and ‖𝑣‖_𝐻^1/2_(𝛾)=min_{𝑉∣𝛾⧵𝜕𝛾=𝑣,𝑉∈𝐻}^1/2_(𝑆)‖𝑉‖_𝐻^1/2_(𝑆) (see [1, pages 77, 99]). Spaces 𝐻^1/2(𝑆) and 𝐻^−1/2(𝑆) on closed Lipschitz surface 𝑆 and their norms are defined, for example, in [1, page 98].

Let Δ be Laplacian in 𝑅³, then for the equation Δ𝑢(𝑥)−𝑘²𝑢(𝑥)=0,𝑘=const≥0, (2) consider the single-layer potential 𝑈[ℎ𝑥)=1](4𝜋_𝑆ℎ𝑦exp−𝑘𝑥−𝑦𝑥−𝑦𝑑𝑠_𝑦, (3) with the density ℎ∈𝐻^−1/2(𝑆). The function (3) is defined for 𝑥∈𝑅³⧵𝑆. According to Theorem  6.11 in [1], the potential 𝑈[ℎ](𝑥) belongs to 𝐻¹_loc(𝑅³) and does not have jump on 𝑆, when approaching 𝑆 from 𝐺 and from 𝑅³⧵𝐺 it has the same trace 𝑈[ℎ]|_𝑆∈𝐻^1/2(𝑆). The overline means closure. Moreover, potential 𝑈[ℎ](𝑥) belongs to 𝐶^∞(𝑅³⧵𝑆) (see [1, page 202]), obeys (2) in 𝑅³⧵𝑆, and satisfies conditions at infinity 𝑢=𝑂|𝑥|⁻¹,|∇𝑢|=𝑜|𝑥|⁻¹,|𝑥|⟶∞. (4)

Lemma 1. Let ℎ∈𝐻^−1/2(𝑆), 𝑘>0, and 𝑆 is a boundary of an open bounded Lipschitz domain 𝐺. Then there is such a constant 𝑐>0, that inequality 𝑈[ℎ]∣_𝑆,ℎ_𝐿2(𝑆)≥𝑐‖ℎ‖²_{𝐻−1/2(𝑆)} (5) holds.

Proof. Note that normal vector exists on the Lipschitz surface almost everywhere [1, page 96]. Let 𝐵_𝑟 be an open ball of the radius 𝑟 with the center in the origin and 𝐺⊂𝐵_𝑟. By 𝑛 denote outward (with respect to 𝐺) unit normal vector on 𝑆 where exists and outward unit normal vector on 𝜕𝐵_𝑟. Writing down Green's formula [1, page 118] for the function 𝑈[ℎ](𝑥) in 𝐵_𝑟⧵𝐺 and in 𝐺, we obtain ‖∇𝑈[ℎ]‖²_{𝐿2(𝐵𝑟⧵𝐺)}+𝑘²‖𝑈[ℎ]‖²_{𝐿2(𝐵𝑟⧵𝐺)}=−𝑈[ℎ]|_𝑆,𝜕𝑈[ℎ]𝜕𝑛⁺_𝐿2(𝑆)+𝑈[ℎ],𝜕𝑈[ℎ]𝜕𝑛_{𝐿2(𝜕𝐵𝑟)},‖∇𝑈[ℎ]‖²_𝐿2(𝐺)+𝑘²‖𝑈[ℎ]‖²_𝐿2(𝐺)=𝑈[ℎ]|_𝑆,𝜕𝑈[ℎ]𝜕𝑛⁻_𝐿2(𝑆). (6)(7) By (𝜕𝑈[ℎ]/𝜕𝑛)⁻ and (𝜕𝑈[ℎ]/𝜕𝑛)⁺, we mean traces of normal derivative of the function 𝑈[ℎ](𝑥) on 𝑆 when approaching 𝑆 from 𝐺 and from 𝑅³⧵𝐺, respectively. According to Theorem  6.11 in [1], traces (𝜕𝑈[ℎ]/𝜕𝑛)⁺ and (𝜕𝑈[ℎ]/𝜕𝑛)⁻ of the normal derivative of the function 𝑈[ℎ](𝑥) exist and belong to 𝐻^−1/2(𝑆). Remind that under conditions of the lemma, the function 𝑈[ℎ](𝑥) has the same trace 𝑈[ℎ]|_𝑆∈𝐻^1/2(𝑆) when approaching 𝑆 both from 𝐺 and from 𝑅³⧵𝐺. Since spaces 𝐻^−1/2(𝑆) and 𝐻^1/2(𝑆) are dual, the scalar products are defined in 𝐿₂(𝑆) in right sides of (6) and (7). Tending 𝑟→∞ in (6) and taking into account that the potential 𝑈[ℎ](𝑥) satisfies conditions (4), we obtain ‖∇𝑈[ℎ]‖²_{𝐿2(𝑅3⧵𝐺)}+𝑘²‖𝑈[ℎ]‖²_{𝐿2(𝑅3⧵𝐺)}=−𝑈[ℎ]|_𝑆,𝜕𝑈[ℎ]𝜕𝑛⁺_𝐿2(𝑆). (8) By Theorem  6.11 in [1], the jump of the normal derivative of potential 𝑈[ℎ] on 𝑆 is given by the following formula: 𝜕𝑈[ℎ]𝜕𝑛⁻−𝜕𝑈[ℎ]𝜕𝑛⁺=ℎ. (9) Adding (7) and (8), we obtain ‖∇𝑈[ℎ]‖²_{𝐿2(𝑅3⧵𝑆)}+𝑘²‖𝑈[ℎ]‖²_{𝐿2(𝑅3⧵𝑆)}=𝑈[ℎ]∣_𝑆,ℎ_𝐿2(𝑆). (10) Since 𝐿₂(𝑅³⧵𝑆)=𝐿₂(𝑅³) and 𝑈[ℎ]∈𝐻¹_loc(𝑅³), then taking into account the theorem on equivalence of Sobolev spaces [1, Theorem  3.16], we observe, that there is such a constant 𝑐₁>0, for which inequality holds 𝑐₁‖𝑈[ℎ]‖²_𝐻1𝑅3≤min𝑘²,1‖∇𝑈[ℎ]‖²_{𝐿2𝑅3⧵𝑆}+‖𝑈[ℎ]‖²_{𝐿2𝑅3⧵𝑆}≤𝑈[ℎ]∣_𝑆,ℎ_𝐿2(𝑆). (11) Using inequality for single-layer potential from [1, page 227] (it follows from [1, Lemma  4.3]), for some constant 𝑐₂>0, we obtain ‖ℎ‖²_{𝐻−1/2(𝑆)}=𝜕𝑈[ℎ]𝜕𝑛⁺−𝜕𝑈[ℎ]𝜕𝑛⁻²_𝐻^−1/2_(𝑆)≤𝑐₂𝑈₀[ℎ]²_𝐻1(𝑅3). (12) Here 𝑈₀[ℎ](𝑥)=𝛿(𝑥)𝑈[ℎ](𝑥), where 𝛿(𝑥)∈𝐶^∞(𝑅³) is a cutoff function, such that 𝛿(𝑥)≤1 for all 𝑥∈𝑅³, 𝛿(𝑥)≡1 in an open bounded domain containing 𝐺, and 𝛿(𝑥)≡0 in the exterior of some ball with the center in the origin. Clearly, 𝑈₀[ℎ]²_𝐻1(𝑅3)≤𝑐₃‖𝑈[ℎ]‖²_𝐻1(𝑅3) (13) for some constant 𝑐₃>0, so ‖ℎ‖²_{𝐻−1/2(𝑆)}≤𝑐₂𝑐₃‖𝑈[ℎ]‖²_𝐻1(𝑅3). (14) Using (11), we obtain 𝑐‖ℎ‖²_{𝐻−1/2(𝑆)}≤𝑈[ℎ]∣_𝑆,ℎ_𝐿2(𝑆)𝑐,𝑐=¹𝑐₂𝑐₃. (15) Lemma is proved.

Let us formulate the Dirichlet problem for (2) in the exterior of nonclosed Lipschitz surfaces 𝛾.

Problem 𝒟. Find a function 𝑢(𝑥)∈𝐻¹_loc(𝑅³)∩𝐶²(𝑅³⧵𝛾), that obeys (2) in 𝑅³⧵𝛾, satisfies the boundary condition 𝑢∣_𝛾=𝑓∈𝐻^1/2𝛾 (16) and conditions at infinity (4).

Note that Lapalce equation Δ𝑢=0 is a particular case of (2) as 𝑘=0. So, the Dirichlet problem for Laplace equation is included in the Problem 𝒟.

Boundary condition (16) implies that the function 𝑢(𝑥) has the same trace 𝑢∣_𝛾, when approaching 𝛾 from 𝐺 and from 𝑅³⧵𝐺, and this trace has to satisfy condition (16).

Let us construct the solution of the problem. We look for a solution in the form of a single-layer potential 𝑢(𝑥)𝑔𝑥)==𝑈(_𝛾𝑔𝑦exp−𝑘𝑥−𝑦4𝜋𝑥−𝑦𝑑𝑠_𝑦=_𝑆𝑔𝑦exp−𝑘𝑥−𝑦4𝜋𝑥−𝑦𝑑𝑠_𝑦 (17) with the density 𝑔∈𝐻^−1/2(𝛾)⊂𝐻^−1/2(𝑆). The function (17) is defined as 𝑥∈𝑅³⧵𝛾.

It follows from aforementioned properties of a single-layer potential (3) that the potential 𝑈[𝑔](𝑥) belongs to 𝐻¹_loc(𝑅³), has a trace on 𝑆∶𝑈[𝑔]∣_𝑆∈𝐻^1/2(𝑆), and has a trace on 𝛾∶𝑈[𝑔]∣_𝛾∈𝐻^1/2(𝛾). Furthermore, the potential 𝑈[𝑔](𝑥) belongs to 𝐶^∞(𝑅³⧵𝛾) (see [1, page 202]), satisfies (2) in 𝑅³⧵𝛾, and conditions at infinity (4). Therefore, for any function 𝑔 from the space 𝐻^−1/2(𝛾), the potential 𝑈[𝑔](𝑥) satisfies all conditions of the Problem 𝒟, except for the boundary condition (16). We have to find the function 𝑔∈𝐻^−1/2(𝛾) to satisfy the boundary condition (16). Substituting (17) into the boundary condition (16), we arrive at the operator equation 𝑈𝑔∣_𝛾=𝑓∈𝐻^1/2𝛾. (18) Here by 𝑈[𝑔]|_𝛾, we mean the trace of the function (17) on 𝛾, this trace belongs to 𝐻^1/2(𝛾). To prove solvability of (18), we have to study properties of the operator in the left side of the equation.

Operator 𝑈 is bounded when acting from 𝐻^−1/2(𝑆) into 𝐻^1/2(𝑆) by Theorem  6.11 in [1], so when acting from 𝐻^−1/2(𝛾)⊂𝐻^−1/2(𝑆) into 𝐻^1/2(𝑆) it is bounded as well. If a set of functions is bounded (in norm) in 𝐻^1/2(𝑆), by a constant, then set of restrictions of these functions to 𝛾 is bounded (in norm) in 𝐻^1/2(𝛾) also and by the same constant. Therefore, the operator 𝑈 is bounded when acting from 𝐻^−1/2(𝛾) into 𝐻^1/2(𝛾). Since 𝑔∈𝐻^−1/2(𝛾)⊂𝐻^−1/2(𝑆), we have for 𝑘≥0𝑈𝑔∣_𝑆,𝑔_𝐿2(𝑆)=𝑈𝑔∣_𝛾,𝑔_𝐿2(𝛾)≥𝑐𝑔²_{𝐻−1/2(𝑆)}=𝑐𝑔²𝐻^−1/2(𝛾). (19) If 𝑘>0, then this estimate follows from Lemma 1, while if 𝑘=0, then this estimate is proved in Corollary  8.13 in [1]. Therefore, for some constant 𝑐>0, we have 𝑈𝑔∣_𝛾,𝑔_𝐿2(𝛾)≥𝑐𝑔²𝐻^−1/2_(𝛾). (20) Note, that the operator 𝑈 acts from 𝐻^−1/2(𝛾) into 𝐻^1/2(𝛾) and is bounded, while spaces 𝐻^−1/2(𝛾), 𝐻^1/2(𝛾) are dual in the sense of scalar product in 𝐿₂(𝛾). Inequality (20) implies that the operator 𝑈 is positive and bounded below. Consequently, from Lemma  2.32 in [1, page 43], it follows that the operator 𝑈 is invertible (it has bounded inverse operator). Therefore, (18) has unique solution 𝑔∈𝐻^−1/2(𝛾) for any function 𝑓∈𝐻^1/2(𝛾). The potential (17), constructed on this solution, satisfies all conditions of the Problem 𝒟. From above considerations it follows the theorem.

Theorem 2. The solution of the Problem 𝒟 exists and is given by formula (17), where 𝑔∈𝐻^−1/2(𝛾) is a solution of (18), which is uniquely solvable in 𝐻^−1/2(𝛾).

Let us prove the uniqueness of a solution to the Problem 𝒟.

Theorem 3. The Problem 𝒟 has at most one solution.

Proof. Let 𝑢(𝑥) be a solution of the homogeneous Problem 𝒟. Consider the ball 𝐵_𝑟 of enough large radius 𝑟 with the center in the origine. Suppose that 𝐺⊂𝐵_𝑟 and 𝐺∩𝜕𝐵_𝑟=∅. The overline means closure, while 𝜕𝐵_𝑟 is a sphere, the boundary of the ball 𝐵_𝑟. Since 𝑢∈𝐻¹_loc(𝑅³), the Green’s formulae [1, Theorem  4.4, page 118] ‖∇𝑢‖²_𝐿2(𝐺)+𝑘²‖𝑢‖²_𝐿2(𝐺)=𝑢∣_𝑆,𝜕𝑢𝜕𝑛⁻_𝐿2(𝑆),‖∇𝑢‖²_{𝐿2(𝐵𝑟⧵𝐺)}+𝑘²‖𝑢‖²_{𝐿2(𝐵𝑟⧵𝐺)}=−𝑢∣_𝑆,𝜕𝑢𝜕𝑛⁺_𝐿2(𝑆)+𝑢,𝜕𝑢𝜕𝑛_{𝐿2(𝜕𝐵𝑟)} (21)(22) hold for the function 𝑢. By 𝑛 on 𝜕𝐵_𝑟, the outward (regarding to 𝐵_𝑟) unite normal vector is understood, while by 𝑛 on 𝑆, the outward (regarding to 𝐺) unite normal vector is understood (where exists). By (𝜕𝑢/𝜕𝑛)⁻ and (𝜕𝑢/𝜕𝑛)⁺, we denote the traces of the normal derivative of the function 𝑢(𝑥) on 𝑆 when approaching to 𝑆 from 𝐺 and from 𝑅³⧵𝐺, respectively. Since the function 𝑢(𝑥) belongs to 𝐻¹_loc(𝑅³), the traces of this function exist on 𝑆 when approaching both from 𝐺 and from 𝑅³⧵𝐺. According to the formulation of the Problem 𝒟, these traces are the same, they are denoted by 𝑢|_𝑆 and belong to 𝐻^1/2(𝑆) (see [1, Theorems  3.37, 3.38, page 102]). Since, in addition, the function 𝑢(𝑥) obeys (2) outside 𝑆, the traces (𝜕𝑢/𝜕𝑛)⁺ and (𝜕𝑢/𝜕𝑛)⁻ of the normal derivative of the function 𝑢 exist and belong to 𝐻^−1/2(𝑆) by Lemma  4.3 in [1]. Since spaces 𝐻^−1/2(𝑆) and 𝐻^1/2(𝑆) are dual, the scalar product in 𝐿₂(𝑆) in the right sides of (21) and (22) is defined. Note that 𝑢|_𝛾=0∈𝐻^1/2(𝛾), since 𝑢 is a solution of the homogeneous Problem 𝒟. Moreover, (𝜕𝑢/𝜕𝑛)⁺=(𝜕𝑢/𝜕𝑛)⁻ on 𝑆⧵𝛾. Adding (21) and (22), we obtain ‖∇𝑢‖²_{𝐿2(𝐵𝑟⧵𝑆)}+𝑘²‖𝑢‖²_{𝐿2(𝐵𝑟⧵𝑆)}=𝑢,𝜕𝑢𝜕𝑛_{𝐿2(𝜕𝐵𝑟)}=_𝜕𝐵𝑟𝑢𝜕𝑢𝜕𝑛𝑑𝑠. (23) Using conditions (4) at infinity, we obtain from (23) as 𝑟→∞lim_𝑟→∞‖∇𝑢‖²_{𝐿2𝐵𝑟⧵𝑆}+𝑘²‖𝑢‖²_{𝐿2𝐵𝑟⧵𝑆}=‖∇𝑢‖²_{𝐿2(𝑅3⧵𝑆)}+𝑘²‖𝑢‖²_{𝐿2(𝑅3⧵𝑆)}=0. (24) Since 𝑘≥0, we have that 𝑢≡𝑐₁ in 𝐺 and 𝑢≡𝑐₂ in 𝑅³⧵𝐺, where 𝑐₁ and 𝑐₂ are some constants. Furthermore, since 𝑢∈𝐶²(𝑅³⧵𝛾), we observe that 𝑐₁=𝑐₂ and 𝑢≡const in 𝑅³⧵𝛾. Taking into account conditions at infinity (4), we have const=0, so 𝑢≡0 in 𝑅³⧵𝛾. Thus, the homogeneous Problem 𝒟 has only the trivial solution. In view of the linearity of the Problem 𝒟, the inhomogeneous Problem 𝒟 has at most one solution. The theorem is proved.

In conclusion we note that the paper [10] treats the Dirichlet problem for the Laplace equation in planar domains with cracks without compatibility conditions at the tips of the cracks. The well-posed classical formulation of the problem is given. It is shown that classical solution exists and unique, while weak solution in 𝐻¹_loc space does not exist typically.

In addition, the Dirichlet problem for the Laplace equation in a planar domain with cracks with compatibility conditions at the tips of the cracks has been studied in [11] (bounded domain) and in [12] (unbounded domain). The Dirichlet problem for the Helmholtz equation in both bounded and unbounded planar domains with cracks with compatibility conditions at the tips of the cracks has been treated in [13, 14]. Furthermore, problems in [11–14] have been reduced to the uniquely solvable integral equations of the 2nd kind and index zero. Moreover, theorems on uniqueness and existence of a classical solution have been proved in [11–14], and integral representations for solutions in the form of potentials have been obtained.