The Dirichlet Problem for the Equation Δu − k 2 u = 0 in the Exterior of Nonclosed Lipschitz Surfaces

We study the Dirichlet problem for the equation Δu − k2u = 0 in the exterior of nonclosed Lipschitz surfaces in R. 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.

Weak solvability of elliptic boundary value problems with Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary conditions in Lipschitz domains has been studied in [1][2][3][4][5][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 Δ −  2  = 0 in the exterior of nonclosed Lipschitz surfaces (cracks) in  3 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  3 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  = ( 1 ,  2 ,  3 ) in  3 consider bounded Lipschitz domain  with the boundary , that is,  is closed Lipschitz surface.Let  be nonempty subset of the boundary  and  ̸ = .Assume that  is a nonclosed Lipschitz surface with Lipschitz boundary  in the space  3 , and assume that  includes its limiting points, or, alternatively, assume that  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,  is not a connected set.Notice that  is a closed set.Let us introduce Sobolev spaces on  as follows: Spaces  Let Δ be Laplacian in  3 , then for the equation consider the single-layer potential with the density ℎ ∈   () when approaching  both from  and from  3 \ .Since spaces  −1/2 () and  1/2 () are dual, the scalar products are defined in  2 () in right sides of ( 6) and (7).Tending  → ∞ in ( 6) and taking into account that the potential [ℎ]() satisfies conditions (4), we obtain By Theorem 6.11 in [1], the jump of the normal derivative of potential [ℎ] on  is given by the following formula: Adding ( 7) and ( 8), we obtain Since  2 ( 3 \ ) =  2 ( 3 ) and [ℎ] ∈  1 loc ( 3 ), then taking into account the theorem on equivalence of Sobolev spaces [1, Theorem 3.16], we observe, that there is such a constant  1 > 0, for which inequality holds Using inequality for single-layer potential from [1, page 227] (it follows from [1, Lemma 4.3]), for some constant  2 > 0, we obtain Here  0 [ℎ]() = ()[ℎ](), where () ∈  ∞ ( 3 ) is a cutoff function, such that () ≤ 1 for all  ∈  3 , () ≡ 1 in an open bounded domain containing , and () ≡ 0 in the exterior of some ball with the center in the origin.Clearly, for some constant  3 > 0, so Using (11), we obtain Lemma is proved.
Let us formulate the Dirichlet problem for (2) in the exterior of nonclosed Lipschitz surfaces .
Note that Lapalce equation Δ = 0 is a particular case of (2) as  = 0. So, the Dirichlet problem for Laplace equation is included in the Problem D.
Boundary condition (16) implies that the function () has the same trace |  , when approaching  from  and from  3 \ , 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  2) in  3 \, and conditions at infinity (4).Therefore, for any function  from the space H−1/2 (), the potential []() satisfies all conditions of the Problem D, except for the boundary condition (16).We have to find the function  ∈ H−1/2 () to satisfy the boundary condition (16).Substituting (17) into the boundary condition (16), we arrive at the operator equation Here by []|  , we mean the trace of the function (17) on , this trace belongs to  1/2 ().To prove solvability of (18), we have to study properties of the operator in the left side of the equation.
Operator  is bounded when acting from  −1/2 () into  1/2 () by Theorem 6.11 in [1], so when acting from H−1/2 () ⊂  −1/2 () into  1/2 () it is bounded as well.If a set of functions is bounded (in norm) in  1/2 (), by a constant, then set of restrictions of these functions to  is bounded (in norm) in  1/2 () also and by the same constant.Therefore, the operator  is bounded when acting from If  > 0, then this estimate follows from Lemma 1, while if  = 0, then this estimate is proved in Corollary 8.13 in [1].Therefore, for some constant  > 0, we have Note, that the operator  acts from H−1/2 () into  1/2 () and is bounded, while spaces H−1/2 (),  Let us prove the uniqueness of a solution to the Problem D.

Theorem 3. The Problem D has at most one solution.
Proof.Let () be a solution of the homogeneous Problem D. Consider the ball   of enough large radius  with the center in the origine.Suppose that  ⊂   and  ∩   = 0.The overline means closure, while   is a sphere, the boundary of the ball   .Since  ∈ hold for the function .By  on   , the outward (regarding to   ) unite normal vector is understood, while by  on , the outward (regarding to ) unite normal vector is understood (where exists).By (/) − and (/) + , we denote the traces of the normal derivative of the function () on  when approaching to  from  and from  3 \ , respectively.Since the function () belongs to  1 loc ( 3 ), the traces of this function exist on  when approaching both from  and from  3 \ .According to the formulation of the Problem D, these traces are the same, they are denoted by |  and belong to  1/2 () (see [1,Theorems 3.37,3.38,page 102]).Since, in addition, the function () obeys (2) outside , the traces (/) + and (/) − of the normal derivative of the function  exist and belong to  −1/2 () by Lemma 4.3 in [1].Since spaces  −1/2 () and  1/2 () are dual, the scalar product in  2 () in the right sides of (21) and ( 22) is defined.Note that |  = 0 ∈ Using conditions (4) at infinity, we obtain from (23) as Since  ≥ 0, we have that  ≡  1 in  and  ≡  2 in  3 \, where  1 and  2 are some constants.Furthermore, since  ∈  2 ( 3 \ ), we observe that  1 =  2 and  ≡ const in  3 \ .Taking into account conditions at infinity (4), we have const = 0, so  ≡ 0 in  3 \.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  1 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][12][13][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][12][13][14], and integral representations for solutions in the form of potentials have been obtained.