CARLEMAN’S FORMULA OF A SOLUTIONS OF THE POISSON EQUATION IN BOUNDED DOMAIN

We suggest an explicit continuation formula for a solution to the Cauchy problem for the Poisson equation in a domain from its values and values of its normal derivative on a part of the boundary. We construct the continuation formula of this problem based on the Carleman–Yarmuhamedov function method.


Introduction
In this paper, we continue the research provided in [12]. We propose an explicit formula for the reconstruction of a solution of the Poisson equation in a bounded domain from its values and the values of its normal derivative on a part of the boundary, i.e., we give an explicit continuation formula for a solution to the Cauchy problem for the Poisson equation.
Problem 2. Let f 1 and f 2 be given on S. Find conditions on f 1 and f 2 that are necessary and sufficient for the existence of a solution to system (1.1) satisfying (1.2) and from the class H(Ω ρ ).
It is well-known that the Cauchy problem (1.2) for the Poisson equation (1.1) is ill-posed [3,5]. Hadamard [17] noted that a solution to Problem 1 is not stable. The possibility of introducing a positive parameter σ, depending on the accuracy of the initial data, was noticed by M.M. Lavrentev [23]. The uniqueness of the solution follows from the general theorem by Holmgren [6]. It has applications in many different areas such as plasma physic, electrocardiography, and corrosion non-destructive evaluation (e.g., [7,9,10,13,19]). Traditionally, regularization techniques, such as Tikhonov regularization [44] and the quasi-reversibility approach [22], were used to provide robust numerical schemes [18].
We suppose that a solution to the problem exists (in this event, it is unique) and is continuously differentiable in the closed domain, and the Cauchy data are given exactly. In this case, we establish an explicit continuation formula. This formula enables us to state a simple and convenient criterion for the solvability of the Cauchy problem.
The result established here is a multidimensional analog of theorems and Carleman-type formulas [4] by G.M. Goluzin, V.I. Krylov, V.A. Fok, and F.M. Kuni in the theory of holomorphic functions of one variable [14,16].
The method for obtaining these results is based on an explicit form of the fundamental solution of the Poisson equation which depends on a positive parameter that vanishes together with its derivatives on a fixed cone and outside it, as the parameter tends to infinity, while the pole of the fundamental solution lies inside the cone. Following to M.M. Lavrent'ev, a fundamental solution with these properties is called a Carleman function for the cone [8,23]. Having constructed a Carleman function explicitly, we write a continuation formula. The existence of a Carleman function follows from S.N. Mergelyan's approximation theorem [28]. However, this theorem shows no way for writing the Carleman function explicitly.
The Carleman function of the Cauchy problem for the Laplace equation and some close problems, in the case when ∂Ω ρ \ S is a part of a conic surface, was constructed in [45]. Mergelyan [28] suggested a method to construct the Carleman function of the Cauchy problem for the Laplace equation in the case when S is a part, with a smooth boundary, of the boundary of a simplyconnected domain. Based on [28] and approximative theorems, the Carleman matrix for elliptic systems was constructed in [41].
In [1], some theorems of existence of the Carleman matrix and a solvability criterion for a wider class of boundary value problems for elliptic systems were established. It was proved earlier in [1,41] that, for every Cauchy problem for elliptic systems, the Carleman matrix exists if the Cauchy data are given on a boundary set of positive measure.
Following Tikhonov [21,43], we call the family of functions U σδ (x) the regularized solution to the Cauchy problem for equation (1.1). The regularized solution determines the stability of the approximate method.
In the paper, based on results from [23,[45][46][47][48] on the Cauchy problem for the Laplace and Helmholtz equations, we construct the Carleman-Yarmuhamedov function in an explicit form. We use it to prove the Carleman formulas and a criterion for the solvability of the Cauchy problem.

Construction of a Carleman-Yarmukhamedov function
According to [45], we define the Carleman-Yarmukhamedov function Φ(y, x) by the equality Here, K(w) is an entire function of complex variable that takes real values for real w (w = a+ib, a and b are real numbers) such that K(a) = ∞, |a| < ∞, (2.4) From (2.2) and (2.3), it follows that, for y = x, the integral in (2.1) converges absolutely. If K(w) ≡ 1, then the function Φ(y, x) is the classical fundamental solution to the Laplace equation, i.e., Φ(y, x) ≡ Φ 0 (r) = 1/(4πr).
where Φ 0 (r) = 1/(4πr) and the function G(y, x) is harmonic in the variable y in R 3 , including y = x.
From Theorem 1 it follows that the function Φ(y, x) of the variable y is a fundamental solution of the Poisson equation. Therefore, for the function U (y) ∈ H(Ω ρ ) and for every point x ∈ Ω ρ , the Green's formula is valid [15]: where f (x) ∈ C λ (Ω ρ ), λ ∈ (0, 1), is bounded, i.e., the former integral on the right-hand side of (2.5) satisfies equation (1.1) in the domain.

The Mittag-Leffler entire function
The continuation formulas below are expressed explicitly in terms of the Mittag-Leffler entire function; therefore, we now present its basic properties without proof. These properties as well as detailed proofs can be found in [11,Chapter 3,§2], [47].
The Mittag-Leffler entire function is defined by the series where Γ is the Euler gamma-function. Hereinafter, we suppose that ρ > 1. Let be the contour in the complex w-plane that consists of the ray arg w = −β, |w| ≥ 1, the arc −β ≤ arg w ≤ β of the circle |w| = 1, and the ray arg w = β, |w| ≥ 1, which is passed so that arg w does not decrease. The contour γ splits the complex domain C into the two simply connected infinite domains Ω − and Ω + lying to the left and to the right of γ, respectively. We suppose that π 2ρ < β < π ρ , ρ > 1.

Carleman formulas
Let the Mittag-Leffler entire function be the function K(w) in (2.1): Denote by Φ σ (y, x) the corresponding fundamental solution and by Φ σ (y − x) its derivative with respect to the variable σ: It follows from Theorem 1 that Ψ σ (y − x) satisfies the Poisson equation in R 3 . Then ϕ σ (y, x, u) e −as−au 2 u 2 + r 2 du, Lemma 1 [47]. Let M be a compact set in G ρ , and let δ be the distance from M to ∂G ρ . Then, for σ ≥ 0, the following inequalities are valid for x ∈ M and y ∈ R 3 \G ρ (|y ′ | ≥ τ y 3 ): where the constants C 4 and C 5 are independent of x, y, and σ.
Theorem 2. Let f be bounded and locally Hölder continuous in Ω ρ , U (y) ∈ H λ (Ω ρ ), and where f 1 (y) and f 2 (y) are given functions of the class C(S). Then the Carleman formulas are valid for every x ∈ Ω ρ , where i = 0, 1, j = 1, 2, 3, and the convergence in (4.5) is uniform on compact sets in Ω ρ .
P r o o f. From Green's formula (2.5), for every x ∈ Ω ρ , we obtain ∂Ω ρ = S ∪ (∂Ω ρ \ S). According to [47], let us estimate Lemma 1 yields the assertion of Theorem 2. Indeed, if M is a compact set in Ω ρ then M ⊂ G ρ . Therefore, the inequalities in Lemma 1 for Φ σ (y − x) and its derivatives remain also valid in the case where x ∈ M ⊂ Ω ρ and y ∈ ∂Ω ρ \S ⊂ ∂G ρ (in this case, δ is the distance from the compact set M ⊂ Ω ρ to ∂Ω ρ ). Now, let σ tend to infinity. The proof of Theorem 2 is complete.
We can write (4.5) in the following equivalent form: where (4.8) The functions Ψ σ (y − x) and Φ 0 (r) are defined by equalities (4.2) and (4.1), respectively. The proof of (4.7) follows from the formulas and x ∈ Ω ρ , i = 0, 1, j = 1, 2, 3; moreover, the differentiation under the integral sign is legal and , and let f be bounded and locally Hölder continuous in Ω ρ . Then for the existence of a function U (y) ∈ H λ (Ω ρ ) ∩ C(S 0 ) such that U (y) = f 1 (y), ∂U ∂n (y) = f 2 (y), y ∈ S 0 , (4.9) it is necessary and sufficient that the following improper integral converge (uniformly on compact sets in G ρ ) for each x ∈ G ρ : where J(σ, x) is defined by (4.8). If (4.10) is satisfied, then harmonic continuation is performed by equivalent formulas (4.5) and (4.7).
Therefore, (2.5) is valid for arg w; moreover, if y ′ = x ′ , then Re w < 0, and this inequality also holds. Consequently, Φ σ (y − x) and Ψ σ (y − x) satisfy estimates (3.2)-(3.5) from Lemma 1, where δ ≥ ετ 1 . Define S ε = G ε ρ ∩ S; in this case, the part S ε ⊂ S together with the part T ε of the cone surface ∂G ε ρ form a closed piecewise smooth surface S ε ∪ T ε (with the consistent direction of the outer normals) which is the boundary of a simply connected bounded domain. Represent the integral on the right-hand side of (4.8) as the sum of two integrals according to the representation S = S ε ∪ (S \ S ε ). Since Ψ σ (y − x) is a regular solution of the Poisson equation, by Green's formula, the integral over the part S ε is equal to the integral over T ε ; moreover, Ψ σ (y − x) satisfies inequalities (4.7) and (4.9) for y ∈ T ε and x ∈ M , and the extended function U (y) together with its gradient is bounded by a constant depending on ε. Therefore, the modulus of the integral over the part S ε does not exceed the quantity const 1 + δ 2 σ 2 , σ ≥ 0, with a constant depending on ρ, ε, δ, and the diameter of the domain Ω ρ . Since |y| ≥ τ (y 3 − ε), y 3 ≥ ε, when y ∈ S \ S ε and x ∈ K and f 1 (y), f 2 (y) ∈ C(S 0 ) ∩ L(S), these inequalities remain valid for the modulus of the integral over S \ S ε (of course, with other constants). Hence, we have (4.10).
Sufficiency: Under the assumptions of the theorem, define functions U (x), x ∈ G ρ \ S 0 , by the right-hand side of (4.7). Consider the first term on the right-hand side of (4.7). Since Ψ σ (y) satisfies the Poisson equation in G ρ for σ ≥ 0, the function J(σ, x) satisfies the Poisson equation with respect to x in G ρ for σ ≥ 0. Therefore, we conclude from (4.10) that the first term on the right-hand side of (4.7) satisfies the Poisson equation in G ρ as the limit of the uniformly converging sequence of the solutions of the Poisson equations U n (x) = n 0 J(σ, x)dσ, n = 1, 2, . . . .
The second and third terms are the potential difference of the volume, single, and double layers and represent one solution of the Poisson equation in Ω ρ and another in Ω ′ ρ = G ρ \ Ω ρ . Therefore, the right-hand side of (4.7) defines two different solutions of the Poisson equations U + (x) and U − (x) in Ω ρ and Ω ′ ρ . If x 1 and x 2 are two points on the normal at x ∈ S 0 symmetric with respect to x, then lim x 1 →x U + (x 1 ) − U − (x 2 ) = f 1 (x), lim x 2 ) = f 2 (x), x ∈ S 0 ; moreover, the limit relations hold uniformly in x on each compact part S 0 . If max y 3 < x 3 , where y ∈ S and x ∈ G ρ , then Re w = y 3 −x 3 < 0 and Φ σ (y−x) and its derivatives satisfy inequalities (4.6) and (4.3). Now, from formula (4.5), which is equivalent to (4.7), we see that U − (x) = 0 and U − (x) ≡ 0, x ∈ Ω ρ , by the uniqueness theorem. It is clear that U − (x) extends smoothly to Ω ′ ρ ∪ S 0 . Then U + (x) extends smoothly as a function of the class C 1 (Ω ρ ∪ S 0 ) (see [29]). Consequently, Now, we set U (x) = U + (x), x ∈ Ω ρ ∪ S 0 . Theorem 3 is proved.