CC BY
6URAL MATHEMATICAL JOURNAL, Vol. 7, No. 2, 2021, pp. 110-120
DOI: 10.15826/umj.2021.2.008
CARLEMAN'S FORMULA OF A SOLUTIONS OF THE POISSON EQUATION IN BOUNDED DOMAIN
Ermamat N. Sattorov^, Zuxro E. Ermamatova^
Samarkand State University, Samarkand boulevard 15, Samarkand, Uzbekistan t [email protected], [email protected]
Abstract: 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.
Keywords: Poisson equations, Ill-posed problem, Regular solution, Carleman—Yarmuhamedov function, Green's formula, Carleman formula, Mittag—Leffler entire function.
1. 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.
Let us introduce the following notation: R3 is a three-dimensional real Euclidean space,
x = (xi,x2,x3), y = (yi,y2,y3) € R3, x' = (Xi,X2), y' = (yi,y2) € R2, s = a2 = |y' - x'|2 = (yi - xi)2 + (y2 - X2)2, r2 = s + (y3-x3)2 = \y-x\2, T = tgP> 1, GP = {y : \y'\ < ry3, y3 > 0}, rJGp = {y : \y'\ = ry3, y3 > 0}, Gp = Gp U rJGp, e, ei, and e2 are sufficiently small positive constants,
G£p = {y.\y'\<r{y3-e)}, dG£p = {y.\y'\=T{y3-e)}, G£p = G£pUdG%
and Qp is a bounded simply connected domain whose boundary dQp in R3 consists of a part of the conic surface T = dGp and a smooth surface S lying inside the cone Gp. The case p = 1 is the limit case. In this case, Gi is the half-space y3 > 0,dGi is the hyperplane y3 = 0, and Qi is a bounded simply connected domain whose boundary consists of a compact connected part of the hyperplane y3 = 0 and a smooth surface S in the half-space y3 >0, Qp = QpLidQp, and So is the interior of S. The Poisson equation or potential equation [15]
3 r)2jj
-Atf^ _£_ = /(*) (i.i)
i=i 1
is a classical example of second-order elliptic partial differential equations and a mathematical model for some important physical phenomena. Let Hx(Qp) be the set of real functions of the class
C2,x(Qp) fl C1(Q,P) satisfying the Poisson equation. Let a function / be Holder continuous with exponent A € (0,1), i.e., / € Cs>x{Tip) and s € Z+.
Problem 1. Assume that we know the Cauchy data for a .solution to equation (1.1) on the surface S:
U{y) = h{y), ^M = f2{y), yes (1.2)
where n = (n1,n2,n3) is the outward unit normal to the surface dQp at a point y, and f1 and f2 are continuous functions. Given fi(y) and f2(y) on S, find U(x), x € Qp.
Problem 2. Let f1 and f2 be given on S. Find conditions on f1 and f2 that are necessary and sufficient for the existence of a solution to system (1.1) satisfying (1.2) and from the class H(Qp).
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 a, 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 dQp \ 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 simply-connected 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 Uas (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, 45-48] 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.
In recent decades, interest in the classical ill-posed problems of mathematical physics has been preserved. This direction of investigation of the properties of solutions to the Cauchy problem for the Laplace equation was started in [2, 20, 23, 24, 42] and was further developed in [25-27, 30-40].
2. Construction of a Carleman—Yarmukhamedov function
According to [45], we define the Carleman-Yarmukhamedov function $(y,x) by the equality
-27r2K(0)$(y,x) = flm dU , w = iVs + v? + ya - x3. (2.1)
J L w \ v s + u2 0
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) = to, |a| < to, K(0) = 0, VR > 0, 3CR > 0
sup (|K(w)| + |Imw||K'(w)| + |Imw|2|K''(w)|) < to.
|Rew\<R, Imw<-CR
For real w, since K(w) is real, we have K(w) = K(w). Then (2.1) implies that Vii > 0
sup {|K(w)| + (1 + |Imw|)|K'(w)| + (1 + |Imw|2)|K''(w)|} < to. (2.2)
\Re w\<R
Now we write (2.1) in the form
-^Wm,*)-]^-^"™ -BeA-(,„)}^. (2.3)
where
fK(w)\ _ 1 f K(w) K(w) } _ TvK(w) - wK(w) \ w J 2i \ w w J 2i(r2 + u2) (J/3 — x-i) Im K(w) — \/s + u2 Re K(w) r2 + u2
(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 $(y,x) is the classical fundamental solution to the Laplace equation, i.e.,
$(y,x) = $0(r) = 1/(4nr).
Theorem 1 [45]. The function $(y,x) defined by (2.1) or (2.3)-(2.4) is representable in the form
$(y,x) = $o(r) + G(y, x), (2.6)
where $0(r) = 1/(4nr) and the function G(y,x) is harmonic in the variable y in R3, including y = x.
From Theorem 1 it follows that the function $(y,x) of the variable y is a fundamental solution of the Poisson equation. Therefore, for the function U(y) € H(Qp) and for every point x € Qp, the Green's formula is valid [15]:
U(x) = y (y)dy - J
'TT, ,d$(y,x) _ , dU(y)
dSy, (2.5)
where f (x) € CA(Qp), A € (0,1), is bounded, i.e., the former integral on the right-hand side of (2.5) satisfies equation (1.1) in the domain.
3. 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
^ n
w = Er(iln/Py p>0' weC> EiM = eW>
where r is the Euler gamma-function. Hereinafter, we suppose that p > 1. Let
n
7 = 7(1 ,P), 0 < /3 < -, p> 1,
p
be the contour in the complex w-plane that consists of the ray arg w = —5, |w| > 1, the arc —5 < argw < 5 of the circle |w| = 1, and the ray argw = 5, |w| > 1, which is passed so that arg w does not decrease. The contour 7 splits the complex domain C into the two simply connected infinite domains Q- and Q+ lying to the left and to the right of 7, respectively. We suppose that
-</?<-, p> 1. 2p p
Under these conditions, the following integral representations are valid:
Ep(w) = pewP + ■0p(w), w € Q+, Ep(w) = ■0p(w), E'p(w) = ^p(w), w € Q-,
where
Y Y
Since Ep(w) takes real vales for real w, we obtain
Re0p(w) = + ^ = -A [
W ; 2 2m J (C-w)((-w) ^
Y
0p(w) - 0p(w) _ plmw f ë>p
2î ~ 2ttî J ((-w)((-w)'
lmÎr^l = JL f 2eCP^-ReZ\dÇ. Imw 27tî J (( — w)2(( — w)2 ''
Y
dQp
Hereinafter, we take
in the definition of the contour 7(1,^). It is clear that, if
n
—+t-2 < largwl <7T, (3.2)
2p
then w € and Ep(w) = ^p(w). Define
p i <?<?
= & /
(( -w)ki( -w)k
Y
—t d(, k = 1,2,..., p = 0,1,...
The following inequalities are valid for n/(2p) + e2 < |arg w| < n:
^£ ITW |£>)i £ TTFF- (3'3)
l*W«OI S T^rn, k = 1,2,--., (3.4)
where Ci,C2, and C3 are constants independent of w. Take in (2.1)
n n £2 n 2p 2 p
Then Ep(w) = ^p(w), where (w) is defined by (3.1). Moreover, note that cosp^ < 0 and the integral converges:
f |Z|pecosp^\c\p|dZ| < to, p = 0,1,... . (3.5)
4. Carleman formulas
Let the Mittag-Leffler entire function be the function K(w) in (2.1):
2
K(w) = eaw Ep(aw),
where
p > 1, w = iVs + u2 +V3- x3, K{0) = Ep{0) = 1, a > 0 a > 0.
Denote by (y,x) the corresponding fundamental solution and by (y — x) its derivative with respect to the variable ct:
ma(y-x) = -~^-{y-x). It follows from Theorem 1 that (y — x) satisfies the Poisson equation in R3. Then
—2n2$CT(y — x) = J Im
du
eaw2 ep(ctw)
w
0
X 2
2 j e-as-au2
a/S +
(4.1)
= ea(y3-x3)- / y^y^u)- dU,
u2 + r2
where
^a(y - x,u) =
{ys ~ x3) V u2 + s
Im Ep(aw) — Re Ep(aw)
cos(z/\/ s + v?)
+
Im + ^ Re Ep(aw)
v s + u2
sin(z/\/ s + t/,2), z/ = 2a(y-3 — x3),
Va(y -X) = ^(y- X) = Am eau^E'p(aw)
du
+ u*
(4.2)
Lemma 1 [47]. Let M be a compact set in Gp, and let 5 be the distance from M to dGp. Then, for a > 0, the following inequalities are valid for x € M and y € R3\GP (|y'| > ry3):
l$a (y — x)| +
dyk
+
9
dxj dyk
$a (y — x)
r > 5> 0, i, = 0,1, = 1,2,3.
l^a (y — x)| +
^-^(y-x)
dyk
+
d
dxj dyk
^a (y — x)
<
<
c\(p,5)r 1 + aS '
C5(p,S)r l + a5 '
(4.3)
(4.4)
r > 5> 0, i = 0,1, k,j = 1,2,3, where the constants C4 and C5 are independent of x,y, and a.
Theorem 2. Let f be bounded and locally Holder continuous in Qp, U(y) € HA(Hp), and
U(y) = fi(y), y) = /2(y), ye 5,
where fi(y) and f2(y) are given functions of the class C (S). Then the Carleman formulas
d*U (x) (x)
— hm
dxi
a^œ dxi
lim
G —^^O
/{/i(y)
<9* ¿^(y - x) dxj dn
— /2(y)
¿^(y - x) dxi
dSy
(4.5)
are valid for every x € Qp, where i = 0,1, j = 1,2,3,
- "(T)
dx0
dx0
and the convergence in (4.5) is uniform on compact sets in Qp.
Proof. From Green's formula (2.5), for every x € we obtain
dU(x) r . .d^*(y - x) , f r, . , di (y - x) , . , d* ^ . . .
dQp = S U (dQp \ S). According to [47], let us estimate
dyj dxj dyj
a
Lemma 1 yields the assertion of Theorem 2. Indeed, if M is a compact set in Qp then M c Gp. Therefore, the inequalities in Lemma 1 for (y — x) and its derivatives remain also valid in the case where x € M c Qp and y € dQp\S c dGp (in this case, 5 is the distance from the compact set M c Qp to dQp). Now, let ct tend to infinity. The proof of Theorem 2 is complete. □
We can write (4.5) in the following equivalent form:
X
dU(x) _ f d f Adi$o(r)
dxj
JL
-J(a,x)+ f f(y)-i
dxj
dy
fi(v)-z-r—z--My)'
dxj dn
dxi
dSy, x € Qp,
(4.7)
where d
Ox' y ' J J Ox' " J
x € Qp, i = 0,1, j = 1, 2, 3,
d°U dx0
= U,
dxi
dx0
dn
=
dxi
dx0
9V dx0
dSy
= j.
(4.8)
The functions (y — x) and $0(r) are defined by equalities (4.2) and (4.1), respectively. The proof of (4.7) follows from the formulas
lim -tp(*,x) = +JLP{X)
a—dxj ' 0 dxj 9ct dxj
and
91 9P{a'X) - J f(yf^{y~X)dy J
dxj 9ct
dxj
^n?^1' x) hi!J]—{!J -r}
dSy
x € Qp, i = 0,1, j = 1, 2, 3;
moreover, the differentiation under the integral sign is legal and
cj dP(a,x) _ dx%- da dx%- ''
Theorem 3. Let S c C2, fi(y) € C 1(S0) n L(S), f2(y) € C(S0) n L(S), and let f be bounded and locally Holder continuous in Qp. Then for the existence of a function U(y) € HA(Qp) n C(S0) such that
U{y) = fi{y), y) = f2(y), ye So, (4.9)
it is necessary and sufficient that the following improper integral converge (uniformly on compact sets in Gp) for each x € Gp:
J (ct, x)d<r
< to,
(4.10)
where J(ct, x) is defined by (4.8). If (4.10) is satisfied, then harmonic continuation is performed by equivalent formulas (4.5) and (4.7).
p
X
X
Proof. Necessity: Let
U(y) € H(Qp) П C U So) П L(S)
satisfy (4.10). Let M be a compact set in Gp, and let e > 0 be such that M с Gpe С Gep С Gp. It is clear that the distance from M to dGp is at least er1 and the distance from dG^ to dGp is er1. Now, let y € R3 \ Gp (|y'| < т(y3 - e) and y3 > e) and x € M (|x'| < т(x3 - 2e) and x3 > 2e). Then argw = arg(aw) = arg(гту/и2 + s + ту3 — rx3) and
rw = ir y u2 + s + ту3 — rx3 туз - ТХз \y'\ - \x'\ - £T
= \JU2 + S +
туз — ТХз
2p Л^Т
s
u > 0, p > 1,
VWT
I <1-^1, У /Ж, arg(a±tg—)
п
> a < 1. 2p
Therefore, (2.5) is valid for arg w; moreover, if y' = x', then Re w < 0, and this inequality also holds. Consequently, — x) and ^a(y — x) satisfy estimates (3.2)-(3.5) from Lemma 1, where 5 > £T\. Define S£ = Gp n S; in this case, the part S£ C S together with the part T£ of the cone surface dGp form a closed piecewise smooth surface S£ U T£ (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£ U (S \ S£). Since ^o-(y — x) is a regular solution of the Poisson equation, by Green's formula, the integral over the part S£ is equal to the integral over T£; moreover, (y — x) satisfies inequalities (4.7) and (4.9) for y € T£ and x € M, and the extended function U(y) together with its gradient is bounded by a constant depending on e. Therefore, the modulus of the integral over the part S£ does not exceed the quantity
const 1 + ô2a2'
a> 0,
with a constant depending on p,e, and the diameter of the domain Qp. Since |y| > t(y3 — e), y3 > e, when y € S \ S£ and x € K and /i(y), f2(y) € C(So) n L(S), these inequalities remain valid for the modulus of the integral over S \ S£ (of course, with other constants). Hence, we have (4.10).
Sufficiency: Under the assumptions of the theorem, define functions U(x), x € Gp \ S0, by the right-hand side of (4.7). Consider the first term on the right-hand side of (4.7). Since (y) satisfies the Poisson equation in Gp for a > 0, the function J (a, x) satisfies the Poisson equation with respect to x in Gp for a > 0. Therefore, we conclude from (4.10) that the first term on the right-hand side of (4.7) satisfies the Poisson equation in Gp as the limit of the uniformly converging sequence of the solutions of the Poisson equations
(x) = J J (a, x)da,
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 Qp and another in Q'p = Gp \ Qp. Therefore, the right-hand side of (4.7) defines two different solutions of the Poisson equations U+(x) and U-(x) in Qp and If x1 and x2 are two points on the normal at x € So symmetric with respect to x, then
lim [U+(x1) — U-(x2)] = fi(x), lim
— (r1)-—(r2) dn {X ' dn {X '
= f2(x), x e So ;
s
moreover, the limit relations hold uniformly in x on each compact part S0. If max y3 < x3, where y € S and x € Gp, then Re w = y3 —x3 < 0 and (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 € Qp, by the uniqueness theorem. It is clear that U-(x) extends smoothly to QpUS0. Then U + (x) extends smoothly as a function of the class C 1(Qp U S0) (see [29]). Consequently,
dU+
U+(x) = fi(x), -^-(x) = f2(x), x € S0. Now, we set U(x) = U +(x), x € Qp U S0. Theorem 3 is proved. □
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