INVERSE PROBLEMS OF FINDING THE LOWEST COEFFICIENTIN THE ELLIPTIC EQUATION Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-4-528-542 УДК 517.946

Inverse Problems of Finding the Lowest Coefficient in the Elliptic Equation

Alexander I. Kozhanov*

Sobolev Institute of Mathematics Novosibirsk, Russian Federation Novosibirsk State University Novosibirsk, Russian Federation

Tatyana N. Shipina^

Siberian Federal University Krasnoyarsk, Russian Federation

Received 30.12.2020, received in revised form 14.03.2021, accepted 20.042021

Abstract. The article is devoted to the study of problems of finding the non-negative coefficient q(t) in the elliptic equation

utt + a2 Au — q(t)u = f (x, t)

(x = (xi,... ,xn) € Q С Rn, t € (0,T), 0 < T < A — operator Laplace on xi, ... , x„). These problems contain the usual boundary conditions and additional condition ( spatial integral overdetermination condition or boundary integral overdetermination condition). The theorems of existence and uniqueness are proved.

Keywords: elliptic equation, unknown coefficient, spatial integral condition, boundary integral condition, existence, uniqueness.

Citation: A.I. Kozhanov, T.N. Shipina, Inverse Promlems of Finding the Lowest Coefficient in the Elliptic Equation, J. Sib. Fed. Univ. Math. Phys., 2021, 14(4), 528-542. DOI: 10.17516/1997-1397-2021-14-4-528-542.

The problems studied in this work belong to the class of nonlinear inverse coefficient problems for elliptic differential equations.

Various aspects of the theory of linear and nonlinear inverse coefficient problems for differential equations are well covered in the world literature — see, for example, monographs [1-8], articles [9-19]. Directly for elliptic equations inverse coefficient problems were studied in [15-19] (a more detailed bibliography can be found in [17]).

The nonlinear inverse coefficient problems for elliptic equations studied in this work, the results obtained in it will be essentially differ either in the formulations (in particular, in the given redefinition conditions), or in the results from the statements and results from the works of predecessors.

The problems studied in this work have a model form. More general cases and also possible generalization of the obtained results will be discussed at the end of the article.

* kozhanov@math.nsc.ru ttshipina@sfu-kras.ru © Siberian Federal University. All rights reserved

1. Statement of the problems

Let Q be a bounded domain of variables (xi,..., xn) of space Rn, r is the boundary of Q. We assume that r is a compact infinitely differentiable manifold. Next, Q is a cylinder Q x (0, T) of finite height T, S = r x (0,T) is the lateral surface of Q. Let f (x,t), u0(x), u\(x), N(x) and n(t) be given functions defined for x G Q, t G [0, T]; let a be given positive number.

Inverse Problem I: Find functions u(x,t) and q(t) connected in the cylinder Q by the equation

utt + a2 Au — q(t)u = f (x,t) (1) provided that u(x, t) satisfies the conditions

u(x, 0) = uo(x), u(x,T) = ui(x), x G Q; (2)

u(x,t)\s = 0, (3)

/ N(x)u(x, t) dx = fj,(t), t G (0,T). (4)

JQ

Inverse Problem II: Find functions u(x,t) and q(t) connected in the cylinder Q by the equation (1) provided that u(x,t) satisfies (2), (4) and also the condition

du(x, t)

dv

0. (5)

s

Inverse Problem III: Find functions u(x,t) and q(t) connected in the cylinder Q by the equation (1) provided that u(x,t) satisfies (2), (5), and also the condition

j N(x)u(x,t) dsx = n(t). (6)

In Inverse Problems I and II conditions (2) and (3), (2) and (5) are the conditions of a correct boundary value problem for second-order differential elliptic equation in a cylinder Q, whereas condition (4) is space-integral overdetermination condition. In Inverse Problem III conditions (2) and (5) are also the conditions of a correct boundary value problem for second-order differential elliptic equations, whereas condition (6) is an boundary-integral overdetermination condition.

All constructions and arguments in this paper will be carried out using the Lebesgue spaces Lp and Sobolev spaces Wp. The necessary information about the functions from these spaces can be found in the books [20-22].

The goal of this article is to prove the existence and uniqueness of regular solutions to the problems under study, that is, of solutions having all the weak derivatives in the sense of Sobolev involved in the equation.

2. Solvability of the inverse Problems I h II

Perform some auxiliary constructions for Inverse Problem I. Given the function w(x,t), we

define the function $(t; w): $(t; w) = a2 J N(x)Aw(x,t) dx.

a

t T — t

Put vo(x,t) = tUi(x)+--t—uo(x), fi(x,t) = f (x,t) - a2Avo(x,t),

fo(t) = f N(x)fi(x,t) dx, <p(t) = , m = f(V(t) - fo(t)],

J a tl(t)

fo(x,t) = f1(x,t) + ty(t) + v(t)$(t; vo)]vo(x,t). Consider the boundary value problem: Find a function v(x,t) that is a solution to equation

vtt + a2Av - [<f(t)$(t; v + vo) + t)]v = f2 (x, t) + p(t)vo(x, t)$(t; v) (1')

and satisfies condition

v(x, 0) = v(x,T)=0, x e Q, (2')

and also the condition (3). Using a solution v(x,t) of this boundary value problem we can establish the solvability of the inverse problem I.

Integro-differential equation (1') is called loaded equation [23, 24].

n 1

Put po = max \f(t)\,^o = min ^(t), N1 = 2^1 [«0,, + «LJ dx, N = - ^20N1T ||N U^

l0,T i 1°>T 1 ¿=in 2

N3 = E I f0xt dxdt, N4 = , N5 = a0UN^(q) (TN4)1/0 + |$(0,«o)| + \^(0,«i)\.

i=1 Q a (1 - N0)

Theorem 2.1. Suppose the fulfillment of conditions

o 0 -1

N(x) e Lo(Q), j(t) e C0([0,T]); f (x,t) e Lo(0,T; W 0(Q)) n L^(0,T; Lo(Q)),

o 0 1 o 0 1

u0(x) e W03(Q) n W 0(Q), u1(x) e W03(Q) n W 1(Q), Au0(x) = Au1(x) = 0 for x e T;

po > 0, ^0 > 0, N0 < 1, N5 < —;

Po

j(0) = N(x)uo(x) dx, j(T) = / N(x)«1(x) dx. JQ JQ

Then the inverse problem I has a solution {u(x,t), q(t)} such that u(x,t) e W°(Q), Au(x,t) e W1(Q), q(t) e L^([0,T]), q(t) > 0 for t e [0,T].

Proof. We establish the solvability of the boundary value problem (1'), (2'), (3) in the space W°(Q). We use the regularization method and method of cut-off functions.

Let y be a number from the interval ^0, ^ . Define the cut-off function GY (£):

( e, if \e\ < Y, Gy (0 = < Y, if e > Y, I —Y, if e < -Y.

Next, let e be a positive number. Consider the boundary value problem: find a function v(x,t) that is a solution to equation

vtt + a0Av - ty(t) + p(t)Gy (&(t; v + vo))]v - eA2v = fo (x, t) + p(t)vo(x, t)$(t; v) (1'F)

in the cylinder Q and satisfies conditions (2') and (3') and also the condition

Av(x,t)\s = 0. (7)

Show that for a fixed number e this problem has a solution belonging to W4'0(Q). Let's use the fixed point method.

Let w(x,t) be a function from the space W4'0(Q). Consider the boundary value problem: find a function v(x, t) that is a solution to equation

vtt + a0Av - ty(t) + v(t)Gy(&(t; w + vo))]v - eA2v = fo(x, t) + p(t)vo(x, t)$(t; v) (1'e w)

in the cylinder Q and satisfies conditions (2'), (3), (7).

This problem is the first boundary value problem for linear loaded quasi-elliptic equation. Using method of continuation in the parameter (see [25]), it is not difficult to establish its solvability in the space W24'2(Q).

Let A be a number from the segment [0,1]. Consider the boundary value problem: find a function v(x, t) that is a solution to equation

vtt + a2Av - ty(t) + p(t)G7(&(t; w + vo))]v - eA2v = f2(x, t) + Ap(t)vo(x, t)$(t; v) (1'E, w, x)

in the cylinder Q and satisfies conditions (2'), (3) and (7). For A = 0, this problem is solvability in the space W24'2(Q) (this is not difficult to prove using the classical Galerkin method with the choice of a special basis [21]). Next, all possible solutions v(x,t) of the boundary value problem (1'e,w, a), (2'), (3), (7) at a fixed e satisfy estimate

(l - l|N||ia(n)) / (Avt)2 dxdt + O2 ¿/(Avxi)2 dxdt+

\ / J Q i=1 Q

+e (A2vf dxdt < Cj f2 dxdt, (8)

QQ

with the constant C1 defined only by e. In order to prove this estimate we multiply the equation (1'e w A) by the function -A2v and integrate on the cylinder Q. Using ^(t) + t)GY(&(t; w + vo)) > 0 and applying Holder's and Young's inequality, and also inequality

/ [Av(x,t)]2 dx < T/ [Avt(x,t)]2 dxdt

Ja Jq

we obtain the estimate (8). From estimate (8) and the second main inequality for elliptic operator (see [21]) it follows that all possible solutions v(x,t) to boundary value problem (1'E w a), (2'), (3), (7) for a fixed e satisfy the a priori estimate

l|v||<.2(Q) < C2 llf2llL2(Q) (9)

with the constant C2 defined only by the domain Q, the functions n(t), N(x), uo(x) and u1(x) and numbers a, T,e. According to the theorem on the method of continuation in a parameter [25, ch. III, Sec. 14], solvability of the boundary value problem (1'E w o), (2'), (3), (7) in W24 ' 2(Q) and estimate (9) imply that the problem (1'E, w), (2'), (3), (7) has a solution v(x,t) lying in the space W24'2(Q).

Held arguments signify that the boundary value problem (1'E w), (2'), (3), (7) generates the operator A, taking the space W2'2(Q) to itself: A(w) = v. We show that for the operator A, all the conditions of Schauder's fixed point theorem are satisfied.

Observe first of all that from the estimate (9) it follows that the operator A takes a closed ball of radius Ro = C2Hf2HL2(Q) of space W24'2(Q) to itself.

We now show that the operator A will be continuous on a closed ball of radius R of the space W2 ' 2(Q).

Let [wm(x,t)}'^=1 be a sequence of functions from this ball converging in the space W2 '2(Q) to the function w(x,t). Let vm(x,t), v(x,t) be images of functions wm(x,t) and w(x,t) under action of the operator A. There are equalities

vmtt - vtt + a2A(vm - v) - eA2(vm - v) - [<f(t)G7(®(t; wm + vo)) + ^(t)](vm - v) =

= <f(t)[G7(&(t; wm + vo)) - Gy(&(t; w + vo))]v + f(t)vo(x,t)$(t; vm - v), (x,t) e Q,

vm(x, 0) - v(x, 0) = vm(x, T) - v(x, T) = 0, x e Q,

vm(x,t) - v(x,t)\s = A(vm(x,t) - v(x,t))\s = 0.

These equalities mean that the functions vm(x,t) - v(x,t) are solutions to the first boundary value problem for the linear quasi-elliptic "loaded" equation (1'E^w). Note that the function GY(£) satisfies the Lipschitz condition and v(x,t) e W24'2(Q). Repeating the proof of the estimate (9) and applying the Holder's inequality, we get inequality

- ^W^Q) < C3]\wm - w^L2(Q) (10)

with constant C3, defined by the functions n(t), N(x), uo(x) and u1(x), as well as the numbers a, T, and e. From this inequality and from the convergence of the sequence {wm(x,t)}^=1 in space W^'2(Q) to the function w(x,t) it follows that the sequence {vm(x,t)}^^i=1 converges in the same space to the function v(x,t). This means that the operator A is continuous on a closed ball of radius Ro of the space W24'2(Q).

We show that the operator A is compact on a closed ball of radius Ro of the space W24'2(Q).

Let {wm(x,t)}^=1 be a family of functions from this ball. Let {vm(x,t)}'^=1 be a family of images of functions wm(x,t) under the action of the operator A. Boundedness of families {wm(x,t)}^=1 in the space of W24'2(Q) and the classical embedding theorems [20-22] imply that there is a subsequence {wmk (x,t)}^=1, strongly convergent in the space L2(Q). Repeating for the difference vmk (x,t) - vmk+l (x,t) proof of the estimate (10), it is easy to obtain, that the sequence {vmk(x,t)}cj=1 is the fundamental in the space W24'2(Q). And this means that the operator A is compact on the ball of radius Ro of the space W24'2(Q).

So, the operator A takes a ball of radius Ro of the space W24 2 (Q) to itself. The operator A is continuous and compact on this ball. According to Schauder's theorem, in the indicated ball there is at least one function v(x,t), for which holds A(v) = v. This function v(x,t) e W2,'2(Q) is solution of the boundary value problem (1E), (2'), (3), (7). Show that the solutions v(x,t) satisfy a priori estimates uniform in e.

Consider the equality

- {vtt + a2 Av - [^(t) + p(t)GY($(t; v + vo))]v - eA2v}A2v dx dt =

Q

- f2A2vdxdt - p(t)vo(x,t)$(t; v)A2v dxdt.

QQ

Integrating by parts and applying the Cauchy-Bunyakovsky and Young inequalities, we conclude that this equality implies the estimate

f 2 n f ( n f \ 1/2 (1 - N2) (Av) dxdt + ^V (Avx. )2 dxdt < n1/2 V (Avx. )2 dxdt) Jq 2 i=JQ \i=JQ * )

It is easy to show that there are estimates

V f (Avxi)2 dx dt < N, f (Avt)2 dx dt < = N4.

7=1 Jq a4 Jq a2(1 - n2)

Summing up the inequalities and also using the second main inequality for elliptic operators, we obtain that solutions solutionsv(x,t) to the boundary value problem (1'E), (2'), (3), (7) satisfy the estimates

\$(t; v + vo)\ < N5, (11)

|v|w2(q) + l|Av||w2i(Q) + VeilA2vHL2(Q) < C4, (12)

with the constant C4 b (12) defined by the functions j(t), N(x), u0(x) h u1(x), and the numbers a h T.

The estimate (12) and the reflexivity of a Hilbert space imply that there exist sequences {em}m=1 of positive numbers and {vm(x,t)}'^i=1 of solutions to the boundary value problem (1'em), (2'), (3), (7) and also a function v(x,t) such that, as m ^ x>, the convergences

em ^ 0, vm(x,t) ^ v(x,t) strongly in W°(Q),

emA2vm(x,t) ^ 0 weakly in L0(Q)

hold.

Obviously, the limit function v(x,t) will be a solution to the boundary value problem (10), (2'), (3), and due to estimate (12) for this solution will be the inclusions v(x,t) e W°(Q), Av(x,t) e W1(Q).

Let us fix the number y: Y = ~. Let us define the functions u(x,t) and q(t):

Po

u(x, t) = v(x, t) + vo(x, t), q(t) = ^(t) + p(t)§(t; u).

Estimate (11) and the inequality from the condition of the theorem for the number N5 mean that the equality GY(§(t; u)) = §(t; u) holds, and that q(t) > 0 Vt e [0, T]. Obviously, the functions u(x,t) and q(t) will be related in the cylinder Q by equation (1). Let's show that for the function u(x,t) the overdetermination condition (4) will be satisfied.

We multiply equation (1) by the function N(x) and integrate over the domain Q. Taking into account the form of the functions p(t), ^(t) , §(t; u) and consistency conditions for of the functions u0(x), we obtain that the function a(t) satisfies the problem

a''(t) - q(t)a(t)=0, a(0) = a(T) = 0. (13)

Since q(t) ^ 0, then a(t) = 0. This means that the function u(x,t) satisfies the overdetermination condition (4). The theorem is proved. □

The study of the solvability of the inverse problem II differs only in insignificant details from the above study of the solvability of the inverse problem I. Let

Ne = V2p0 [4(x) + u2(x)] de) 7 \\N\\l2(n),

1/2

N7 = [pjn dx) \\h\\L2(Q) + l$(°,uo)| + \mui)\.

Theorem 2.2. Suppose the fulfillment of conditions

N (x) G Wi(Q), p(t) G C2([°,T]); f (x,t) G L2(°,T; W22(Q)),

uo(x) e Wi(Q), mix) e W3(Q); po > 0, '0 > 0, N6 < 1, N7 < ^,

Po

j(0) = N(x)uo(x) dx, j(T)= / N(x)ui(x) dx. JQ JQ

Then inverse problem II has a solution {u(x, t), q(t)} such that u(x, t) e W^iQ), q(t) e LTO([0, T]), q(t) > 0 for t e [0,T].

3. Solvability of the inverse Problems III

We introduce the function &i(t; w): $i(t; w) = a2 j N(x)Aw(x,t) dsx, where w(x,t) is some

r

given function.

Introduce the notations Fo(t) = f N(x)f (x,t) dsx, 'i(t) = p(t)[j''(t) - Fo(t)],

r

h(x,t) = fi(x,t) + ['i(t)+ p(t)$i(t; vo)]vo(x,t).

Consider the boundary value problem: Find a function v(x,t) that is a solution to equation

vtt + a2Av - [p(t)$i(t; v + vo) + '(t)]v = h(x,t) + V>(t)vo(x,t)$i(t; v) (14)

in the cylinder Q and satisfies conditions (2') and (5). A solution v(x,t) to this problem will provide an opportunity construct the required solution to the inverse problem III. The function w(x) e W^Q.) satisfies the inequality

/ w2 (x) dsx < co JT JQ

2

w (x

(x)+Y1 wXXi(x)

dx

(15)

with a constant c0 determined only by the domain Q (see [20, 21]). Let us specify again that the function v0(x,t) satisfies the inequality

n n 10 p

Y] / v0ixi(x,t) dx <2 Y] / [uoxi (x) + u\x% (x)]dx-„•_i JQ „--i JQ

i=i"Q i=i As before, we define the required constants:

'i = min 'i(t), Ns = ^ Q f2x% dx dtj

1/2

N9 =

cort NT 2n.2

4a2

L2(r)'

(16)

N10 = co^N, Nu = 2N

4a2

a2 - 2N9 ''

Nu =

NsNn 1 - N10 '

N13 = V2po JQ [(Auo)2 + (Aui)2] dx (NçN2! + NwN^2 + ( Q faf dxdt^j

1/2

N14 = a2\\N\\2L2{T)(coT )1/2

N2 + N3

a2

1/2

+ |$i(0; uo)| + |$2(0; ui)\.

Theorem 3.1. Suppose the fulfillment of conditions N(x) G L2(r), n(t) G C2([0,T]); f (x,t) G L2(0,T; Wi(Q)), uo(x) G W3(Q), ui(x) G W3(Q); df (x,t) dAu0(x) dAul(x)

dv dv dv

= 0 for x G r;

po > 0, > 0, Nw < 1, a2 - 2N9 > 0, N14 < —.

Po

n(0) = j N(x)uo(x) dx, n(T) = j N(x)ui(x) dx.

Then inverse problem III has a solution {u(x,t),q(t)} such that u(x,t) G W^(Q), Au(x,t) G W1(Q), q(t) G L^([0,T]), q(t) > 0 for t G [0,T].

Proof. Let y be a number from the interval ^0,

, e > 0.

Consider the boundary value problem: Find a function v(x,t) that is a solution to equation vtt + a2 Av - tyi(t) + p(t)Gy($i(t; v + vo))]v - eA2v = f2(x, t) + p(t)vo(x,t)^i(t; v) (14'e) in the cylinder Q and satisfies conditions (2'), (5), and

d(Av)

dv

d(A2v)

s dv

= 0. (17)

S

Using the fixed point method and the method of continuation in a parameter, it is easy to show that for a fixed e and for satisfying the conditions of the theorem, this problem has a solution v(x,t) such that v(x,t) G W2;(Q), A2v(x,t) G L2(Q). Let us show that the function v(x,t) satisfies a priori estimates uniform in e.

Multiply equation (14'e) by the function A2v(x,t) and integrate over the cylinder Q. We obtain the equality

/ (Avt)2 dxdt + a2 / (Avx.)2 dxdt + e (A2v)2 dx dt+ JQ i=lJ Q Jq

+ f \^i(t) + y(t)C7 ($i(t; v + vo))](Av)2 dxdt = V / fat Avxi dxdt-

JQ i=\-' Q

n »

-Y, <f(t)$i(t; v)voxt Avxt dxdt. (18)

Q

Let us introduce the notation: I1 = J(Avt)2 dxdt, I2 = J2 f (Avxi)2 dxdt.

Q i=l Q

Taking into account the notation introduced above and using the Young's and Cauchy -Bunyakovsky's inequalities it is easy from the equality (18) go to inequality

h + y I2 < NsIy2 + *î(t; v) dt. (19)

f $2(t; v) dt < N912 + Nwh. (20)

o

There is an estimate

!■ T

F*2 . , - . ,., . . ' V io

Summing up, we obtain a consequence of inequalities (19) h (20):

2

Ii + 212 < NSI1/2 + N912 + Nioli. (21)

Elementary calculations allow us to derive from (21) the estimate

I2 < Nh, (22)

and further, the estimate

Ii < N12. (23)

Equality (18) and estimates (22), (23) imply the boundedness of the first term on the left side of(1<):

e vft dxdt < C5. (24)

Q

Here the constant C5 is determined by the functions f (x,t), uo(x), ui(x), N(x), n(t), numbers a and T (the exact value of the number C5 is not important).

Multiply equation (14'e) by the function A2v(x,t) and integrate over the cylinder Q. We obtain the equality

n „ r.

\2 , „2 /A 2 x2

У2 / (AvXit)2 dxdt + a2 (A2v)2 dxdt+

i=ij q jq

+ ^ / \^1(t) + ф(t)GY (Ф1 (t; v + v0))](AvXi )2 dxdt + £ (A3v)2 dxdt =

jq г jq

-iJ Q

= Af2 A2vdxdt + I p(t)^i(t; v)AvoA2v dx dt. (25)

QQ

An inequality similar to the inequality (18) holds:

/ [Avo(x,t)]2 dx < 2 [(Auo)2 + (Aui)2] dx. JQ JQ

Using this inequality, Holder's inequality and estimates (20), (22), (23), we obtain from (25) the inequality:

a2 j (A2v)2 dxdt ^ j (A2v)2 dxdt^ ^I (Af^ dxdt^ +

it \ 1/2 ' +V2voU [(A«o)2 + (A«i)2] dx) (N9N2i + NioNi2)1'2 .

This inequality and again from equality (25) imply the estimates

I (A2v)2 dxdt < , (26)

Jq a

С n f JJ" 2

£ (A3v)2 dxdt + V (AvXit)2 dxdt < N3. (27)

Jq ,_i Jq ъ a

Estimates (22)-(24), (27), estimates for solutions of elliptic equations (see [21]) and also the reflexivity of a Hilbert space imply that there exist sequences {em}^=1 of positive numbers and

{vm(x,t)}'^=i to the boundary value problems (14'Em), (2'), (5), (17) and also a function v(x,t) such that, as m ^ œ, the convergences

£m ^ 0, vm(x,t) ^ v(x,t) weakly in W^Q),

Avm(x,t) ^ Av(x,t) weakly in Avm(x,t) ^ Av(x,t) strongly in L2(r), emA3vm(x,t) ^ 0 weakly in L2(Q) hold. The limit function v(x,t) satisfies the equation

vtt + a2Av - [tyi (t) + y(t)Gy (№i(t; v + vo))]v = h(x,t) + y(t)v0(x,t)$i (t; v),

and the conditions (2'), (5). The function v(x,t) belongs to W2(Q) and Av(x,t) G W2(Q), A2v(x,t) G L2(Q), AvXit(x,t) G L2(Q), i = 1,... ,n. The following inequalities

\№i(t; v + vo)| < \№i(t; v)| + \№i(t; vo)| < a2\\N\\L2(r) ^(Av)2 ds^ 7 + \№(0; uo)\ + \№(0; ui)\ <

r n li/2

f (Av)2 dx + V f (AvXi)2 dx

J J i=iJQ

i (Avt)2 dxdt + V i (Avxit)2 dx JQ „-_i JQ.

< a c.

0 \N \\L2(r)

< ^(coT)V2\\N\\L2(r) + \№(0; ui)\ < a2(coT)i/2\\N\L2(r)

+ \$(0; uo)\ + №0; ui)\ < i/2

+ \№(0; uo)\+

N2

a2

i3 + Ni2

iQ i/2

+ \№(0; uo) \ + \№(0; ui)\ = Nu (28)

hold.

^1

Let y = —. Due to the condition N14 < — it follows from (28) that GY(&1(t; v + v0)) = ¥o ¥o

= v + v0). Let us define the functions u(x,t) h q(t):

u(x, t) = v(x, t) + vo(x, t), q(t) = ^i(t) + ¥(t)&i (t; u).

It is these functions that give the required solution to the inverse problem III (which is shown as in the proof of Theorem 2.1). The theorem is proved. □

4. Uniqueness of solutions

The following theorems give conditions under which the inverse problems I-III can only have one solution.

Let WRo = | v(x,t) : v(x,t) G W2(Q), vrai m ax ^ J v2(x,t) dx^ < Ro^.

Theorem 4.1. Let {ui(x,t),qi(t)}, {u2(x,t), q2(t)} be two solutions of the inverse problem I such that ui(x,t) G WRo, qi(t) G Lœ([0, T]), qi(t) ^ 0 for t G [0, T], i = 1, 2. Suppose the fulfillment of the conditions

N (x) G L2 (i), n(t) G C2([0,T]), f (x,t) G L^(0,T; L2(i)); fo > 0, foR^WN ^(j) < 1.

Then the functions ui(x,t) and u2(x,t) coincide almost everywhere in Q, the functions qi(t) and q2(t) coincide for almost all t from the segment [0, T].

Proof. The function w(x,t) = ui(x,t) — u2(x,t) satisfies the following problem wtt + a2 Aw — qi (t)w = p(t)§(t; w)u2, (x, t) G Q; w(x, 0) = w(x, T) = 0, x G Q; w(x,t)\s = 0.

We multiply the equation by the function Aw(x,t) and integrate over the cylinder Q. Taking into account the nonnegativity of the function qi(t) and the boundary conditions, applying

Holder's inequality, we obtain the inequality f (Aw)2 dxdt < 0. This inequality implies that the

Q

functions ui(x,t) and u2(x,t) coincide almost everywhere in Q. But then the functions qi(t) h q2(t) coincide for almost of all t from the segment [0, T]. The theorem is proved. □

Theorem 4.2. Let {ui(x,t),qi(t)}, {u2(x,t), q2(t)} be two solutions of the inverse problem II such that ui(x,t) G WRo, qi(t) G L^([0,T]), qi(t) ^ 0 for t G [0, T], i = 1, 2. Suppose the assumptions of Theorem 2.2 are fulfilled. Then the functions ui(x,t) and u2(x,t) coincide almost everywhere in Q, the functions qi(t) and q2(t) coincide for almost all t from the segment [0,T ].

The proof of this theorem is quite similar to the proof of Theorem 4.1. Let

WRo = i v(x,t) : v(x,t) G W2(Q), Av(x,t) G W2,(Q), vr&im&xiy] v2x (x,t) dx] < Ro

I [o -T \i=ijQ '

Theorem 4.3. Let {u\(x,t), qi(t)}, {u2(x,t), q2(t)} be two solutions of the inverse problem III such that Ui(x,t) G WRo, q^(t) G L^([0,T]), q^(t) ^ 0 for t G [0, T], i = 1,22. Suppose the fulfillment of the conditions

N (x) G L2(r), v(t) G C2([0,T]), f (x,t) G L2(Q) n L^(0,T; L2(r));

Vo > 0, vo(coRo)1/2\\N||La(r) < mm , -J^^j .

Then the functions u\(x,t) and u2(x,t) coincide almost everywhere in Q, the functions qi(t) and q2(t) coincide for almost all t from the segment [0, T].

Proof. The function w(x,t) = u\(x,t) — u2(x,t) satisfies the following problem

wtt + a2 Aw — qi(t)w = v(t)$i(t; w)u2, (x,t) G Q; (229)

w(x, 0) = w(x,T) = 0, x G i; (30)

dw(x, t)

dv

Equalities (29) and (31) imply, in particular, the property

dAw(x, t)

= 0. (31)

S

dv

= 0. (322)

S

Further, using the procedure for approximating the function w(x,t) by smooth functions while maintaining the property (32), it is easy to show that the equality holds (formally obtained by multiplying equation (29) by the function A2w and integrating over Q)

/ (Awt)2 dxdt + a2Y, / (AwXi)2 dx dt + / q(t)(Aw)2 dx dt = JQ i=1J Q Jq

= J p(t)$i(t; u2xi Awx^ dxdt. (33)

We obtain an estimate for the right-hand side of the inequality (33). Using the Cauchy-Bunyakovsky's and Holder's inequalities, the condition (32) and estimate

f (Aw)2 dxdt < ^ f (Awt)2 dxdt, JQ 2 JQ

we obtain the inequality

/ (Awt)2 dxdt + a2^ / (AwXi )2 dxdt < JQ i=iJ Q

< 1 a2T2<fo(coRo)1/2\\N\\l2(f) f (Awt )2 dxdt +3 a2Vo(co Ro)1/2\\N 11^ V f (Awxt )2 dxdt. 4 JQ 2 i=iJ Q

This inequality and the conditions of the theorem imply the identities Awt(x,t) = 0, AwXi = 0 for (x,t) G Q, i = 1,... ,n, and further follows the identity w(x,t) = 0 for (x,t) G Q. The last identity means that he functions u\(x,t) and u2(x,t) coincide almost everywhere in Q. But then the functions qi(t) h q2(t) coincide for almost of all t from the segment [0,T]. The theorem is proved. □

5. Comments and appendices

1. Let us show that the set of input data of inverse problems I-III, for which all conditions of the existence and uniqueness theorems are satisfied, is not empty.

Let u0(x) and ui(x) be given nonnegative functions in Q such that, in addition to the conditions of Theorem 2.1, they satisfy the conditions

duo(x) dui(x) dv dv

= 0 for x G r,

/ uo(x) dx = i ui(x) dx = 1. JQ JQ

Similar functions exist. For example, u0(x) = a0[p(x)]mo, u\(x) = a.i[p(x)]mi, where p(x) is the distance from the point x G Q to the boundary r, m0 > 3, m\ > 3. The multipliers a0 and a\ are selected so that the required integral equalities hold. Or u0(x) and u1(x) can be finite in Q.

Let N(x) = 1, n(t) = 1, f (x,t) = f0(x), f0(x) < 0, x G Q. Then

^(t) = - f0(x) dx = ^0 > 0, $(0,u0) = $(0,ui) = 0. JQ

Obviously, the condition N2 < 1 of Theorem 1 will hold for small numbers T, the number N5 can also be made arbitrarily small by decreasing the number T. Hence, for the given functions f (x,t), u0(x), u1(x), n(t) and N(x) for small T all conditions of the Theorem 1 will be satisfied.

Condition N6 < 1 of Theorem 2.2 will be be executed if the functions u0(x) and u\(x) or the measure of the region i are small, the condition for the number N7 will be run automatically.

The non-emptiness of the set of input data for which all conditions of Theorem 2.2 are satisfied is also easy to show. Take as u0(x), u^(x), N(x) and p(t) are identically constant functions, f (x,t) is negative function in Q. Condition N6 < 1 of Theorem 2.2 will be be executed if the functions u0(x) and ui(x) or the measure of the domain l are small, the condition for the number N7 будет will be run automatically.

Conditions of Theorem 3 are satisfied for small numbers T, if the functions u0(x), u^(x), N(x) and n(t) are identically constant functions, f (x,t) > 0, (t,x) e Q and overdetermination conditions hold.

Obviously, the conditions of the uniqueness theorems (Theorems 4.1-4.3) will obviously be satisfied for small numbers R0.

2. Inverse problems I-III can also be studied for equations that are more general than (1). Thus, the Laplace operator can be replaced an arbitrary second-order elliptic operator with variable coefficients, into the equation (1) low-order terms with first-order derivatives can be added. The essence of the results obtained is a more general form of the equation (1) will not change, but the number of calculations will increase.

3. If the conditions of existence theorems are satisfied, then for solutions u(x,t) of inverse problems I, II, or III it is easy to establish estimates for quantities defining the sets WRo or WRo. The constants in these estimates will be determined by the input data. Using further conditions of the respective theorems of uniqueness, it will be easy to obtain theorems that give both the existence and the uniqueness of solutions to inverse problems I, II, or III.

4. Conditions (3) or (5) in inverse problems I, II, or III can be inhomogeneous. Assuming that there are continuations of the given boundary data into the cylinder Q and using the technique of proving Theorems 2.1-2.2, Theorem 3.1, it will be possible to obtain the solvability of the inverse problems with nonzero boundary data.

The work is supported by the Russian Foundation basic research (grant 18-01-00620).

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Обратные задачи восстановления младшего коэффициента в эллиптическом уравнении

Александр И. Кожанов

Институт математики им. С.Л. Соболева Новосибирск, Российская Федерация Новосибирский Государственный Университет Новосибирск, Российская Федерация

Татьяна Н. Шипина

Сибирский федеральный университет Красноярск, Российская Федерация

Abstract. Изучается разрешимость обратных задач восстановления неотрицательного коэффициента q(t) в эллиптическом уравнении

utt + a2 Au — q(t)u = f (x, t)

(x = (x1,...,xn) € О С Rn, t € (0,T), 0 < T < A — оператор Лапласа, действующий

по переменным xi, ... , xn). Вместе с естественными для эллиптических уравнений граничными условиями в изучаемых задачах задают также одно из дополнительных условий — либо условие пространственного интегрального переопределения, либо же условие граничного интегрального переопределения. Доказываются теоремы существования и единственности решений.

Ключевые слова: эллиптические уравнения, неизвестный коэффициент, пространтсвенное интегральное переопределение, граничное интегральное переопределение, существование, единственность.