On an inverse problem for quasi-linear elliptic equation Текст научной статьи по специальности «Математика»

УДК 517.95

On an Inverse Problem for Quasi-Linear Elliptic Equation

Anna Sh. Lyubanova*

Institute of Space and Information Technology Siberian Federal University Kirenskogo, 26, Krasnoyarsk, 660026

Russia

Received 12.11.2014, received in revised form 03.12.2014, accepted 20.12.2014 The identification of an unknown constant coefficient in the main term of the partial differential equation -кМф(и) + g(x)u = f (x) with the Dirichlet boundary condition is investigated. Here ф(и) is a nonlinear increasing function of u, M is a linear self-adjoint elliptic operator of the second order. The coefficient к is recovered on the base of additional integral boundary data. The existence and uniqueness of the solution to the inverse problem involving a function u and a positive real number к is proved.

Keywords: inverse problem, boundary value problem, second-order elliptic equations, existence and uniqueness theorem, filtration.

This paper proceeds the investigation started in [5,6] and is concerned with an inverse problem for the second order differential equation

-div(k + y (x,u) = f (0.1)

with the Dirichlet boundary data

u = в(х) (0.2)

3Q

where k(x, u) is a matrix of functions, ф(и) and y(x, u) are scalar functions. We assume that Q is a bounded domain in Rn with the boundary dQ С C2.

Applications of this problem deal with the recovery of unknown parameters indicating physical properties of a medium (the heat conductivity, the permeability of a porous medium, etc.). Various aspects of such problems are discussed in [1-8] (see also the references given there). Of special interest is the problem of finding the leading coefficients of (0.1) given the additional boundary data on дQ or a part of dQ. In [1,2,7,8] this problem is investigated in the case where ф(и) = u, y(x,u) = 0, k(x,u) = k(x)E, a function k(x) is unknown, E is the identical matrix.

In this paper we study the problem of the identification of the constant coefficient k in equation

-k{d«v(M(x)V^(u)) + m(x)^(u)} + g(x)u = f(x), x G Q, (0.3)

under the boundary data (0.2). Here is a known function, M(x) is a matrix of functions mj (x), i,j = 1, 2, ...,n. The functions m(x), f (x), g(x), e(x) are given. In physical context the constant coefficient k is interpreted as the average conductivity of a medium.

As an additional data for the recovery of the coefficient k we take the condition of overdetermination

k / wde = <p (0.4)

Jsq dN

where du/dN = (M(x)Vu, n)R is the conormal derivative, n is a unit outward normal to dQ, ш is a given function, ^ is a given real number. Physically, (0.4) describes, for instance, the total flux of the liquid through the surface of the rock [3]. Inverse problems with similar nonlocal boundary

* lubanova@mail.ru © Siberian Federal University. All rights reserved

conditions for elliptic equations were considered in [5,6]. The results of [6] are concerned with the existence and uniqueness of the solution to the problem (0.2)-(0.4) where ^(p) = (p — 1)-1 |p|p-2p and p > 2 is a real number. The inverse problem for linear elliptic equation (0.3) with ^(p) = p is discussed in [5].

The goal of our paper is to establish the existence and uniqueness of a solution to the nonlinear problem (0.2), (0.3), (0.4). We discuss it in Section 2. In Section 3 we present examples and comments to the results of Section 2 and the inverse problem involved.

1. The preliminaries

We start with preliminary results for the direct problem (0.2), (0.3). The results concern the existence, uniqueness and certain properties of the solution to this problem.

From now on we keep the notations: || • ||R, (•, •)R is the norm and the inner product of Rn; || • ||, (•, •) is the norm and the inner product of L2(O); || • |j, (-, •) 1 is the norm of Wj(O), j = 1, 2,

o

and the duality relation between W2 (O) and W2-1(O), respectively; a(x) is the solution of the Dirichlet problem

-div(M(x)V^(a)) + m(x)^(a) = 0, x G O, a|dfi = ft(x); (1.1)

b(x) is the solution of the Dirichlet problem

-div(M(x)Vb) + m(x)b = 0, x G O, = w(x). (1.2)

We introduce the linear operator M : W2(O) ^ (W2(O))* of the form M = -div(M(x)V) + m(x)I where I is the identity operator and reckon that the following assumptions hold throughout the paper.

I. The operator M is elliptic. That is, m(x), mj (x), dmj/dx; , i, j, l = 1, 2, ...,n are bounded, m(x) > 0 and there exist positive constants m1, m2 such that for any £ G Rn

n n n

m^£i2 < E mij(x)£i£j < m^£2. (1.3)

i=1 i,j = 1 i=1

II. The operator M is self-adjoint. That is, mj(x) = mjj(x) for i, j = 1,..., n.

III. The function ^(p) is a continuous mapping of (-to, onto itself . For any p1,p2 G

(-TO, p1 = p2,

(^(P1) - V>(/°2))(p1 - p2) > 0, (1.4)

that is, the function ^(p) is strictly monotone.

From the assumption III it follows that there exists an inverse ^-1(p) of ^(p). The inverse ^-1(p) is strictly monotone and continuous on (-to,

By the assumptions I-III the problem (1.1) has a unique solution a and ^(a) G W22 (O) when ^(ft) G W23/2(dO). If in addition ft G O), then, by the continuity of ^ and Theorem 13.1 of [9, Chapter 3], ^(a) G LTO(O). This means that a G LTO(O).

Lemma 1.1. Let the assumptions I-III are fulfilled. Let also k be a given positive number, f G L2(O), ^(ft) G W21/2(dO), g G C(O) and g ^ 0 in O. Then the following assertions are valid.

1. There exists a unique solution u of the direct problem (0.2),(0.3) such that ^(u) G W21(O).

2. If V>(P) e W23/2 (dQ) and

|V(p)| > c|p|p (1.5)

for any p e (—ro, where p ^ 1, c > 0 are constants, then u e L2p(Q) n W = {v | ^(v) e

W|(Q)}.

3. If g = 0 and ^(P) e W23/2(dQ), then ^(u) e W22(Q). /fin addition P e LTO(dQ), then u e LTO(Q) n W.

Proof. 1. Let a = ^(a). From (1.1) it follows that a solutions the equation Ma = 0 and satisfies the boundary condition a|dQ = ^(P). Multiplying (0.3) by V>(u) = ^(u) — a in terms of the inner product of L2(Q) and integrating by parts in the first term we obtain

k j | (MV<a(u), W(u))r + m(x)^>2(u)|dx + j g(u — a)?/>(u)dx = (—ga + f, V'(u)). (1.6)

The estimation of the right side of (1.6) with the help of the Friedrichs inequality

1/2 o . ||VV||Rdx; for v eW

in

leads to the relation

/ I \1/2 0

< c0mes1/n^ J ||Vv||RdxJ for v eW^ (ft) (1.7)

k (MV^i(u), V"0(w))r + m(x)-02(w)|dx + j g(u — a)-0(w)dx ^

km1 t . Il2 , c0mes2/nQ.. „l|2

^ ~2T |Vi^(u)|Rdx + 0 2kmi ||ga — f ||2

whence, by (1.3), (1.4),

||VV>(w)||Rdx < m~/ { (MVi/>(m), W(w))r + mt^2(u^dx <

c2mes2/nQ,, „ll2 „

< 0 k2m1 Iga — f |2 = C (1.8)

Here the positive constant c0 depends only on n. (1.7) and (1.8) implies

II^a(u)| < C0mkes2/"n ||ga — f || (1.9)

km1

and

||^(u)|1 < ||a|1 + C1/2max{1,c0mes1/nn} = C1. (1.10)

To prove the existence and uniqueness of the solution of the problem (0.2), (0.3) we denote V>(u) by h. In view of the definition of V'(u) the function h satisfies the equation

M h = Mh + g^-1(h + a) — ga = f — ga = f (1.11)

o

and h|dQ = 0. One can show that the operator M as a map from W21 (Q) into W2-1(Q) is

o

demi-continuous, coercive and strongly monotone. Hence, there exists a solution h eW21 (Q) of equation (1.11) (see Theorem 2.1 [10, Chapter III]). Moreover, the solution h is unique. Indeed,

o

let h1 and h2 be two solutions of (1.11) in W21 (Q). Then by (1.3) and the strong monotonicity of M we have

0 = (Mh1 — Mh2, h1 — h^1 > m1|V(h1 — h2)|2,

which implies h1 = h2.

Coming back to the original problem we conclude that the problem (0.2), (0.3) has a solution

u = — 1(h) G W and this solution is unique.

2. In the hypotheses of the lemma from (1.9) it follows that u G L2p(O) and

c2mes2/nQ,

||u||L2P(n) < ||V(u)|| < U km1 ||ga - f || + ||^(a)| = C2.

Multiplying (0.3) by M^(u) in terms of the inner product of L2(O) we obtain the equality

k||M^(u)||2 = -(gu, M^(u)) + (f,M^(u)), from which it follows by (1.9) and the Cauchy inequality that

1 r 1 k

k||M^(u)||2 < - ygy2c(S)yuy22p(Q)mes(p-1)/pQ+ ||/||2 + -||M^(u)||2 <

1

< -^ k

|2 ,C2mes(p-1)/pQ + ||/||2

k

2

whence

/2 r n 1/2

||M^(u)| < ^ [|g|2C(^)C2mes(p-1)/pO + ||f ||2] = C2. (1.12)

In view of (1.8), (1.10) and Theorem 5.1 of [11, Chapter 2] the last inequality implies

||V(u)||2 < c(||M^(u)|| + ||V(u) - ^(a)|1) + ||^(a)|2 < c(C2 + C1) + ||V(a)||2(c + 1) (1.13)

where the constant c depends on n, m1, m2, vraimax^ |dmj/dx;|, i, j, l = 1, 2,..., n and mesO. Thus, the solution u G L2p(O) n W.

3. In the hypotheses of this part of the lemma M^(u) G W22(O) and the estimate (1.13) takes the form

||V(u)||2 < c(k-1|f|| + C1) + ||V(a)||2(c +1). If in addition ft G (O), then the boundedness of u follows from the assumption IV and Theorem 13.1 [9, Chapter 3]. □

Lemma 1.2. Let the assumptions of Lemma 1.1 be fulfilled and ga - f ^ 0. Then the solution u of the problem (0.2), (0.3) satisfies the inequality u ^ a for almost all x G O.

Proof. Let us consider the function w = a - u. By (0.3) and (1.1), we have

-kdiv(M(x)(V^(a) - V^(u))) + g(x)w = ga - f.

We multiply this equation by ^a - ^u where

u if u ^ a, a if u > a,

in terms of the inner product of L2(O) and integrate by parts in the first term of the left side. This yields

k

J Q

MV(>(a) - v(u)), v(>(a) - ^(u))) + m#(a) - ^(u))2

dx +

+ (gw - ga + f)(^(a) - ^(u))dx = 0. (1.14)

JQ

By (1.3), (1.4) and the nonnegativity of ga - f it follows from (1.14) that

km J (V^(a) -V^(u))2dx < 0, Jq

which implies V(a) - ^(u)) = 0. In view of (0.2) and (1.1) a|dQ = u|dQ. Hence a - u = 0 that is u ^ a for almost all x G O. □

2. The main result

In this section we prove the existence and uniqueness theorem for the problem (0.2), (0.3), (0.4). By a solution of this problem is meant the pair involving a function u G W and a positive real number k which satisfy (0.2)-(0.4).

Hereafter we keep the following notation: for v1,v2 G W21(O)

(Mvi, M = / [(MVvi, Vv2) R + mviv^ dx. Jn

Theorem 2.1. Let the assumptions I-III and the condition (1.5) be fulfilled. Let also

(i) f G L2(O), V(ft) G W23/2(dO), w(x) G W23/2(dO), g(x) G C(O);

(ii) w(x) be nonnegative in dO;

0 ^ g(x) ^ g1 = const < x G O; (2.1)

* = (MV^(a), Vb)M > 0; (2.2)

F(x) = ga - f > 0; (2.3)

$ = ^ - (ga - f, b) > 0. (2.4)

Then the problem (0.2)-(0.4) has a solution (u(x),k). Moreover, u(x) G L2p(O) and

u(x) ^ a(x) (2.5)

for almost all x G O. If in addition g = 0 or the inequalities

B = *-1(M^(a), ^(a))JM2(Mb, b)M2 < 2 (2.6)

and ( ) (

$ > 2c„m71/2mes1/nO 11 Fl l(2 - B)-1(Mb b\

' M

$ > 2com-1/2mes1/nQ ||F||(2 - B) 1(M6,6)M/2 (2.7)

are valid, then the solution of the problem (0.2)-(0.4) is unique.

Proof. We multiply (0.3) by b in terms of the inner product in L2 (O) and integrate by parts in the first term of the resulting equality twice. In view of (0.3) we obtain

+ k* + / g(x)ubdx = fbdx. (2.8)

JQ J n

By (2.2) and the definition of $ in (2.4) we can rewrite (2.8) as

k = ($ + (g(a - u),b))*-1. (2.9)

If g = 0, then k = $/* > 0 is known function. By Lemma 2.1 the problem (0.2), (0.3) with such k has a unique solution u G L2p n W.

Let now g = 0 and (2.1), (2.4) holds. We introduce the operator A mapping the set R+ of the positive real numbers into R according to the following role: for any y G R+

A(y) = ($ + (g(a - uy),b))*-1 (2.10)

where uy is a solution of the direct problem (0.2), (0.3) with k = y. According to Lemma 1.1, the problem (0.2), (0.3) has a unique solution uy G W and hence the value of A(y) defined by (2.10) is meaningful for any y G R+. Therefore (2.9) can be consider as the operator equation

k = A(k). (2.11)

Following the idea of [12, Chapter 1] one can show that the problem (0.2)-(0.4) has a solution if and only if the operator equation (2.11) is solvable.

According to Lemma 1.2 in the hypotheses of the theorem 3.1 uy ^ a that is (2.5) is valid. By (2.2) and (2.5),

0 < ko . * < A(y) < M + |(f,b)*+ g1K||||b| (2.12)

for y G R+. The left inequality in (2.12) enables to estimate ||uy||. As shown in Lemma 1.1, the estimate (1.12) is valid for —(uy). Therefore for every y > k0

p— i....... -i , „, p—i

rc2mes2/nO,

|uy|| < c-1(mesO)^ ||-(uy)|| < c-1(mesO)^ 0 2 llga - f || + ||^(a)H = C3. (2.13)

L ko mm 1

Then for y ^ ko

ko < A(y) < M + |(f,b)l + g1C3|b| , Ko. (2.14)

This inequality implies that the operator A maps [ko, Ko] into itself.

Furthermore, A is a continuous operator on [ko, Ko]. Indeed, let y1, y2 G [ko, Ko] and uy1, uy2 be solutions to problem (0.2), (0.3) with y1 and y2 instead of k, respectively. In accordance with the definition of the operator A, we have

A(y1) - A(y2) = *-1 (g(uy1 - uy2), b). (2.15)

On the other hand, the difference W = uy1 - uy2 obeys equation

y1M(^(uy1) - -0(uy2)) + gW = (y 1 - y2)M^(uy2) (2.16)

and the boundary condition (uy1 - uy2)|dQ = 0. Multiplying (2.16) by — (uy1) - — (uy2) in terms of the inner product of L2(O) and integrating by parts in the first term of the resulting equation yields

y^M(^(uy!) - ^(uy2)), ^(uy1) - -(^2)) 1 + (gW, ^(uy1) - ^(uy2)) =

= -(y1 - y2XM^(uy2), ^(uy1) - ^(uy2^ 1. (2.17)

The second term of the left side of (2.17) is nonnegative by the assumption (1.4). Multiplying (0.3) for uy2 by —a = — (uy2) - -0(a) in terms of the inner product of L2(O) and integrating by parts in the first term one can obtain

y2 {M^(uy2) -(uy2^M + (g(uy2 - a), -a) = y2 (-(a))M + (f - ga, -a)

from where in view of (1.4), (1.9) and the Cauchy inequality it follows that

c2 mes2 / n O

<M-(uy2), -(uy2)>M < <M-(a), -(a)); + "S-||ga - f ||2. (2.18)

y2 m1

The right-hand side of (2.17) can be estimated with the help of (2.5) and (2.18).

|y1 - y2|KM-(uy2),-(uy1) - -(uy2^ 1| < M(-(uy1) - -(uy2)),-(uy1) - -(uy2^ 1 +

+

"2mes2/nO

<M-(a),-(a)>m + o T ||ga - f ||2 . (2.19)

y2 m1

|y1 - y2|

2

2y1

Then, by (2.5), (2.18) and (2.19), the equality (2.17) yields

yi(M(V(«yi) - V(«y2)), V(«yi) - V(«y2))x < (M^(a), V(a)

'M

2

c0 mes

2/»n

k2mi

|yi - y2|

2

+-^2-Iis« - f y2 ' ' = C4|y2 - yi|2. (2.20)

yi

Next, multiplying (2.16) by b in terms of the inner product of L2(Q) and integrating by parts with the use of the fact that

f d (yi^Ki) - )) Wds = 0, (2.21)

Jan dN

we obtain

(g(uyi - «y), b) = -yi(M(V(uyi) - b)M - (yi - y2Kb)M,

whence, by (2.18) and (2.20),

i/2

Kg(uyi - uy2 ),b) 1 < yi(M^KO - V(uy2)),V(uyl) - M +

+ |yi - y2KMVK2UK2))Mfj <Mb,b)M < C5|yi - y21. (2.22) Joining (3.15) and (2.22) we are led to the inequality

|A(yi) - A(y2)| < C |yi - y21

proving the continuity of A on [k0,K0]. By Brouwer's theorem, equation (2.11) has a solution k G [k0,K0]. This in turn implies the existence of the solution {u(x),k} of problem (0.2)-(0.4). The solution satisfies (2.5), (2.13), (2.14). Moreover, by Lemma 1.1 the estimates (1.8)-(1.12) holds with k0 instead of k in the constants Ci and C2.

The only point remaining concerns the uniqueness of the solution. Let (ui,ki) and (u2,k2) be two solutions of problem (0.2)-(0.4). By (0.2), (ui -u2)|dn = 0. Subtracting (0.3) for (u2, k2) from (0.3) for (ui,ki), we obtain

kiM(V(«i) - V(«2)) + g« = -(ki - k2)M^(«2). (2.23)

Multiplying this difference by V(«i ) - V(«2) in terms of the inner product of L2(Q), integrating by parts and applying the arguments proving (2.20) to the resulting identity one can derive a similar inequality. Namely,

i/2

(M(V>(ui) - V(U2)), V>(«i) - ^M) 1 <

/ , s , c2mes2/nQ,, ll2i Ifci — k2|2

[(M^(a), V(a)>m + 0 k2m Iis« - f II2 k2 2 • (2.24)

<

Furthermore, we multiply (2.23) by b in terms of the inner product of L2(Q). Integrating by parts and taking into account (2.21) for y1 = k1 and y2 = k2 we are lead to the identity

(g(«i - «2), b) = -ki(M(V(«i) - VM), b)M - (ki - k2) [<M(V(«2) - V(a), b)M + *]. (2.25) Subtracting (2.9) for (u2,k2) from (2.9) for (u1 ,k1), we obtain

ki - k2 = Aki - Ak2 = (g(Ui ^«2),b). (2.26)

Without loss of generality we can suppose that ki > k2. Then (g(u — u2), b) ^ 0. Let us rewrite (2.25) as

(g(ui — «2),^ + ^(ki — k2) =

= —(kiM(-(ui) — -M) + (ki — k2)M(-M — -(a), b)M. (2.27)

We can estimate the right term of (2.27) with the help of (1.8), (2.14), (2.24). — ki(M(-(ui) — -(u2)), b)M — (ki — k2)(M(-(u2) — -(a), b)M <

< ki(M(-(ui) — -(«2)), -(ui) — -(«2))iM2(M6, b)M + + (ki — k2)(M(-(u2) — -(a), -(u2) — -(a))JM2(Mb, <

<

|"2co^mes1/nQ ^ _ /y + i/2j ^ 6>M/2(k1 - k2). (2.28)

Thus, (2.26)-(2.28) lead to the inequality

Ak1 _ Ak2 < 2

2c0mes/^||ga _ / || + *-1(M-(a), -(a))M/2J (Mb, b>M/2(ki _ k2)

$mi

proving the contractibility of the operator A by (2.6) and (2.7), from which and (2.26) it follows that ki — k2 =0 and in view of (1.4), (2.24) ui — u2 =0 for almost all x G 0. □

The condition -(u) > c|u|p for p > 0 provides that u G L2p(0) and -(u) G W22(0). This condition can be ignored under additional assumptions on f, ft and -.

Theorem 2.2. Let the assumptions I-III and the hypotheses (i) and (ii) of Theorem 3.1 be fulfilled. Let also ft G LTO(0), f ^ 0 and ft ^ p0 = min{0,p*} for almost all x G d0 where a real number p* is such that -(p*) = 0. Then the problem (0.2), (0.3), (0.4) has a solution (u(x),k). Moreover, u(x) G LTO(0) and

p0 < u(x) < a(x) (2.29)

for almost all x G 0. If in addition g = 0 or the inequalities (2.6) and (2.7) are valid, then the solution of the problem (0.2), (0.3), (0.4) is unique.

Proof. The proof of the theorem repeats the proof of Theorem 3.1 almost at all. We need only establish (2.29) and show that the solution of the direct problem (0.2), (0.3) belongs to W for every k > 0 because in the hypotheses of the theorem this fact is not evidenced by Lemma 2.1.

We first note that by Lemma 2.1 the direct problem (0.2), (0.3) has a unique solution u and -(u) G W2(0) for every k > 0. By Lemma 2.2 the solution satisfy the right inequality of (2.29). Moreover, by the maximum principle for linear elliptic equations, -(a) ^ -(p0) and hence a ^ p0 for almost all x G 0. In order to prove the left part of (2.29) we define the function

u u < po,

u = <

po u > po

and multiply (0.3) by h = -(u) — -(p0) in terms of the inner product of L2(0). Integrating by parts in the first term of the resulting identity gives

k(Mfo, h)M + / {km^(p0) + gU _dx = 0. JQ

In the hypotheses of the theorem the second term is nonnegative. Therefore, by (1.3), the last equality implies Vh = 0 from which we conclude that h = 0 for almost all x G Q because h|an =0. In view of monotonicity of V this means that u ^ p0 for almost all x G Q.

As mentioned above, if ft G Q), then V(a) G LTO(Q) by the continuity of V and Theorem 13.1 of [9, Chapter 3]. In the hypotheses of the theorem this means that a G LTO(Q). Hence u G LTO(Q) and ||u||L~(n) < max{|p0|, ||a||L~(n)} in view of (2.29). Repeating the arguments led to (1.12) one can obtain the inequality

/2 r

I|MV>(u)i| < [ilslfe) max{|po|2, ||a||^(Q)}mes2Q +

i/2

= C2'.

By (1.8), (1.9) and Theorem 5.1 of [11, Chapter 2] the last inequality implies

||V(u)II2 < c(C2' + Ci) + ||V(a)||2(c +1),

which proves that the solution u belongs to W. □

Certain inverse problems can be reduced to the problem (0.2)-(0.4). This allows to establish the existence and uniqueness theorems for such problems with the help of Theorems 2.1 and 2.2. In particular, the problem of the identification of the constant coefficient v in equation

-vdiv(M(x)VVi(v)) + g(x)V2(v) = f (x) (2.30)

with the boundary condition v|dn = ^(x) and the condition of overdetermination

f dVi (v) v ^ w = Pi Jen dN

are reduced to the problem (0.2)-(0.4) if the function V2(p) is an injection in (-ro, +ro). Indeed, we can define a function u = V2(v) and rewrite this problem as (0.2)-(0.4) with k = v, V(p) = Vi(V2-1(p)), ft = V2(m) and p = ^1.

The inverse problem (0.2), (0.3) with the condition of overdetermination given only on a part r of the boundary dQ, that is

k/^ = p, (2.31)

Jr dN V ^

can also be reduced to the problem (0.2), (0.3), (0.4). If the function w G W3/2(r) is finite on r and supp w c r then one can extend w into dQ setting w = 0 in dQ \ r and consider the integral in (2.31) over the whole boundary dQ. In this case Theorem 2.1 is formulated as follows.

Theorem 2.3. Let the assumptions I-III and (1.5) be fulfilled, f G L2(Q), V(ft) G W23/2(dQ),

3/2 —

w(x) G W2 (r), g(x) G C(Q). Let also w(x) be nonnegative and finite on r, suppw c r, w(x) = 0 on dQ \ r and (2.1)-(24) hold. Then the problem (0.2), (0.3), (2.31) has a solution (u(x), k). Moreover, u(x) satisfies (2.5) for almost all x G Q. If in addition g = 0 or (2.6), (2.7) are fulfilled, then the solution of the problem (0.2), (0.3), (2.31) is unique.

Theorem 2.2 can be reformulate for the problem (0.2), (0.3), (2.31) in a similar manner.

Theorem 2.4. Let the assumptions I-III be fulfilled, f G L2(Q), V(ft) G W23/2(dQ), w(x) G W23/2 (r), g(x) G C(Q), ft G LTO(Q). Let also (21)-(24) hold, f > 0 for almost all x G Q, the function w(x) be nonnegative and finite on r, supp w c r, w(x) =0 on dQ \ r and ft ^ p0 for almost all x G dQ. Then the problem (0.2), (0.3), (2.31) has a solution (u(x),k). Moreover, u(x) G LTO(Q) and (2.29) is fulfilled for almost all x G Q. If in addition g = 0 or the inequalities (2.6) and (2.7) are valid, then the solution of the problem (0.2), (0.3), (2.31) is unique.

2

3. Examples and comments

As mentioned above the interest to the identification of coefficients in the elliptic equations, among them (0.3), is due to its extensive applications. Certain examples of such problems were considered in [6]. In particular, the special case of (0.3)

—kdiv(M(x)|u|p-2Vu) = 0 (3.1)

describes the steady-state isothermal gas filtration [13] where p = 3, M(x) = (mpi)-iE, u is the pressure of a liquid in the pores, is the viscosity of liquid, k and m are the average permeability and the porosity of the rock. In this case -(p) = (p — 1)-i|p|p-2p and g = 0. The equation (4.1) satisfies the assumptions I—III. If the other hypotheses of Theorem 2.1 coincide, then the inverse problem (3.1), (0.2), (0.4) has a unique solution.

In [14] the physical model of the non-equilibrium effects in a simultaneous flow of immiscible fluids in porous media is presented. For the steady-state flow with capillary counter-current imbibition the basic equation for the effective water saturation assumes the form

kA-(u) = 0 (3.2)

where u is the actual water saturation, the coefficient k depends on the air permeability of the porous medium, its porosity (the relative volume occupied by the pores), the surface tension and a (conventional) contact angle at the triple water-oil-solid boundary,

,( ) f /i(s)/2(s) J,( )d

np) = _ TTTTTTTT J (s)ds Jo Ji(s) + J2(s)

The dimensionless nonnegative quantities fi and f2 are called the relative permeabilities and satisfy the inequality 0 < f < 1. The function Leverett J is the dimensionless capillary pressure. fi(s) is a monotonic non-decreasing smooth function. The function f2(s) is a monotonic non-increasing smooth one. Finally, the nonnegative function J(u) monotonically decreases. The function - is continuous and strictly increases on (—ro, Therefore under the hypotheses

of Theorem 3.2 the inverse problem (0.2), (0.4), (3.2) has a unique solution (u(x),k).

Some kinds of equations (2.30) are involved in modeling the electric fields of semiconductors. In the absence of an external electric field the stationary nonlinear dissipative equation for the

potential u of the self-consistent electric field of a semiconductor takes the form [15]

2

£ £2

—kA(u + - u2 +--u3) + A|u|q u = 0

26

where the positive parameters k and A depends on the electric susceptibility of the semiconductor, the parameter £ > 0 depends on the temperature of free electrons, q > 0. These equations can be reduced to (0.3) with the appropriate function -(p) satisfying the assumption IV.

The last point of our considerations is the stability of the solution to (0.2), (0.3), (0.4) with respect to y. One can prove it much as the contractibility of the operator A was established in Theorem 3.1.

References

[1] G.Alessandrini, R.Caburro, The Local Calderon Problem and the Determination at the Boundary of the Conductivity, Comm.. Partial Differential Equations, 34(2009), 918-936.

[2] A.P.Calderon, On an inverse boundary value problem, Seminar on Numerical Analysis and

its Applications to Continuum Physics, Rio de Janeiro, Brazil, 1980, Soc. Brazil. Mat., Rio de Janeiro, 1980, 65-73.

A.Hasanov, A.Erdem, Determination of unknown coefficient in a nonlinear elliptic problem related to the elasto-plastic torsion of a bar, IMA J. Appl. Math., 73(2008), no. 4, 579-591.

M.V.Klibanov, A.Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, Netherlands, 2004.

A.Sh.Lyubanova, Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87(2008), 1121-1128.

A.Sh.Lyubanova Identification of a constant coefficient in a quasi-linear elliptic equation Journal of inverse and ill-posed problems, 22(2014), 341-356.

A.Nachman, B.Street, Reconstruction in the Calderon Problem with Partial Data, Communications in Partial Differential Equations, 35(2010), 37-390.

G.Nakamura, K.Tanuma, A nonuniqueness Theorem for Inverse boundary Value Problem in Elasticity, SIAM J. Appl. Math., 56(1996), 602-610.

O.A. Ladyzenskaja, N.N. Uralceva, Linear and Quasilinear Equations of Elliptic Type, Academic Press, New York, 1973.

H.Gajewski, K.Groger, K. Zacharias, Nichtlinear Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbucher und Monographien, II, Abteilung, Mathematische Monographien, vol. 38, Akademie-Verlag, Berlin, 1974.

J.-L.Lions, E.Magenes, Problemes aux Limites Non Homogenes et Applications, vol. 1, Travaux et Recherches Mathematiques, no. 17, Dunod, Paris, 1968.

A.I.Prilepko, D.G.Orlovsky, I.A.Vasin, Methods for solving inverse problems in mathematical physics, Marcel Dekker Inc., New York, 2000.

G.I. Barenblatt, V.M.Entov, V.M.Ryzhik, Flow of fluids through natural rocks, Kluwer Academic Publ., 1990.

G.I.Barenblatt, J.Garcia-Azorero, A.De Pablo, J.L.Vazquez, Mathematical Model of the Non-Equilibrium Water-Oil Displacement in Porous Strata, Applicable Analysis, 65(1997), 19-45.

A.G.Sveshnikov, A.B.Alshin, M.O.Korpusov, Yu.D.Pletner, Linear and nonlinear equations of the Sobolev type, Physmatlit, Moskow, 2007 (in Russian).

Об одной обратной задаче для квазилинейного эллиптического уравнения

Анна Ш. Любанова

Исследуется задача идентификации неизвестного постоянного коэффициента в старшем члене уравнения с частными производными -кМф(и) + g(x)u = f (x) при граничном условии Дирихле. Здесь ф(и) - нелинейная возрастающая функция от u, M - линейный самосопряженный эллиптический оператор второго порядка. Коэффициент к восстанавливается по дополнительным интегральным данным на границе. Доказывается существование и единственность решения обратной задачи, включающего функцию u и положительное действительное число к.

Ключевые слова: обратная задача, краевая задача, эллиптическое уравнение второго порядка, теорема существования и единственности, фильтрация.