Inverse problems of recovering the boundary data with integral over determination conditions

We consider the parabolic equation n∂                n

Lu = ut - ∑    aij (t, x)ux + ∑ai (t, x)ux + a0 (t, x)u = f , i,j=1∂xi               j i=1 i where x ∈ G ⊂ Rn is a bounded domain with boundary Γ of the class C2 (see the definition, for instance, in [1, Sect. 1]), t∈ (0,T) . Put Q = (0,T)×G and S = (0,T)×Γ . The equation (1) is furnished with the following initial and boundary conditions:

u | _S = g , u | _{t =0}= u ₀( x ).

Put _∂ ^∂ Nu = ∑ _i ⁿ _{, j =1} a_ij ( t , x ) u_xj ( t , x ) ν _i , where ν = ( ν ₁, ν ₂

,...,v_n ) is the outward unit normal to 5 . The

inverse problem is to find a g = ∑ _i ^m ₌₁ q_i ( t ) Φ _i ( t , x ) , where overdetermiantion conditions

solution u to the problem (1)–(2) and a function g of the form the vector <7 = (qv q2, ^, qm) is unknown, with the use of the

G ( x , t )ф_к ( x ) dx = Y k ( t ), к = 1,2, ^ , m.

Inverse problems of recovering boundary regimes, in particular, the convective heat exchange problems are conventional (see, for instance, [2–11]). They arise in different problems of mathematical physics such as the problems of control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coverings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, the study of composite materials, etc. Mathematical models describing these prosesses and the corresponding inverse problems in both one-dimensional and multidimensional cases are described, for example, in [2]. The essential attention here is paid to numerical methods of solving inverse problems in question and some uniqueness theorems together stability estimates for solutions. We refer also to the monograph [3] mainly devoted to numerical

Математика

methods of determining a solution, where in the one-dimensional case different inverse problems for parabolic equations and problems of recovering the boundary regimes as well are studied. The overdetrermination conditions are the values of a solution at some points lying inside the spatial domain. These problems are studied in different settings in dependence on the type of the ovedetermination conditions. It is often the case when these problems are ill-posed in the Hadamard sense. In particular, it is true in the case of the overdetermination conditions in the form of values of a solution at separate points or on some surfaces lying in the spatial domain (see [2]). At the present article we examine the problems with overdetermination conditions in the form of some integrals with weights of a solution over a spatial domain. Note that these conditions arise in applications and they are often used in literature. Inverse problems of recovering the right-hand side or coefficients of an equation with integral ovedetermination conditions are studied in the articles [12–18], the monographs [19, 20], and some other articles. In particular, the existence and uniqueness theorem of a generalized solution to the problem (1)–(3) (from the class u ∈ W ₂^0,1( Q ) ) in the case of m = 1 and the Neumann boundary condition was obtained in [9] and a similar result for a heat-and-mass transfer system including the Navier–Stokes system and a parabolic equation for the concentration of an admixture was obtained in [10]. The article [11] is devoted to a regular solvability ( u ∈ W ₂^1,2( Q ) ) in the case of m = 1 and the Robin boundary conditions. The case of the Dirichlet boundary condition happens to be more complicated than the case of the Neumann (Robin) boundary conditions and was not studied before. The present article is devoted to this case. Under some conditions on the data we prove well-poseness of this problem. The article in some sense is an extension of [21], where the Robin boundary conditions are treated. Some our auxiliary statements are taken from this article.

Preliminaries

Let E be a Banach space. Denote by L_p ( G ; E ) ( G is a domain in R ⁿ ) the space of strongly measurable functions defined on G with values in E and the finite norm || || u ( x ) || _E || L ( g ) [1]. We also employ the Holder spaces (see the definition for instance, in [22]) C ^{α , β} ( Q ), C ^{α , β} ( S ) , C^k ( G ) and the Sobolev spaces W_p^s ( G ; E ) , W_p^s ( Q ; E ) (see [21, 23]). If E = R or E = R ⁿ then the latter space is denoted by W_p^s ( Q ) . Given an interval J = (0, T ) , put W_p^{s , r} ( Q ) = W_p^s ( J ; L_p ( G )) ∩ L_p ( J ; W_p^r ( G ) . Respectively, W_p^{s , r} ( S ) = W_p^s ( J ; L_p ( Γ )) ∩ L_p ( J ; W_p^r ( Γ )) . All spaces and the coeffciecients of the equation (1) are assumed to be real. Let ( u , v ) = ∫ _Gu ( x ) v ( x ) dx , Q ^γ = (0, γ ) × G and S ^γ = (0, γ ) × Γ .

We endow the space W_p^s (0, τ ; E ) ( s ∈ (0,1) , E is a banach space, with the norm

|| q ( t ) ||         =( | q || p _m     + <  q >  p _ ) 1/ ^p , <  q >  p- =[     ¹ q ⁽ t ¹)— q ⁽ t 2 ) ^{1 E} dt_xdt₂ . If E = R then we

W _p ^s (0, τ ; E )           ^L p (0, ^τ ; ^E )            ^s , ^{τ                s} , ^τ    0 0 | t 1 _- t 2 | 1 ^{+ sp}       1 2

obtain the conventional Sobolev space     W_p^s (0, τ ) .    For     s ∈ (1/ p ,1] ,    we put

Wɶps(0,τ) = {q∈Wps(0,τ) : q(0) = 0} . This class is a Banach space with the norm || • II „     . We can

W _p ^s (0, τ )

equivalence results from

define also the equivalent norm

| q ( t ) | p_s    = | q | ^p p (0.)+ <  q > p r. The

_Wp s _{(0, τ )}     t s L _p (0, τ )           s , τ

Lemma 1 of the subsection 3.2.6 [1]. Similarly, we can define the spaces W ^ɶ _p ^s (0, τ ; L_p ( G )) , W ^ɶ _p ^{s ,2 s} ( Q ^τ )

comprising , the functions v ( t , x ) in W_p^s (0, τ ; L_p ( G )) and W_p^{s ,2 s} ( Q ^τ ) , respectively, such that v (0, x ) = 0 .

The new norms || • |L,         , | • |L,7, T are defined naturally with the use of the above norm

W _p ^s (0, τ ; L _p ( G ))        W _p ^{s ,2 s} ( Q^τ )

W p ^s (0, τ ) .

Lemma 1. Let s ∈ (1/ p ,1) and p ∈ (1, ∞ ) . Then the following statements are valid.

• 1)    Let q ( t ) ∈ W_p^s (0, τ ) ( τ ∈ (0, T ]) . Then q ∈ C ([0, τ ]) after a possible change on a set of zero measure . If q (0) = 0 and q ɶ is an extension by zero of q for t ≤ 0 then

II q II - c II q II ,                                                             (4)

Wps ( - T + τ , τ )    1       Wps (0, τ )

where the constant c ₁ is independent of τ ∈ (0, T ] and q .

• 2)    The product q ⋅ v of functions in W_p^s (0, τ ) ( τ ∈ (0, T ]) belongs to W_p^s (0, τ ) and if q ∈ W_p^s (0, τ ) and v ∈ W_p^s (0, τ ) then qv ∈ W_p^s (0, τ ) . Moreover, the following estimate holds:

• ¹    qv Wp(0 T ) ^- c 2 ¹ q ^ p (0 T ) ⁽< v > s T + ¹ v ¹ '(0 т ) )                          ⁽⁵⁾

where the constant c ₂ is independent of q , v, and τ ∈ (0, T ] .

• 3)    If a function v is strictly bounded from zero on [0, τ ] , i. e. δ ₀ = inf _{t ∈ [0, τ ]} | v ( t ) |> 0 then the ratio

q / v of functions in W_p^s (0, τ ) ( τ ∈ (0, T ]) belongs to W_p^s (0, τ ) again and

|| q / v || ₅        - c ₃ || q || _s       || v || _s ,                                                        (6)

Wps (0, τ )    3       Wps (0, τ )      Wps (0, τ )

where the constant c ₃ is independent of q but it depends on δ ₀ and tends to ∞ as δ ₀ → 0 .

• 4)    Assume that q ( t ) ∈ W ^ɶ _p ^s (0, τ ) ( τ ∈ (0, T ]) , v ( t ) ∈ W_p^s (0, T ) , and Φ ( t , x ) ∈ W_p^{s ,2 s} ( S ) . Then qv ∈ W ^ɶ _p ^s (0, τ ) , q Φ ∈ W ^ɶ _p ^{s ,2 s} ( S ^τ ) , and

|| qv ||            - c || q ||            || v ||             ,                                                         (7)

W ^ɶ _p ^s (0, τ )           W ^ɶ _p ^s (0, τ )      W _p ^s (0, T )

H q Ф 11-525 Г- c ₄ || q ||         || Ф || 25    ,                                     (8)

W ^ɶ _p ^s ,2 ^s ( S^τ )    4       W ^ɶ _p ^s (0, τ )       W _p ^s ,2 ^s ( S )

where the constant c is independent of τ ∈ (0, T ].

The proof can be found in [21].

We describe now the conditions on the data used below. Fix a number s = 1 - 1/2 p and assume that p >3/2 .

The conditions on the coeficients aij∈C([0,T];W∞1(G))∩Lq(G;Wps(0,T)),∇xaij∈Lq(G;Wps(0,T)),(9)

where i , j = 1,2, … , n , 1/ p + 1/ q = 1 .

ai∈Lq(G;Wps(0,T)),(i=0,1,…,n),∇xai∈Lq(G;Wps(0,T))(i=1,2,…,n).(10)

Suppose also that there exists a constant δ ₀ > 0 such that

∑n aijξiξj ≥δ0|ξ|2, ∀(t,x)∈Q,∀ξ∈Rn.(11)

• i ,    j =1

The conditions on the data of the problem f∈Lp(Q), u0(x)∈Wp2-2/p(G),(12)

g∈Wps,2s(S),g(0,x)=u0|Γ,(13)

ϕk∈W∞2(G),Φk∈Wps,2s(S),ψk∈Wps+1(0,T),(f,ϕk)∈Wps(0,T),k=1,2,…,m.(14)

As a consequence of Theorem 9.1 in [22, Ch. 4] (see also Theorem 10.4 in [22, Ch. 7]) we have the following theorem.

Theorem 1. Assume that G is a bounded domain with boundary of the class C ² and the conditions (9)–(13) hold . Then there exists a unique solution u to the problem (1)–(2) such that u ∈ W_p ^1,2( Q ). A solution meets the estimate

^ u I WP ,2 _{( Q} ) ^- C ⁽ll f ^ Lp (Q ) + ^ u 0 I Wp 2 - 2/ p ₍ G ) + ^ g I Wp ,2 s ₍ s ) ^).

As a consequece of Theorem 1 we have that

Математика

Theorem 2. Assume that G be a bounded domain with boundary of the class C ² and the conditions (9)–(13) hold, where f ≡ 0 and u ₀ ≡ 0 . Let γ ∈ (0, T ] . Then there exists a unique solution u to the problem (1)–(2) such that u ∈ W_p ^1,2 ( Q ^γ ). A solution meets the estimate

II u "  1 7 V

W 1 p ,2( Q γ )

< c " ^g " Wp,2 ^s ( s ^y )

, where the constant c is independent of γ ∈ (0, T ] and g .

| g ( t , x ), t e ( - T + y , Y )

Proof. Extend the function g by zero for t < 0 and put g =                        . Obviously,

[ g (2 y - t , x ), t e [ Y , T + y ]

g e Wp,2s (S). By Theorem 1, we can construct a solution to the problem (1)-(2), where u0 = 0, f = 0, and g = g such that u e W1,2(Q). Theorem 1 yields the estimate " u " 12    < c " g " 2    . Estimate

W _p ( Q )          W _p ( S )

the right-hand side. Lemma 1 implies that

" ^g " Wp ,2 ^s ( S ) < " ^g " Wp ,2 ^s (( - T + Y , T + Y ) хГ ) <  c ⁽ " ^g " Wp ,2 ^s (( - T +YY ) хГ )) ⁺ "  ^g " Wp ,2 ^s (( y , T + Y ) хГ )) ) <  c ¹ "  ^g " Wp ,2 ^s ( S^Y ) ^.

We employ the additivity of the Sobolev space with respect to the partition of a domain (see Remark 3 of Subsect. 4.4.1 in [1]) and the definition of the corresponding norm.

Basic results

In addition to the above conditions we require that

|det B ( t ) | ≥ δ ₀ >0 ∀ t ∈ [0, T ],

where B (t) is the matrix with entries b„ = [ ^L(x) ф(t, x) d г, ^ \ =0 (k = 1,., m ); ij Γ ∂N     j             k Γ u0(x)ϕk(x)dx=ψk(0),ψkt(0)=(u0,L*0ϕk)- u0(x)∂ϕi(x)dΓ+(f,ϕk)|t=0, G                                   Γ     ∂N where   k = 1,., m

and     L ^*₀    is

a formally adjoint to the operator

L 0 ,

L o ® = Z

∂ aω -

¹, j =1 d x i ^{4 x} j    ^Z

n

n aω i=1 i xi

-

a ₀ ω ;

• (A) the functions Ф 1 (0, x ),..., Ф _m (0, x ) are linearly independent on Г and u ₀( x ) |_г belongs to the span of these functions.

We can note that (16) is a necessary solvability condition of the inverse problem.

Theorem 3. Assume that G is a bounded domain with boundary of the class C ² and the conditions (9)-(12), (14)-(16), and (A) hold . Then there exists a unique solution ( u , q ) ( q = ( q 1 , . , q_m )) to the problem (1)-(3) such that u e W p ^,2( Q ), <7 e W p (0, T ) . A solution satisfies the estimate

m

" u " Wp ,2( Q ) ⁺ "  q " Wp (0, T ) < ^{C ( I f} " ^L p ⁽ Q ) ⁺ "  u ⁰ " W p^{-2/ p} ( G ) ⁺ i =1 ^{( I} ^ ‘ " Wp ⁺ s (0, T ) ⁺ "  ^{( f} , ^ ) " Wp (0, T ) ^)).

Proof. Let u e W p ^,2( Q ) be a solution to the problem (1)-(3), where g = Z m= q li Ф i . The conditions (15) and (A) imply that there exists a unique collection of constants q_i (0) such that u ₀ 1_Г = Z m =i q i (0) Ф i (0, x ). Put Z m =i q i (0) Ф i ( t , x ) = g 0 ( t , x ) and denote by ve Wp>’²(Q ) a solution to the problem (see Theorem 1)

Lv = f , v | _S = g ₀( t , x ), v | _{t =0}= u ₀( x ).                                   (17)

Let q e W ps (0, T ). In view of our conditions Ф j - e W p ^{,2 s} ( S ). Lemma 1 yields q_i ( t ) Φ _i ( t , x ) ∈ W_p^{s ,2 s} ( S ) and thus g ∈ W_p^{s ,2 s} ( S ) . Make the change of variables u = v + ω . The function ω ∈ W_p ^1,2( Q ) is a solution to the problem

L to =^0, B to ^l s = g - g 0 = g; , to\ t =o =0.                               ⁽¹⁸⁾

The condition (3) transforms into the form

\ G toV k ( x ) dx = / k — j G ^v ( t , x )Фк ( x ) dx = / k , k = 1,2, ^ , m.

By (16), / 7 k (0) = 0 and we have at least that / 7_t ( t ) e W p (0,T ). Below we demonstrate that / 7 k ( t ) e Wp⁺⁵ (0, T ) and / к '(0) = 0 . Multiply the equation in (18) by ф _к ( x ) and integrate the result over G . We obtain that ( ω _t , ϕ _k ) = ( L ₀ ω , ϕ _k ). Using (18), (19), and integrating by parts, we infer

/ к '( t ) = ( to , L ^ k ) - Z ⁷ i ( t ) | Ф i ^d^ i (7)d^d Г , к = 1, ^ , m , < i ( t ) = <  ( t ) — <  (0). i =1     Γ      ^∂ N

The last inequality can be written in either of the forms

m

Z

or

7 _,_.!

B9a = F + R(

II bi IL     L 111 -.

^ijW_p^s(0,T)       ^jL_p(Γ;W_p^s(0,T)) ⁱC¹(G)

As was noticed in the proof of Lemma 1, the embedding theorems state that W_p^s(0,T) ⊂C([0,T]) .

Hence, we can assume that b_ij∈C([0,T]) . In view of (15), we can write

<a = B¹F⁺R0⁽<a ), R0⁽<a ) = B 1 R⁽<a )

We can determine the vector <a from this equation. Indeed, consider the segment [0,5] c [0, T] and estimate the quantity || R_o(< ) ||      . The second and third statements of Lemma 1 and the

^{0 a}W_p^s (0,δ)

conditions on the coeffcients and the functions ϕ_k imply that the entries of the inverse matrix B^-1 also belong to the class W_p^s(0,T) . In this case the estimate (7) and Lemma 1 yield

m

1 R0(

W_p ( , )    _k=1                 W_p ( , )

Estimate the norm of the expression (ω, L^*₀ϕ_k ) . The Minkowski and Holder inequalities and

Lemma 1 ensure that

|| (to,L*^) I ,o^^lltoH _     || L*^ ||       .      .(24)

II V     0Yk)\\Lp(O,d) II Ur (G;W,$5(0,5))ll 0yk "tn(G;W5(0,T))

pp    qp

In view of our conditions on the coefficients, the last factor is estimated by some constant independent of δ . Estimate the first factor. We have f |to||p, dx-( f5to J G      Wwp (0,5)       JGJ 0 t5p

dtdx₊δ δ|ω(t1,x)-ω(t2,x)|^{p G 0 0}        | t₁

-t2 |^1+sp

dt₁dt₂dx.

The Newton–Leibnitz formula validates the inequality

II - toll      ₅< c 5¹/²II to II      5.

s L (Q^δ)    ^{1 t}L (Q^δ)

pp

Estimate the second summand on the right-hand side of (25). To this end, we first make the change of variables t1 = T15, 1₂= T25 and next use the inequality Ц to(x,t) ||W₅(01)< c I(O_T Ц_L (₀,1) (6) (x ,T)) = to( x ,5т)) followed by the inverse change of variables. As a result, we arrive at the inequality

δδ ∫∫∫ G00

|ω(t₁, x) -ω(t₂, x) |^p

|t1 -t2 |^1+sp

δ dt1dt2dx ≤ c1∫∫ |ωt (t, x) |dtdxδ1/2.

G0

Математика

Thus, from (25)–(27) it follows that f II to|| ps dx < c2 61/2 ||to, II 5C c2 6 2 || toll 12 5 .(28)

Jg" Wp (0,6)        2      11 t 'Lp (Q6) 2 11 Wp ,2( Q6)

Therefore, taking (23) and Theorem 2 into account, we obtain the estimate

I R 0( qa ) ^ '(0 ,)< c 3 c6 I to^ 1.2( 6< < c4^ I g ^s ,2 s( ^ -^

_p( , )                       _p (Q )                     _p (Q )

Lemma 1 implies the inequality

|| Ro (qa) |b < c 61/2 || q || .$     ,(30)

ii 0vta/i^p(0,6)    5 ii^aWp(0,6)’v where the constant c5 is independent of 6 and q a . Fix 6 > 0 such that 61/2c5 < 1. In this case the operator R0 is a contraction and thereby the equaiton (22) is uniquely solvable in the space Wps (0,δ) , of course under the condition that /k' e Wp (0, T). We have that /k' e Wp (0, T). Show that /0k = iGV(t,x)^k(x)dxe Wp+ s(0,T) and /0k'(0) = /k'(0), i- e., /kt e Wp(0,T). Multiply the equation in (17) by ϕk and integrate the result over G . We infer

/ok'(t) = ^(to,LA) - Eqi⁽⁰⁾ [^ф_;^d^k^x^{)dГ + (}f^,Vk^),,k = 1, -^,m-              ⁽³¹⁾

i=1     Γ     ^∂N

In view of the condition (14) the right-hand side in this equality belongs to W_p^s (0,T) and the relations (16) and (31) for t = 0 yield /0k '(0) = /k'(0). Thus, /k e Wp (0, T) for all k and thus the equation (22) is uniquely solvable on the segment [0,δ] . Find a solution ω∈W_p^1,2 (Q^δ) to the problem (18). Prove (19). Multiply the equation in (18) by ϕ_k and integrate the results over G . Using (17), (18), and integrating by parts, we obtain the equality

(to, Vk) = (to, LoVk)- ^mqi(t) [ф, ^^k(x)dг, k = 1, -, m- i=1 Γ     ∂N

The vector-function qa satisfies the system (20), subtracting its k -th equation from this equality and cancelling, we arrive at the equality ^totvkdx = /k', k = 1,-,m, whose integration with respect to t G and the intial condition validate (19) on [0,δ] .

We now demonstrate that this solution is extendible to the whole segment [0,T] . We have defined the vector-function q a only on [0,6] - Extend q a by zero for t < 0 and denote f q a (t), t e (0,6)                                                                 ,        , qb = <                   - The coordinates of qb are denoted by q1,-,qm - This vector-function

[q_a (26 -1), t e [6, T]

belongs to W^s_p^,2s (S) - Make the change q¹ = q _a - qb - The vector-function with the coordinates q¹ is a solution to the system

m

m

Eq1 (t) bki = </k'(t)+(to, l* ф_к) - Eqb (t) bki -

i=1

i=1

By definition of q_b, the right-hand side in this equality and the vector 1 vanish on [0,6] - Let to* – be a solution to the problem

^Ltoo = 0 ^Btoo |_s = ]mq^bф_t^,to0 |t=o=⁰-i=1

In this case the function ω₁ = ω-ω₀ is a solution to the problem

m

Lto, = 0, Bto^S = Eq¹Фi, to1\t=o=O-                               (34)

i=1

By Theorem 1, у = 0 for t e [0, £]. Thus, the problem of extension of ga is reduced to solving the system m

∑q_i¹(t)b_ki =ψ_1k'(t) +(ω₁, L^*₀ϕ_k),                                  (35)

i=1

m where ^1 k' = ^k'(t) + (уо,L0Фk) - ^gb(t)bki, and у is a solution to the problem (34). A solution to the i=1

system vanishes for t ≤ δ . We obtain the same system with zero Cauchy data at the point t = δ and a new right-hand side F . Next, we repeat the same arguments and estimates on the segment [δ, 2δ] . Without loss of generality, we can assume that the constants arising in estimating the norm of the operator R₀ are the same. Thus, the system (35) is solvable on [δ, 2δ] . Repeating the arguments on [2δ, 3δ] and so on, we can construct a solution on the whole segment [0,T] . The estimate in the claim of the theorem has been actually proven in the proof.

Remark. At first sight, the well-posedness conditions (15) look rather strange and possibly arising in the method of the proof. However, employing other methods leads actually to the same condtitions. It is possible that they are essential.

Acknowledgement

The authors were supported by the grant on development of scientific schools with participation of young scientists of the Yugra State University.