Abstract
We are concerned with both regular and degenerate first-order identification problems related to systems of differential equations of weakly parabolic type in Banach spaces. Several applications to partial differential equations and systems will be given in a subsequent paper to show the fullness of our abstract results.
1. Introduction
The basic aim of this paper consists in extending the results in [1] and in solving some identification problems in a product space , where is a Banach space , endowed with a norm , related to systems of two possibly degenerate first-order differential equations in time. More precisely, we will consider the differential problem: determine a pair of functions and , , such that In particular, the regular choice corresponds to the case where The starting point for this paper is provided by [1] (cf. also [2]) where the identification degenerate problem is studied in the Banach space under assumptions of weak parabolicity (cf. [3]) on the linear closed linear operators and .
Existence of solutions to evolution equations with matrix-valued operator coefficients has been considered very recently by Engel [4] where no degeneration is involved.
Abstract systems of parabolic equations of relevant importance in applications are described in the monograph by Yagi [5].
Here we study both the regular and the degenerate cases using the basic results in [1] and the different methods introduced therein. This will allow us to handle, in particular, systems of elliptic-parabolic equations and systems of degenerate parabolic equations in different functional spaces.
We indicate now the plan of the paper.
In Section 2, we will consider an identification problem for first-order regular systems of differential equations.
In Section 3, we will deal with identification problems for first-order in time systems of PDE’s, treating in particular some nonstandard boundary conditions.
Section 4 contains the main contributions to the degenerate case. The first two results—Theorems 14 and 18—can be easily extended to matrix-valued operators of the form where operators and map from to and , respectively, while operators and map from to and , respectively, and being a suitable Banach space.
Different levels of degeneration for operator matrices not entering the time derivative will be considered and corresponding conditions for the solvability of the related identification problems will be given.
Section 5 contains some extensions of the basic results in [1], which will play a key role for the present paper.
Section 6 contains a number of applications to systems of PDE’s enlightening the concrete applications and the strict conditions to be satisfied by the single equations to guarantee both existence, uniqueness, and the regularity of solutions.
The Appendix section describes a general approach to inverse problems using a perturbation theory of generators. In such a way, the inverse problem under consideration is reduced to a direct problem with a new generator.
A number of specific examples illustrate the extension and the strength of this method.
2. Identification Problems for First-Order Regular Differential Systems in Banach Spaces
Let be a complex Banach space with norm and let The space is endowed with the product norm.
Let , , , be four closed linear operators satisfying the following properties, where denotes the resolvent set of a linear closed operator :(H1) and ;(H2) for all ;(H3) for all ;(H4) for all ;(H5) for all ;(H6), , , ;(H7), .
We consider here the problem consisting in recovering the unknown scalar function in the following differential system in : subject to the initial conditionsand to the additional information First we consider the following resolvent system in , where and : that we rewrite in the vector form: Note that according to our assumption (H3) we get Observe now that if Then for all the linear operator admits an inverse in satisfying the estimate Hence the solution to system (9) is given by Therefore, for all , we deduce the estimates Since , we conclude that the operator matrix satisfies the following bound in the product space : where
Remark 1. If operators and are bounded, then ; thus, is trivially verified.
Remark 2. If , then . Indeed, for any it follows Moreover, the closed graph theorem implies . This and the previous estimate imply The relation , along with H9, implies . Consequently, the set relations and cannot occur simultaneously.
Taking the results in [1] into account (reported and improved a bit in Section 5), we deduce the following Theorem 3.
Theorem 3. Let and be the generators of two differentiable (not necessarily densely defined) semigroups of parabolic type in the complex Banach space X satisfying (H1)–(H3). Let and be two linear closed operators satisfying properties (H4)–(H6) and let , , be two linear functionals with properties (H7). Let be a triplet of real numbers such that (cf. (19)) Let the data satisfy the properties Then the identification problem admits a unique global solution .
Remark 4. If for some and , then , . Thus implies . In this case, owing to (19), we have , so that we need to require .
In most applications, , so that this relation reduces to .
3. Identification Problems for Regular First-Order in Time Systems of PDE’s
In this section, we will deal with some identification problems related to systems of PDE’s.
Problem 5. Let be a bounded domain in with a -boundary .
We want to recover the scalar function in the initial-boundary value problem:
under the additional information
Here
are two second-order linear elliptic differential operators with real-valued -coefficients on such that
being a positive constant.
Assume for the time being that and are multiplication operators defined, respectively, by two -functions and . In this case we have .
Concerning the linear boundary differential operators defined by assume that , , , and , are real-valued functions and vector fields on such that . and are real -tangential operators on , and standing for the conormal derivatives associated with the matrices and , respectively; that is, denoting the unit outward normal vector to at .
Assume further (cf. page 515 in [6]) that the vector field (resp., ) does not vanish on (resp., ) and the function (resp., ) has zeros of even order not greater than (resp., ) along the integral curve of (resp., ) passing through (resp., ) at . In other words, the so-called -condition holds with and , respectively.
It is shown on page 516 in [6] that the operator and defined by satisfy in the resolvent estimates for all complex in a sector , with and suitable large.
We are in the subelliptic case: Choose now and make the following assumptions: Then our identification problem admits a unique global solution .
We consider now the more general case where , , and is the multiplication operator by .
From Remark 4 we get , so that reduces to , that is, to . Since we are compelled to require that either of or must coincide with .
Note that, if , then the corresponding boundary value problem is elliptic and this holds if and only if for all (cf. [7], on page 515). In this case so that .
Choose now and make the same assumptions as in the previous case, except for . The related condition has to be changed to the following: Then the given identification problem admits a unique global solution .
A corresponding result holds when .
Problem 6. We note that in Problem 5 the domain of the operator-matrix is a product of domains. However, Corollary 31 allows to handle also decoupled domains. For this purpose, we will consider a problem related to a reaction diffusion model describing a man-environment epidemic system investigated in [8]. Such a model consists in a parabolic equation coupled with an ordinary differential equation via a boundary feedback operator (cf. also [7]).
In order to obtain stability results the authors linearize the model and arrive at the following evolution system, where and stand, respectively, for the concentration of the infection agent and the density of the infective population at time and point :
where is a bounded domain in with a smooth boundary , is the Laplacian, , , are nonnegative functions, and denoted the outward normal derivative on .
We define , and denote by the multiplication operator induced by the function . Moreover, we introduce the operator matrix
It can be proved (cf. [7, page 126]) that generates an analytic semigroup on .
Consider then the identification problem consisting in finding a triplet , being a scalar function satisfying the direct Problem (38) as well as the additional condition: where , , , are fixed points in . As a consequence of [2, Theorem 3.2] we get
Proposition 7. Let , , , , , . Then the identification Problem (38), (41) admits a unique global strict solution .
Notice that the interpolation spaces are well characterized both in an abstract form and for many boundary conditions (cf. [4, 9]), but in this concrete case it seems to be difficult to translate them as on page 321 in [4]. Therefore, one can use the more restrictive assumptions that can be easily checked.
Problem 8. We solve here an identification problem in Hölder spaces.
Let be a bounded domain in with a smooth boundary . Then , , and denote the Banach space of all functions in whose derivatives of order are all Hölder continuous with exponent . Such a space will be endowed with the natural norm .
We introduce now some notation and assumption. As usual, denotes a multi-index and we associate with it the monomial differential operator , . Finally, for any (fixed) and let the functions , , satisfy, for some positive constant , the relations: Introduce now the linear operator in defined by A remarkable result by von Wahl (cf. [10] on page 241) establishes the resolvent bound for a large , where .
For our application is not restrictive to assume that such a condition is satisfied for all . In this case the resolvent estimate holds with .
Let now be another linear differential operator of order , with properties similar to those of , defined by where .
Likewise as above, the spectral estimate holds with .
Notice that and imply and .
Let now and be linear differential operators, with smooth coefficients (e.g., in ), defined by where and .
In view of Satz II on page 239 in [10] we have where We introduce now the operator matrix We can now apply Theorem 3 with , and Under such hypotheses the identification problem and being two fixed elements in , admits a unique solution provided that the following conditions are satisfied: (H1), , , ;(H2), , ;(H3), , .Notice that, if and , then and condition is satisfied.
Problem 9. Here we solve an identification problem related to the Ornstein-Uhlenbeck operator in , . For this purpose, we refer to the monograph [11]. Such an operator is the prototype of an elliptic operator with unbounded coefficients and is defined on smooth functions by where and denote, respectively, the gradient and the Hessian matrix of , while and are constant matrices, being strictly positive definite and the spectrum of being contained in the left complex half-plane .
It is well known that the realization of in the weighted space , where generates an analytic semigroup with domain , where and is endowed with the norm , where .
Moreover, the spectrum of is the discrete set denoting the (distinct) eigenvalues of .
We note also that (cf. [11, Theorem ]) Whence, since the operator is continuously invertible, we easily deduce the estimates Likewise we get the estimates Finally, recall that the dual space to can be identified with , .
Consider now the linear differential operator defined by where , are uniformly continuous and bounded functions in .
The previous estimates yield Consider then the linear operator where are uniformly continuous and bounded functions in .
Reasoning as above, we conclude that satisfies the estimate Consider now the identification problem: where , , , , and Theorem 3 applies with , . Here , , , so that . Assume further that possess the additional properties: Then the identification Problems (66)–(321) admit a unique strict global solution with , .
We note that a corresponding result still holds if either of the lower order operators is replaced with .
Since is an unbounded domain of cone type, the real interpolation spaces , are well characterized. Exactly we have (cf. [9, Theorem ]): Assume now that our data possess the following properties: Therefore, we can apply Theorem 3 or Proposition 5.1 in [1] and deduce the same conclusion as above.
We can analogously deal with the case when either of the lower order operators is replaced with .
Problem 10. In [12] the following remarkable result is proved. Let be a Hilbert space, let be a densely defined, strictly positive self-adjoint operator, and let be a constant complex-valued matrix. Then the operator , defined by generates a strongly continuous analytical semigroup in if and only if (cf. [12, page 311]) the eigenvalues of the matrix satisfy (i) if is complex and (ii) if is real.
This result allows to deal with a lot of very important problems related to evolution PDE’s. We confine ourselves to describe an identification problem related to one of such models.
Suppose that generates an analytical semigroup in . Then owing to [1, Theorem 2.1, page 45] (as improved in a following paper for Al Horani and Favini) the identification problem admits a unique global solution , provided that Let us apply this result to the following thermoelastic PDE problem with simplified “hinged” homogeneous boundary conditions and Dirichlet thermal boundary conditions (cf. [12, page 317]), where the reference bounded domain has a smooth boundary : determine a triplet of functions and such that where and denotes the duality pairing between and its dual space.
Introduce now the positive self-adjoint operator defined by This definition implies The previous differential equations read now Introduce then the Banach space , the new variable , the vector , and the operator defined by Then system can be rewritten as the single equation: In [12, page 318] it is shown that operator generates an analytic semigroup on if and only if the operator generates a strongly continuous analytic semigroup in the space .
Indeed, in this case we have , where the constant matrix is defined by Consider then the eigenvalue equation and observe that (i) since and , the matrix admits a real negative eigenvalue ; (ii) all the real eigenvalues of are strictly negative; (iii) all the nonreal eigenvalues have negative real parts , since the sum of is . Consequently, all the eigenvalues of have negative real parts so that generates an analytic semigroup in (resp., ).
Let us consider first our identification problem in . Since we must require Moreover, we assume Note that the previous spaces are well characterised as Besov spaces (cf. [9]). More precisely, since is a positive operator we have (cf. [9, page 105]) On the other hand, if is a positive operator, from [9, Theorem (b)] it follows that , , coincides with . Therefore, Using now the notation in [9, page 321], we deduce the following characterization in terms of Besov spaces: cf. [9, Definition , page 317].
Therefore, we assume that our data satisfy the following properties: Under these assumptions the identification Problem admits a unique solution such that , and .
Remark 11. Since is a strictly positive and self-adjoint operator in the Hilbert space , we could treat in the same way the case when , as in Problem 9. Indeed, the realization is that of , with , is strictly positive and self-adjoint [11, Proposition , page 251].
Remark 12. If we consider the identification Problem with the additional information then, setting under the assumptions Problem , admits a unique solution .
4. First-Order Systems of Singular Differential Equations in Banach Spaces and Identification Problems
Here we face identification problems for systems of singular first-order differential equations in the Banach space , both applying the general results described in Section 5 and developing ad hoc methods in order to improve the corresponding consequences in some cases. For this purpose, we need some preliminary lemmas on the resolvent estimates.
Theorem 13. Suppose that the closed linear operators , , , , , in satisfy the conditions: Then the matrix operators and defined by with , , satisfy the estimate where .
Proof. First of all, we need to verify that and are closed linear operators.
Let , , , , , as . We immediately deduce that ad . Moreover,
implies that as for some . Since is closed . But is closed too, so that and . Analogously, if , , , as , , , then and . Since
and thus has a limit as . It follows that tends to so that we can conclude as the above.
We easily see that
so that
Notice that if , then
Therefore, from (97)
we deduce the bound
Since , this completes the proof.
We can now apply Proposition 29 to the identification problem: Suppose that assumptions (95)–(98) are satisfied, with and let . Further, assume with
We have the following.
Theorem 14. Under assumptions (95)–(98), if , , (109) and (110) hold together with , , then the identification Problem (108) admits a unique global solution , , .
Remark 15. When , we can take in (97).
Particular attention deserves the case when , being a reflexive Banach space. Then Let ; that is,
Then . Denote by the projection operator onto along and suppose , , , (112) holds, if , After applying Proposition 32, we get the following result.
Proposition 16. Let be a reflexive Banach space, suppose (95)–(98) hold with , , being as above, , , , . If (112)–(117) hold, then the identification Problem (108) admits a unique global solution , such that , . Moreover, if has a closed range, then (116)-(117) can be dropped out.
Next we extend Theorem 13 to nontriangular operator matrices. Precisely, we consider the system where
We have the following.
Lemma 17. Assume (95)–(98) hold and , with If , then large, when .
Proof. We proceed by a perturbation argument. Write
In view of (119), we get first
and then
Notice now that implies that the linear operator
has a bounded inverse for large and its inverse can be estimated in the norm of by , for example. It follows that has precisely the same bound as in Lemma 17.
Theorem 18. Under assumptions (95)–(98) and (119), if and , , for some , , , and (111)-(112) hold, then the identification problem admits a unique global solution , such that , .
Remark 19. When , we can choose ; thus, the first two conditions in the statement of Theorem 18 involving , , , , reduce to and .
Remark 20. We could easily establish a result at all analogous to the one in Proposition 16, concerning this more general case.
We are now in a position to face the general identification problem: To this end, we will assume that Multiply the second equation in the system by and substract the obtained equation from the first one. We obtain the following system equivalent to (126): together with and the additional information where is related to by so that
Notice that the operator may have no bounded inverse. Our next step is to translate to this case the conditions established in Theorem 14. Assume, in addition to (95), and thus . Compute now as desired.
Therefore, we are in a position to establish the following result.
Theorem 21. Suppose that operators , satisfy (95), (135), (136), and (137) and that have bounded inverses, , and , , where , , , and (111) holds together with the compatibility relation Then the identification Problems (126)–(128) admit a unique strict global solution , with , .
Proof. It is a simple rewriting of the result in Theorem 18 as applied to Problems (130)–(132), taking into account the relation to the starting Problems (126)–(128).
Of course, the preceding results apply to the abstract strongly degenerate elliptic-parabolic system However, in view of this generality, the corresponding assumptions would yield the restrictions and . To overcome this difficulty we will make suitable assumptions on the operators involved.
Clearly, if and has a bounded inverse, the second equation in (141) gives , so that (141)–(143) reduces to the identification problem: Observe that the compatibility relation must hold. On the other hand, all the results in Propositions 29 and 32 apply provided that operators and satisfy the assumptions described there.
If , so that , we arrive at the following identification problem: Notice the extra difficulty arising from the fact that the unknown term is not supposed to be differentiable. However, if has a bounded inverse, we can introduce the new unknown: so that
Then (145) reads equivalently where This is a regular identification problem if, for example, is a bounded operator. Indeed, we have the following.
Corollary 22. Let have a bounded inverse and let be bounded. Let the compatibility relation hold together with . Then Problems (141)–(143) admit a unique strict solution , .
Corollary 22 can, in fact, be refined weakening the assumption on . For this purpose consider the system If operator has an inverse, then Therefore, the pair satisfies the problem where That is, satisfies (148), as desired.
Let . Then, from Theorem 2.1 in [1] we deduce the following.
Theorem 23. Let be the generator of an analytic semigroup of negative type in , , , , , , . Then Problems (141)–(143) admit a unique strict global solution , such that .
Applying Corollary 31, we also obtain the following.
Corollary 24. If where , , , , , , , then Problems (141)–(143) admit a unique strict global solution , with .
Example 25. Consider the identification problem: where is a closed linear operator in the Banach space and .
To apply Theorem 23 we set and we note that , so that . Moreover, It follows that if generates an analytic semigroup of negative type and , , , , , then the identification Problem (155) admits a unique global strict solution , .
We stress that we have been able to determine a triplet of functions and in the following parabolic-elliptic identification problem:
Here operator is defined by where and the coefficients , , satisfy the following assumptions: The same argument applies when is the opposite of the realization in , , of the Ornstein-Uhlenbeck operator , precisely (for the properties in , , cf. Problem 9).
In this case (161) changes to
Example 26. Consider the identification problem: with .
Such a problem is easily reduced to an equivalent problem related to the integral differential equation: Consequently, we have come back to Example 25 with , , being replaced by , , , respectively.
Taking this into account, the same conclusions as Example 25 can be obtained, provided that , , generates a holomorphic semigroup in , , , . Then the identification problem above admits a unique global strict solution , .
Consider now the identification problem: Under the assumption that have a bounded inverse we get . Therefore we obtain the following differential equation for the single unknown : Now it suffices to suppose that the pair satisfies the properties described in [1] or in Section 5 in order to obtain existence and uniqueness for the given identification problem.
If has an inverse , introduce function defined by , so that Assuming that has a bounded inverse, we have Substituting in the first equation of the system, we get That is, we have obtained the following differential identification problem for : This identification problem might be treated similarly under the obvious hypothesis to have a bounded inverse. But if is invertible we come back to the first case.
The assumption requiring to be invertible seems really essential in some sense. As an example, take where and admit bounded inverses. Then implies . Hence, if , then we necessarily have Moreover, the second equation furnishes uniquely . This shows the importance of the invertibility of , as expected. Notice too that no assumption like semigroup generation is required to operators and .
Example 27. Let , be two bounded linear operators in . Observe first that the identification problem is equivalent to the problem Thus we can apply both Theorem 23 and Corollary 22. Operator is now given by , while is given by .
Example 28. Let , be two bounded linear operators in , with , for some .
Consider the identification problem: with a compatibility relation , .
Note that, under our assumptions on and , such a problem is equivalent to the following: where , .
Since, in turn, this identification problem is equivalent to
Theorem 23 and Corollary 24 run as well.
As an example, let , , being a domain in of class . Let be the laplacian in endowed with Dirichlet boundary conditions, , and let , , , . Let and be two real-valued continuous functions on , , being possibly negative. Let be such that for all . Given , our identification problem consists in finding a triplet such that , being given functions in , with , . If , let Then , so that is assumed to generate an analytic semigroup in .
Since our previous abstract assumptions read as Suppose . Trivial computations show that We need for all . Therefore, all the required elements are determined.
Clearly, if generates an infinitely differentiable semigroup we are compelled to require , as in Corollary 24, (cf. [10, 13, 14]).
5. Some Improvements of Known Results
For the reader’s convenience, we report here the main results in [1] with some minor improvements.
Proposition 29. Let , , . Let be a Banach space, let and and be two closed linear operators such that (i) ; (ii) is invertible; (iii) for all , where Let and satisfy the following properties for some , : Then the identification problem admits a unique global solution:
Remark 30. Assumptions (ii) and (iii) can be weakened to (iibis) is invertible for some ;(iiibis) for all .
Indeed, let us introduce the new unknown . Then Problem (189) is equivalent to the following: Proposition 29 applies immediately provided we replace the triplet by . Once and have been determined so are and , with the same regularity.
As a consequence, we have the following result relative to the generators of infinitely differentiable semigroups of parabolic type with nonnecessarily dense domains, satisfying
Corollary 31. Let , , and . Let and the closed linear operator satisfy (192) in . Let , , , , , . Then the identification problem has a unique global solution .
In particular, but very important case, where , and is reflexive, it is possible to weaken the assumptions on the initial data and . Notice that in the statement of Proposition 29 must belong to the range of (or of ) and .
The following extension to Theorem 2.2 in [1] holds.
Proposition 32. Let and be two closed linear operators in the reflexive Banach space , with , and , . Suppose Let and let be the projection of on the null space along . Suppose and Then the identification Problem (189) admits a unique global solution
Remark 33. If has a closed range, conditions (196)-(197) can be dropped out. Indeed, it suffices to apply Theorem 2.2 in [1] to (191) and to observe that reduces to , denoting the restriction of to .
Remark 34. Very recently in [8, Theorem 3.1] the above results have been improved to the case and .
Proposition 35. Let be the generator of an analytic semigroup in the complex Banach space , let , , for some , and let , , , . Then the inverse problem admits a unique solution such that As for Problem (199), also the hyperbolic case, corresponding to the case where generates a -semigroup, has been dealt with in [8, Corollary 2.1].
Proposition 36. Let be the generator of a -semigroup in the complex Banach space , and let , , , , , . Then the inverse Problem (199) admits a unique solution .
In the next section, we are giving specific applications of the results listed in this section.
6. Applications
In this section, we will give several concrete applications of our previous abstract results.
Problem 37. First, we recall some previous results from [15–17]. Let and be two linear differential operators with domains in , , being a bounded region with a boundary of class . —a multiplication operator by a nonnegative function —is defined by Operator is defined either by or by We assume that the coefficients , , enjoy the following properties: and being two positive constants.
Then it is shown in Theorem 2.1 in [17] that the pair satisfies in the sector the following estimate with : Let us consider the following identification parabolic-elliptic problem: where the linear operators , , and enjoy the same properties as with , with and a.e. in , (if we can apply Remark 19), , , , , .
We know that the pairs , , where , satisfy (206) in .
In view of the moment inequality (cfr. [18, page 115]), assumption (97) holds with . Therefore, Theorem 14 applies provided that ; that is, . So, we must necessarily have and respectively.
Let , , where , , Then Problem (207) admits a unique global solution , , .
Using the same scheme, one could handle the more general problem: where , is constant, provided that (cf. Theorem 14).
As a particular case, we can also treat the problem where , , , .
Problem 37bis. We show here how some more regularity of functions and allow to choose a larger exponent in the reference space .
We recall that a function , , , is said to be -regular for some [3, 15, 16] if there exists a positive constant such that If , , , and , then is -regular with .
If is -regular and then (cfr. Theorem 3.3 in [17]) the pair satisfies (206) with , . Note that if and only if .
Let now be -regular, so that satisfies (206) with according to [15]. Moreover, let , , , . Then, . Thus, we must have to find Let , . Then the right-hand side in (215) changes to In particular, the last property is achieved if tends to and is large enough.
Therefore, Theorem 14 applies in the reference space with and whenever the following inequalities hold:
Problem 38. We are concerned with the initial and boundary value problem: where for the sake of simplicity, we have set and have endowed such operators with either the Dirichlet or the Robin boundary condition.
We will assume that , , are real-valued, while , , and , , are scalar functions, the ’s being nonnegative, satisfying the following properties: We observe that properties (221) hold if and for all . This choice implies and in (97). Moreover, the last condition in (222) implies (137) and .
Assume now that the matrix operator is invertible in , . For this purpose, it is enough to assume for all . Further, , , , are given functions on .
Relying on the proof of Theorem 18 we are led to the following equation:Under our assumptions (221)–(224) Theorem 18 applies with , , that is, , if the following additional properties are satisfied when , , , with , : Then the identification Problems (221)–(224) admit a unique global strict solution such that , and .
We observe that if , and , are more regular, the previous result can be extended to the case where and is larger. For this purpose, we assume that is -regular; that is, Since for all , we get Therefore, owing to Theorem 3.3 in [15], we can choose , . This choice implies and , so that in this case runs in the interval .
In the case when and is -regular with , then, owing to Theorem 4.2 in [15], we can choose so that .
Problem 38 bis. We could handle also the case where the determinant vanishes everywhere in . However, in this case, Theorem 18 forces us to assume for all .
In view of assumptions (222) the second equation in system (225) becomes Thus all is reduced to a regular identification problem.
However, using Theorem 23, in some cases we can handle the situation when . For the sake of simplicity we choose and assume ., so that for some constant . Therefore our differential equation becomes
Referring to the notation in Theorem 23, we get Consequently, the linear closed operator generates an analytic semigroup of linear bounded operators provided and for all . In this case the resolvent of admits the representation: The element in Theorem 23 can be easily described.
Example 39. Let and let operator be endowed with either Dirichlet or Robin boundary conditions.
Consider then the following identification problem, where and are real positive numbers:
If we suppose that
all our conditions in the last lines in Problem 37 are verified for , . Moreover, if , then belongs to and from the inequality we deduce
That is, is -regular with , and the arguments in the treatment of Problem 38 apply as well.
Example 40. Consider the identification problem: where and for all . Observe that the determinant of the matrix inside the time derivative vanishes identically in .
As in the proof of Theorem 18 we obtain the equivalent problem:
If then in Theorem 23 specifies to We observe that assumptions on are satisfied if and for all .
We compute now : Finally, observe that in this case we have Assume now that the data satisfy Then Problem (237) admits a unique strict global solution such that , .
We now face the general case where the additional information is Taking advantage of the identity condition (246) takes the form Moreover, solving (238) for , we get the following equation for all : so that Consequently, our identification problem is reduced to the previous one with being replaced by .
Problem 41. Let be a bounded region in with a smooth boundary . We are concerned with recovering function in the following problem related to a degenerate parabolic weakly coupled linear system: We assume that functions and , for a.e. , while , , and . Finally, we assume that the consistency condition is fulfilled.
Recall now that if , where then [3, page 83] operator defined by satisfies the resolvent bound: for all ’s in a sector containing the closed half complex plane . This implies and .
Suppose now that and , so that the corresponding differential operators and defined by satisfy (H1) and (H3) in Theorem 3, with , , , .
Let and be the multiplication operators by and in , respectively.
Theorem 3 runs provided when .
Whence it follows that if , , , , , then Problem (251) admits a unique strict global solution such that .
More generally, consider the degenerate parabolic identification problem: where and satisfy , for all .
The change of unknown functions and leads to the following (equivalent) identification problem: Suppose that —so that and must vanish on if and do. Moreover, we assume , and being defined as above and satisfy , as well as Then Problem (259) admits a unique strict global solution such that .
Problem 42. Consider the following one-dimensional parabolic identification problem of Sobolev type: Here and is a positive constant.
Introduce now the linear operators , , and , , defined by Note that the pairs and satisfy our spectral assumptions with so that we can apply Theorem 14 with provided we make the following assumptions: Then Problems (262)–(265) admit a unique strict global solution such that ).
Problem 43. We want to point out a different approach for solving the identification problem described in Example 28. For the sake of simplicity, we confine ourselves to the problem: with a compatibility relation .
Here is a bounded linear operator in , , are two densely defined linear operators in such that , , and , , are invertible with inverses in and being a sectorial operator with .
From the matrix equation in (265) we obtain the following equation for the new unknown : Introduce now the linear functional defined by Then the pair solves the identification problem: Finally, set It is immediate to check that function solves the nondegenerate identification problem: We can now apply [1, Theorem 2.1] if the following conditions are satisfied: Since it follows that is sectorial, too. Moreover, from we deduce that if and only if .
Since is sectorial and has domain , from [19] (cf. also [20]), it follows . Therefore if and only if .
Consequently, our condition on the data read equivalently Then Problem (265) admits a unique solution . Consequently, since and , then the solution to Problem (269) has the properties , .
Finally, from the latter differential equation in (265) we easily deduce the following representation formula for the pair : We can conclude by stating that Problem (269) admits a unique solution such that .
Remark 44. If is not differentiable with respect to time, functions and need not to be differentiable, even though their sum is. This fact exhibits the degeneracy of our Problem (265).
Remark 45. Consider the identification problem: with a compatibility relation .
Here is a densely defined invertible linear operator in such that , while and .
First we note that our assumptions are not satisfied, since for all . However, a trivial computation yields Therefore, if is uniquely determined by Since then the assumptions on our data imply without any resolvent estimate involving , but .
Application 1. Let , , being a domain in with a boundary of class . Let with , , be given function satisfying the ellipticity condition: Define then the linear differential second-order operator by where .
Introduce also two functions such that and for all and denote by the multiplication operator in corresponding to .
Following the same steps as in [3, pages 79-80], we can readily derive the resolvent estimate Consider then the identification problem: We assume that our data satisfy We note that the spaces are well characterised in [9, page 321].
By virtue of the previous result we deduce that our degenerate parabolic identification problem admits a unique global strict solution such that and .
Application 2. According to [21] the projection operator onto relative to the direct sum representation is well characterised. Now we show how the corresponding projection in product space can be characterised as well.
For the sake of simplicity we confine ourselves to a bit less general case than the one discussed in Proposition 16, choosing here . Hence the corresponding differential system becomes Suppose that and admit bounded inverses satisfying From our assumption we deduce the following formula: Setting , we get Hence
Whence it follows Note now that if is a reflexive Banach space, we have the decomposition: Observe then that Introduce finally the linear bounded operator: where and are the projections on and , respectively.
It is easily checked that and since and Once we have characterised , we obtain the basic formula . Consequently, we have at our disposal all the elements required in Proposition 16.
As an application, we can handle the identification problem: Assume now that are nonnegative functions in , , , while , , and are second-order linear operators such that the first two are isomorphisms from into , while the third one is bounded from into .
Under such conditions, the conditions in Proposition 16 ensuring the solvability of the identification Problem (299) are fulfilled. Such conditions, related to the projection , are expressed by means of the projections and .
Application 3. We generalize here to a system Problem 9 related to the scalar Ornstein-Uhlenbeck equation and use the same notation.
Let be a uniformly continuous and bounded function on such that Consider first, for all and , the resolvent equation: where Then is self-adjoint in and [19] (cf. [20, Section , page 251]): Thus, taking the real and imaginary parts in the equality: we get the relations as follows: They, in turn, imply Since , we deduce implying Since , we can conclude that has a bounded inverse and Let us now consider the following degenerate identification problem related to a matrix-valued Ornstein-Uhlenbeck operator: look for a triplet of functions and such that where We now assume that functions and are uniformly continuous, bounded, and strictly positive in . Moreover, we assume that function has the same properties and satisfies for all . Then condition (309) is satisfied with . Further, the interpolation space is characterised by Assume now that our data enjoy the following properties: Consequently, our degenerate parabolic identification problem possesses a unique global strict solution such that , .
Application 4. We are here concerned with hyperbolic systems.
Let , , , be a Hermitian matrix, that is, , and let , the space of all bounded functions along with all first-order derivatives. Let be a matrix with .
Set and for all define the inner product: where and , so that .
If , is the temperate distribution defined by for all . If and one defines Of course, is identified with the temperate distribution .
Therefore the components of are given by where is meant in the sense of distributions.
We can now introduce the linear first-order differential operator defined by It is well known (cf. [22]) that generates a -semigroup in . Consequently, our abstract identification problem reduces to the following: find a pair such that where , and are given.
Theorem 13 allows to conclude that if (in particular they belong to , and ) the problems (319)–(321) admit a unique solution .
As an example, consider the following identification problem related to Maxwell equations: where , , and in , and are given.
We note that , , and in imply and in , via (322) and (323).
We notice that our assumption concerning and ensure that no electric charges occur in .
Introduce now the vector . Recalling that we observe that (322) and (323) can be rewritten in the unified form: where all the elements in vanish but the following:
Define now
We observe that it is easy to check that is a closed subspace in orthogonal to .
We observe now that, according to our assumptions, the initial value belongs to , while the additional condition can be expressed as where .
Since operator is defined by (318), with , it generates a -semigroup in . Moreover, , where . Consequently, Corollary 2.1 in [8] applies provided , , and . Then the identification Problems (322)–(325) admit a unique solution .
Remark 46. Likewise we could treat identification problems for the Dirac equation in nonrelativistic Quantum Mechanics (cf. [22, page 54]), making use of the function space .
Application 5. Let be the linear operator defined in , by where the potential belongs to for some and . Then it is well known (cf. [23]) that generates a group of unitary operators on . Therefore, Proposition 35 yields the following result.
Corollary 47. Let , , , , , for some and . Then the identification problem admits a unique solution .
Application 6. In a very recent paper [24] Taira exhibits a functional approach to the existence of Markov processes endowed with both Dirichlet and oblique derivative and the so-called first-order Wentzell boundary conditions for second-order uniformly elliptic differential equations with discontinuous coefficients. More exactly, the related differential operator is assumed to be of the form on a bounded domain , , with boundary of class .
The discontinuous real-valued coefficients are assumed to satisfy the following properties:(i), for a.e. and for a.e. , all and some ;(ii);(iii) and for a.e. .
The differential operator is a diffusion operator describing a strong Markov process with continuous paths in such as Brownian motions.
Consider also a boundary operator of the form where denotes the tangential component of the gradient along .
The following assumptions on the coefficients of are made:(iv) is Lipschitz continuous on and for all ;(v) is a Lipschitz continuous vector field on ;(vi) is a Lipschitz continuous on and for all ;(vii) is a Lipschitz continuous on and for all ;(viii) is the unit interior normal at to .
The terms , , , correspond, respectively, to the following phenomena: reflection, drift along the boundary, absorption, and sticking (or viscosity).
Let be the Banach space of real-valued functions continuous on endowed with the sup-norm.
Associated with the formal differential operator we introduce the realization defined by Functions in are said to satisfy Wentzell conditions of the first order.
Finally, recall that a strongly continuous semigroup on is called a Feller semigroup if it satisfies This implies that is nonnegative and contracting on .
Assume now that and is defined by Then Theorem 1.1 in [24, page 717] establishes that generates a Feller semigroup on . Therefore, Theorem 13 applies. Recall now that the dual consists of real measure with finite variation coinciding with .
Finally, consider the identification problem: If , , , , then the previous Problems admit a unique solution .
Application 7. We are here again concerned with the Ornstein-Uhlenbeck operator, but in . For this purpose denote by , , the realization of the the Ornstein-Uhlenbeck operator . Consider now the identification problem:
In view of [11, Proposition , page 280] operator defined by generates a strongly continuous semigroup , which is not analytic.
In this case, we can apply Theorem 13. Indeed, if , , , , , , then the previous Problems admit a unique solution .
Appendices
A. Perturbing Generators: An Interpolation Approach
Let be a complex Banach space with norm .
We make the following assumptions, where denotes the resolvent set of a linear closed operator :(H1) is a (possibly multivalued) linear closed operator, whose resolvent contains the set , , ;(H2), , and , for all ;(H3) for some , where the space , introduced in [3], is defined by .
We want to show that the linear closed operator generates a -semigroup if (resp., an analytical semigroup, if ) of linear bounded operators in . For this purpose, we extend two perturbation results listed in [20].
For all and , consider the spectral equation: Applying operator to both sides in (A.1), we deduce the following equation:
Recall now (cf. [3, page 49]) that Consider now the following resolvent identity:
If , we get the estimate
Since is an equivalent norm on , indeed , we conclude that
Since , we deduce the estimate This implies where, recall that, owing to (H3), , Then, according to (A.8), we deduce that is invertible in and, for all , its inverse is estimated by Hence (A.2) admits, for all and , the unique solution: satisfying the estimate Therefore, for all the operator has a bounded inverse from into , and, hence, from into itself.
We now estimate in . From the equation , for all , we get As a consequence, from (A.13) we deduce the desired estimate
B. A General Approach to the Identification Problem
Let , , and be three complex Banach spaces with norms , , , respectively.
We make the following assumptions, where denotes the resolvent set of a linear closed operator :(H1) is a linear closed operator, whose resolvent contains the set , , and ;(H2), , and , for all ;(H3), standing for the Banach space of all linear bounded sesquilinear operators from to ;(H4);(H5)for each fixed and for all the equation is uniquely solvable in and its solution can be represented by , where the (nonlinear) operator is linear continuous as a function of ; that is , for all ;(H6)there exist Banach spaces and in and , respectively, with , such that We consider now the following problem: determine a pair of functions and such that where , , .
Apply now operator to both sides of our differential equation. Using assumptions (H4), we obtain the following equation: From assumptions (H3) and (H5) we deduce Inserting this expression into the differential equation (B.2), we conclude that the identification Problem (B.2) is equivalent to the unusual following Cauchy problem: Introduce now the linear operator defined by From assumptions (H5) and (H6) we deduce the bounds as follows: Consequently, the linear operator , with , generates a -semigroup in if (resp., an analytical semigroup, if ) of linear bounded operators in with
Remark 48. Note that no compactness is required to operator in order that the perturbed operator generates a -(or an analytical) semigroup.
Application 8. In the case of Application 8 is given by In this case one has
Proof. Suppose and . Then . Hence, Therefore, This implies Hence, if is sufficiently large, On the other hand, Consequently, Therefore, one concludes that there exists a constant such that
We consider the problem Suppose that , , and Since , one has . In view of (B.9) one observes By virtue of Corollary of [25] problem (B.18) one admits a unique solution such that Let be so large that has a bounded inverse. Since by virtue of (B.22) and (B.23) one obtains It follows from that Hence, with the aid of (B.25), one obtains Since , one has This implies Hence, in view of (B.28), one has From this and (B.23) it follows that Thus, the unique solution to problem (B.18) satisfies
It is also possible to show that .
Application 9. Let , , , , , . Let such that . Therefore, equation , with , means Suppose that the infinite matrix defines an invertible operator in , the inverse of which is denoted by , so that .
One also has Moreover, Therefore, all assumptions (H1)–(H6) are verified. Consequently, we conclude that the linear operator , , generates a -semigroup if and (resp., an analytical semigroup, if and ).
Reasoning as in Application 8, we conclude that if , , , then the identification Problem (B.40), with , admits a unique solution such that , , , .
Application 10. We consider here the following identification problem: determine functions , , and , , such that
Before stating precise properties involving operators, functionals and functions in (B.37) we apply (formally) functional to both sides in the differential equation in (B.37). We obtain the following equations for all , , :
Suppose that is an invertible matrix and denote by its inverse.
Then we can solve system (B.38) for . For all , , we obtain
Consequently, our identification problem is equivalent to the following direct problem where
We list now our assumptions. Let , , and assume that each satisfy properties (H1) and (H2). is a bounded multilinear operator from to , , and , , , , , .
As is easily seen there exists a constant such that
Problem (B.40) is solved by transforming it to the following system of integral equations
Applying to both sides one deduces
Hence one is led to the following system of integral equations for new unknown finctions , :
Let
In view of (B.43) and [25, Corollary 3.3] one observes
From and (B.42) it follows that
Since , one can easily show that the sequence converges to a set of functions uniformly in and is a solution to the system (B.46) of integral equations. It is obvious that a set of functions defined by is a solution to the system (B.44). Again in view of [25, Corollary 3.3] one concludes that is a unique solution to (B.40) satisfying
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
A. Favini and A. Lorenzi are members of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM).