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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 247425, 36 pages
Characterization of the Solvability of Generalized Constrained Variational Equations
Departamento de Matemática Aplicada, E.T.S. de Ingeniería de Edificación, Universidad de Granada, c/Severo Ochoa s/n, 18071 Granada, Spain
Received 4 December 2011; Accepted 30 January 2012
Academic Editor: Juan J. Nieto
Copyright © 2012 Manuel Ruiz Galán. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.
It is common knowledge that in studying differential problems, variational methods have come to be essential. For instance, in , for a certain impulsive differential equation, its variational structure as well as the existence and uniqueness of a weak solution is shown. Specifically, given , , , , and , the impulsive linear problem where , is considered, and as a direct application of the classical Lax-Milgram theorem ([2, Corollary 5.8]), possibly the most popular variational tool, it is proven that there exists a unique such that In this paper we replace the Lax-Milgram theorem with a characterization of the unique solvability of a certain type of variational equation with constraints. Such a constrained variational equation arises naturally; for instance, when in the variational formulation of an elliptic partial differential equation, its essential boundary constraints are treated as constraints. This result allows us to consider problems without data functions in a Hilbert space, which is beyond the control of the classical theory ([3, section II.1 Proposition ], [4, Lemma 4.67]). In particular, it extends the class of the said impulsive linear problems admitting a weak solution, since we prove that for any , where and not necessarily , and for all , for some only depending on and , the impulsive equation has one and only one solution.
To help understand specifically the sort of constrained variational inequality under consideration, we have selected a simple but illustrative model problem, which will adequately serve our purposes related to linear impulsive problems. For , , and , let us consider the corresponding Poisson’ equation with nonhomogeneous Dirichlet boundary conditions whose usual weak formulation is By imposing essential boundary conditions weakly, we can equivalently write that variational formulation as a constrained variational equation. To be more concrete, let be the Sobolev space , let , and let and be the continuous bilinear forms given by (“” stands for the Euclidean inner product in ), and let and be the continuous linear functionals defined by Since then the weak formulation above leads us to consider the following variational equation with constraints: find such that With regard to this problem, a more abstract approach has been adopted: let and be the Hilbert spaces, let and be continuous linear functionals, let and be continuous bilinear forms, and let . Under these assumptions, The known classical results are nothing more than sufficient conditions guaranteeing that such a variational equation with constraints has a solution. However, when the function data do not belong to Hilbert spaces, these results do not apply. For this reason we study a more general type of variational equation with constraints, whose most important particular case relies on this construction: given a reflexive Banach space , normed spaces , , and , continuous bilinear forms , , and , and continuous linear functionals and , denoting find, if possible, such that In Theorem 2.2 of Section 2 we characterize when this constrained variational equation, in effect a more general variational inequality with constraints in the framework of locally convex spaces, admits a solution. The particular normed case is discussed in Sections 3 and 4. In the first one, the version for normed spaces of Theorem 2.2 leads to an easier statement and, under hypotheses of uniqueness, it is possible to obtain a stability estimation for the solution. The reflexive case, which immediately follows and extends the classical known results, is illustrated with a non-Hilbertian data example. Section 4 completes the normed space setting, and as an application of Theorem 2.2 we also obtain analogous results for dual normed spaces. Finally, Section 5 is concerned with solving weakly the aforementioned kind of impulsive differential equation, generalizing the linear results in  to the reflexive context.
From now on, we assume that all the spaces are real, although our results are equally valid and easily adapted to the complex case.
2. Variational Inequalities with Constraints in Lcs
We first discuss a characterization of the existence of solutions to some constrained variational inequalities in the general setting of locally convex spaces. In order to state our main result, Theorem 2.2, the Hahn-Banach theorem, is required. Although there is a long history of using the Hahn-Banach theorem, recently a fine reformulation of this fundamental result has been developed in [5, 6] (see Proposition 2.1 below). It is known as the Hahn-Banach-Lagrange theorem and has encountered numerous applications in different branches of the mathematical analysis (see [5–8]). Let us recall that if is a real vector space, a function is sublinear provided that it is subadditive and positively homogeneous. For such an , if is a nonempty convex subset of a vector space, then is said to be -convex if Finally, a convex function is proper when there exists with .
Proposition 2.1 ([6, Theorem 2.9]). Let be a nontrivial vector space, and let be a sublinear function. Assume in addition that is a nonempty convex subset of a vector space, is a proper convex function, and is -convex. Then there exists a linear functional such that
Now we state the main result of this section, along the lines of [5–7]. To this end, some notations are required. For two real vector spaces and , a bilinear form , and , , and stands for the linear functional on and for the analogous linear functional on . In addition, given a real Hausdorff locally convex space , we will write to denote its dual space (continuous linear functionals on ).
The characterization is stated as follows.
Theorem 2.2. Let be a real Hausdorff locally convex space such that its dual space is also a real Hausdorff locally convex space. Suppose that and are real vector spaces, and are convex subsets of and , respectively, with , and are concave functions such that and , and and are bilinear forms satisfying that Then, if, and only if, there exists a continuous seminorm so that Furthermore, if one of these equivalent conditions holds, then it is possible to choose and with .
Proof. We can assume without loss of generality that is nontrivial, which is exactly the same as being nontrivial, thanks to the Hahn-Banach theorem.
Let us first assume that (2.6) is true for some continuous seminorm . The Hahn-Banach-Lagrange theorem (Proposition 2.1) applies, with the sublinear function , the -convex mapping defined as and the proper convex function given by obtaining thus that there exists a linear functional such that, on the one hand, and therefore for some , and on the other hand, satisfyies But we are assuming that Hence that is, Since and and , taking in this last inequality yields while for it implies and we have proven (2.5) as we wish.
And conversely, if satisfies (2.5), then so for the continuous seminorm on we have that
In any case we have stated the inequality .
As we can see, all the topological assumptions fall on . Thus, and are nothing more than convex sets, and there is no topological assumption on them, not even that they are closed. In particular, no continuity is supposed for or .
Let us also note that the condition , which seems to be irrelevant, will entail in the normed case a control of the norm .
Let us also emphasize that Theorem 2.2 captures the essence of the Hahn-Banach theorem from a variational standpoint. Theorem 2.2 extends the Lax-Milgram-type result given in [8, Theorem 1.2]. But the latter in turn is an equivalent reformulation of the Hahn-Banach theorem (Proposition 2.1), as shown in [8, Theorem 3.1]. Because the Hahn-Banach theorem and the Hahn-Banach-Lagrange theorem are equivalent results, Theorem 2.2 is nothing more than an equivalent version of the Hahn-Banach theorem.
3. Constrained Variational Equations in Reflexive Banach Spaces
Both in this section and the next one we turn to the study of constrained variational equations, only in the case of being a normed space and considering a kind of locally convex topology that, in some sense, satisfies a compactness property, which actually ensures that the solutions of the constrained variational inequality (2.5) belong to . To be more precise, in this section we fix the norm topology in and deduce that in an obvious way when is reflexive; that is, its closed unit ball is weakly compact ([2, Theorem 3.17]), which suffices for the applications in Section 5. In the next section we assume that is a dual Banach space, that is, for some normed space , endowed with its weak-star topology , in which, by the way, its closed unit ball is closed ([2, Theorem 3.16]). Continuing with the contents of this section, we provide an estimation of the norm of the solution only in terms of the data. We also generalize to the reflexive framework the classical Hilbertian characterization [3, 4] of those constrained problems (1.9) that admit a solution, indeed obtaining a proper extension of a result in the reflexive context that follows from [9, Theorem 2.1], developed for mixed variational formulations of elliptic boundary value problems.
Thus it is that we consider a real normed space , equipped with its norm topology, and the topology in is taken to be that associated with the canonical norm of . In this way we obtain the said estimation of the norm of a solution to the variational inequality with constraints (2.5). Moreover, we replace the existence of the continuous seminorm in Theorem 2.2 with that of a nonnegative constant.
Corollary 3.1. Suppose that is a real normed space, and are real vector spaces, and are convex subsets of and , respectively, , and are concave functions such that and . If and are bilinear forms so that then the constrained variational problem is solvable if, and only if, for some Moreover, when these statements hold and there exists satisfying , then
Proof. The equivalence between (3.2) and (3.3) clearly follows from Theorem 2.2 for the real Hausdorff locally convex space endowed with its norm topology and considering in its dual space the dual norm topology, and from the fact that if is a continuous seminorm, then it is bounded above by a suitable positive multiple of the norm.
Let us finally suppose that (3.2) or equivalently (3.3) holds. In order to prove (3.4) provided that there exists such that , let us start by fixing an arbitrary element in for which (3.2) is valid. Then, Hence, if then and so But in addition we have that since for with it clearly holds, and when , it suffices to make use of the fact that (3.2) is valid to arrive at the same conclusion. In summary, the continuous seminorm given for each by satisfies (2.6); therefore, Theorem 2.2 implies the existence of for which (3.2) is valid and , that is, which together with (3.7) finally yields (3.4).
Of course, when in Corollary 3.1 we additionally assume that is reflexive, then the existence of such that is equivalent to the existence of a solution in to the constrained variational inequality, that is, Next we focus our effort on proving that Corollary 3.1, with the additional hypothesis of the reflexivity of , provides a result, Theorem 3.8, generalizing the classical Hilbertian theory.
Lemma 3.2. Assume that is a real normed space, and are real vector spaces, and , , and are bilinear forms. If one writes and supposes that then
Proof. Let and . Since , choose so that Then, given , we have that and the announced inequality follows from the arbitrariness of .
In the next result we establish the first characterization of the solvability of a variational equation with constraints.
Theorem 3.3. Let be a real reflexive Banach space, let , and be real normed spaces, and let , , and be bilinear forms with and being continuous. Let and , and write Then, the corresponding constrained variational equation admits a solution; that is, if, and only if, In addition, if one of these equivalent conditions is valid and there exists with , then one can take in (3.20) with
Proof. Let us begin by stating (3.20)(3.22). Let be a solution to the variational equation with constraints (3.20). Then, in particular,
Thus, given there exists such that , so for all
and we have shown (3.22).
To conclude, we prove the converse (3.22)(3.20). Let and , and let , whose existence guarantees (3.22). Then Therefore, there exists such that so Corollary 3.1 (in combination with the reflexivity of ) for , , , and ensures, on the one hand, that the variational system (3.20) has a solution, hence stating the equivalence between (3.20) and (3.22), and, on the other hand, that if one of these conditions holds, then So, for establishing (3.23) and concluding the proof, it suffices to state the equality Suppose, on the one hand, that , , and , with , which in view of Lemma 3.2 implies that . Then Thus and therefore, as is arbitrary, And, on the other hand, if , , and satisfy but , Hence and thus Therefore, it follows from (3.32), Lemma 3.2, and (3.35) that But thanks to (3.28) we can choose with in such a way that so finally and (3.29) is proven.
Remark 3.4. In Theorem 3.3 we do not need to suppose that and are continuous: it suffices to impose as seen in its proof. Since for our applications the bilinear forms and are continuous, for the sake of simplicity, we have assumed this hypothesis. However, in Theorem 4.2 we impose these less restrictive assumptions, in a more general setting.
In connection with uniqueness we have the following elementary characterization, which establishes the equivalence of such uniqueness with that of the corresponding homogeneous variational equation with constraints.
Proof. If satisfies the nondegeneracy condition (3.42), then for any we have , so if (3.20) has two solutions and , then for some , and ; hence
By hypothesis it follows that ; that is, .
And conversely, if there exists , , such that then given a solution of (3.20), is also a solution, which is different than .
Hypotheses more restrictive than those of Theorem 3.3 imply uniqueness of the solution and simplify the control of the norm of the solution.
Corollary 3.6. Let be a real reflexive Banach space, let , , and be real normed spaces, let , and let , , and be bilinear forms such that and are continuous. Let one take and suppose that and that there exist constants with Then, for each , the corresponding variational equation with constraints admits one and only one solution; that is, Furthermore, if one defines then the a priori estimate is valid for the norm of the solution .
Proof. Corollary 3.1 (or [8, Corollary 1.3]) and Theorem 3.3, together with conditions (3.47) and (3.48), imply the existence of a solution whose uniqueness follows from Lemma 3.5. Besides, we deduce from Theorem 3.3 that for the solution , the identity holds. Finally, we have by condition (3.47) that and by Corollary 3.1 (or [8, Corollary 1.3]) and (3.48) that and thus we have the announced bound.
If we assume a condition on stronger than (3.48), the so-called inf-sup or Babuška-Brezzi condition (see [10–14] for some recent developments); that is, there exists such that then we have the stability estimate for the solution.
Taking into account that when , , and conditions (3.42) and (3.47) are satisfied if is coercive on , we deduce the following immediate consequence, which is well known (see [3, section II.1 Proposition 1.1] and [4, Lemma 4.67]) in the particular case of and being Hilbert spaces.
Corollary 3.7. Suppose that and are reflexive real Banach spaces, , , and and are continuous bilinear forms. Suppose in addition that, taking there exists such that and that . Then Besides, the solution satisfies the a priori estimate: In particular, if there exists such that then for the norm of the following estimation: holds.
In Theorem 3.3 and Lemma 3.5 we drive a characterization when the variational equation with constraints (3.20) admits a unique solution, for two fixed functionals and . The known particular cases in the literature are stated for arbitrary functionals and . Now, we derive a characterization along those lines. The particular case , , and was stated in [15, Corollary 2.7].
Theorem 3.8. Let be a real reflexive Banach space, let , , and be real normed spaces, and let , , and be bilinear forms with and being continuous. Let Then, for all and , if, and only if, and there exist so that In addition, if one of these equivalent conditions is satisfied, one has the following stability estimate:
Proof. In view of Corollary 3.6 and Lemma 3.5 we deduce, provided that (3.65), (3.66), and (3.67) hold, that for all and the constrained variational problem (3.64) admits a unique solution, whose norm satisfies estimation (3.68).
And conversely, suppose that for arbitrary and there exists a unique solution of (3.64). Then obviously we have that (3.65) holds, from Lemma 3.5. Moreover, since, in particular, for all there exists with ; then, the uniform boundedness theorem and Corollary 3.1 (or [8, Corollary 1.3]) imply (3.67). In a similar way we can arrive at (3.66): taking we have that for all there exists such that , which according to the Hahn-Banach theorem, the uniform boundedness theorem and Corollary 3.1 (or [8, Corollary 1.3]) are exactly (3.66).
Remark 3.9. We emphasize, in view of Remark 3.4, that in this proper extension of the Lax-Milgram theorem we just need to assume that and not necessarily that and are continuous. Theorem 4.3 is stated in these terms, and in a more general framework.
Let us again take up the elliptic boundary value problem considered in Introduction, in this case a more general one with non-Hilbertian data. We make use of our results with that elliptic boundary value problem for which the classical theory in the Hilbert framework does not apply. Thus we show how Theorem 3.8 (or Theorem 3.3) increases the class of elliptic boundary value problems for which it is known that the corresponding constrained variational equation derived from weakly imposing boundary conditions has a unique solution. But before doing so, we give a technical result, interesting in itself, and recall some common notations. For and stands for the usual norm in the Lebesgue space . In addition, the standard norm in the Sobolev space is given by and the inherited norm on the subspace is equivalent to the norm defined as thanks to the well-known Poincaré inequality ([16, Theorem 6.30]), which asserts that there exists a constant , depending only on and , in such a way that In fact, it is easy to check that we can take and for the optimal is given by (see [17, Section 1.1.3]). As usual, we denote by the dual space of , where is the conjugate exponent of defined through the relation . The dual norm of , when restricted to , will be denoted by and that of by .
The following result is a particular case of [18, Theorem 4], although we include it because we provide an explicit inf-sup constant, which will allow us to improve such a result in the concrete case to be used in Section 5.
Proposition 3.10. Let and , with conjugate exponent , and consider the continuous bilinear form given for each and by Then satisfies the inf-sup condition. More specifically,
Proof. It is sufficient to deal with . The description of the dual space of (see for instance [2, Proposition 8.14]) guarantees that for some , Then and so But taking into account that if with , then so (3.79) is equivalent to or in other words, that is, Therefore, we have that for some Hence, integrating and noting that , and thus, as a consequence of the triangular and the Holdër inequalities, so
The corresponding bilinear form defined for each in as when is a convex bounded plane polygon domain also satisfies the inf-sup condition (see [19, Theorem 2.1]). However we are just interested in the 1D case, since it is the one to be used in the applications of Section 5.
Now we are in a position to return to the mentioned example.
Example 3.11. Assume that , , , , and consider the elliptic boundary value problem
which does not admit the classical treatment ([3, section II.1 Proposition 1.1], [4, Lemma 4.67]), since the involved spaces are not Hilbert, except for . However, Theorems 3.3 and 3.8 apply. Indeed, if we multiply equation in by a test function , and integrate by parts, then we obtain the weak formulation of problem (3.90), that in the particular case coincides with the classical one:
Let us consider the real reflexive Banach space
and the normed spaces
with endowed with its norm, the continuous linear functionals and defined for each and , respectively, as
and the continuous bilinear forms , , and given by
Now , so the variational formulation above is nothing more than the variational equation with the following constraints:
In order to prove that this problem has a unique solution, let us check (3.65), (3.66), and (3.67). The first of these conditions follows from the duality and Proposition 3.10, which imply
which, in view of the fact that , is equivalent to
and the equivalence of the norm and the usual one in yields (3.65).
To prove that condition (3.66) is satisfied, we fix and apply Proposition 3.10 using equivalence of the norm and the usual one in .
In order to conclude, let us deduce the inf-sup condition (3.67), more specifically, that there exists , depending only on and , such that where denotes the dual norm of in , equivalently, Thus, let with . On the one hand, let us assume that . Then we define the function for which Then On the other hand, if , for we have that and and so, As the cases and follow from the preceding ones, we arrive at the inf-sup condition for (3.101) with
Therefore the variational equation with constraints (3.97), that is, the weak formulation obtained by imposing weakly the boundary conditions of (3.90), admits a unique solution , for whose norm the stability estimation is valid, where depends only on and .
With the applications of Section 5 in mind, we are just interested in the 1D case, yet as we commented previously, in [19, Theorem 2.1] an analogue of Proposition 3.10 is established for a convex bounded polygon . In such a case, it is possible to prove that the boundary value problem admits a constrained variational formulation similar to that of Example 3.11, where is the topological boundary of , , , and , the space of traces on of functions in the Sobolev space .
Next we show that the reflexivity of is essential in Theorem 3.8.
Proposition 3.12. If is a nonreflexive real normed space, then there exist real normed spaces , , and , continuous bilinear forms , , and , and and satisfying the assumptions in Theorem 3.8, except obviously the reflexivity of , but for any the constrained variational equation does not hold.
Proof. It suffices to take , , , and the corresponding duality pairings, any continuous linear functional on and . Then, since and , the first condition in the characterization holds. The second one is also satisfied, because However, although for all for each , since but .
The following result, stated in [9, Theorem 2.1], has a certain relation with Theorem 3.8 and ensures that the mixed variational formulation of some boundary value problems is uniquely solvable, when the data functions belong to reflexive spaces, unlike the classical Babuška-Brezzi theory ([3, 11, 20, 21]), developed only for the Hilbert framework: if , , , and are real reflexive Banach spaces and , , and are continuous bilinear forms, and we write Then, for all , there exists a unique such that if, and only if, and there exist such that