Research Article | Open Access
Junichi Aramaki, "Variational Problem Involving Operator Curl in a Multiconnected Domain", Chinese Journal of Mathematics, vol. 2016, Article ID 2459694, 8 pages, 2016. https://doi.org/10.1155/2016/2459694
Variational Problem Involving Operator Curl in a Multiconnected Domain
We shall study the problem of minimizing a functional involving the curl of vector fields in a three-dimensional, bounded multiconnected domain with prescribed tangential component on the boundary. The paper is an extension of minimization problem of the curl of vector fields. We shall prove the existence and the estimate of minimizers of more general functional which contains norm of the curl of vector fields.
In this paper, we consider the following problem which was proposed by Pan [1, p. 9].
Problem A. Minimize the norm of the curl of vector fields in a given space with tangential trace on the boundary being prescribed.
The problem is related to the mathematical theory of liquid crystal, of superconductivity, and of electromagnetic field. When and is a simply connected domain without holes, Bates and Pan [2, 3] showed the existence of minimizer. For the multiconnected domain, the author of  obtained the existence of a minimizer of the Problem A in the case .
In the present paper we shall extend the results to more general functional containing Problem A.
More precisely, let be a Carathéodory function on and is a convex function with respect to ; moreover assume that for a.e. , , and there exist and such that for a.e. and all : Without loss of generality, we may assume that . We furthermore assume the following structure condition: Under (1) with , we have
Let be a bounded domain in with boundary . Let be a given tangential vector field on . Let be the standard Sobolev space of vector fields. From now, we denote the tangential component of a vector field by ; that is, , where is the outer normal unit vector to the boundary . For any given tangential vector field on define a space of vector fieldsThen it is clear that is a closed convex set in . We consider the minimization problem
When , , and is a simply connected domain without holes, the authors of [2, 3] showed that (6) is achieved, and then in the case where , , and is bounded multiconnected domain, the author of  succeeded to show the existence of a minimizer of (6).
(O1) is a bounded domain in with boundary . is locally situated on one side of ; has a finite number of connected components and denoting the boundary of the infinite connected component of .
(O2) There exist manifolds of dimension and of class denoted by such that and they are nontangential to and such that is simply connected and pseudo .
The number is called the first Betti number and the second Betti number of . We say that is simply connected if , and has no holes if . If we define the spacesthen it is well known that and . We note that and are contained in ; moreover, and are closed subspaces of . Also it will be shown in Lemma 4 that and are closed subspaces of . Thus since is a finite-dimensional closed subspace of , has a complement in ; that is, is a closed subspace of , , and (the direct sum). Therefore, for any , there exist uniquely and such that . We denote the projection by .
Define Note that if and , then the tangent trace is well defined as an element of (cf. [5, p. 45]), andMoreover, we note that if , then(cf. Amrouche and Seloula [6, Theorem ]). We will see, in Lemma 2 of Section 2, that We are in a position to state the main theorem.
Theorem 1. Let be a bounded domain satisfying (O1) and (O2), and let be a tangential vector field on . Then is achieved, and the minimizer of in satisfies the following estimate. There exists a constant independent of such that
In this section, we shall give some lemmas as preliminaries.
Lemma 2. Let be a tangential vector field on . Then one has
Proof. Put Since , it is trivial that . For any , the problemhas a unique solution (cf. Girault and Raviart [7, Theorem ]). If we define , then in and . Thus . So we haveThus we have .
Problem B. Find the minimizer such that
In the later, we frequently use the following lemma.
Lemma 3. (i) If , , , and , then , and there exists a constant such thatHere we note that if furthermore is simply connected, we can delete the first term in the right-hand side of (18).
(ii) If , , , and , then , and there exists a constant such thatWe note that if furthermore has no holes, we can delete the first term in the right-hand side of (19).
Lemma 4. The space is a closed subspace of .
Proof. Let in . Then from (19) we haveTherefore is a Cauchy sequence in . Hence there exists such that in , so we have and in as . It is clear that in , and on . This implies that .
3. Proof of the Main Theorem 1
In this section, we give a proof of Theorem 1. The proof consists of some lemmas and propositions. Throughout this section, we assume that is a given tangential vector field on .
Lemma 5. Let . Then the minimization problem has a unique minimizer.
Proof. From Lemma 4, we know that is a closed subspace of . Thus it is well known that (21) has a minimizer. For the uniqueness of the minimizer, it suffices to show that the unit sphere does not contain any line segment for and . (cf. Fujita et al. [9, p. 306 and the remark]). However, this is clear because the functionalis strictly convex.
For , let be a unique minimizer of (21) and define . Then since for any and , , we haveIf we define a spacethen we see that . Then we have the following.
Lemma 6. One can see that
Proof. For any , as the above we can writeWe show the uniqueness of the above decomposition. If we can writewhere and , then . Therefore we haveHenceHere we use the following inequality. There exists a constant such thatfor all . For the proof of this inequality, see DiBenedetto [10, Lemma ] for , and see Miranda et al. [11, (7C’)]. Applying (30) with to (29), we haveFrom these equalities, we have , so .
Now we state a refinement of Fatou’s lemma (cf. Evans [12, pp. 11-12]).
Lemma 7. Assume that . Let weakly in and a.e. in . Then one hasIf furthermorethenwhere denotes the conjugate exponent of ; that is, . In particular, if strongly in and a.e. in , then (34) holds.
Proof. We use an elementary estimate. Let . Then, for any fixed , there exists a constant such that for any (cf. [12, (1.13)]). Definewhere for . Then we have If we apply (35) with and , we haveWe note that the right-hand side is integrable. By the hypothesis, we can see that a.e. in . Therefore by the Lebesgue dominated theorem, we haveTherefore we have Since weakly in , is bounded. Since is arbitrary, we haveIf furthermorethen we have
Lemma 8. is a weakly closed set in .
Proof. Let weakly in . Then we have in , on , andPassing to a subsequence, we may assume that strongly in and a.e. in . Thus from Lemma 7, we have in . Therefore we haveThis implies that .
Lemma 9. There exists a constant such that for all satisfying in andone has
Proof. If the conclusion (47) is false, there exists a sequence satisfying in andsuch that as . After passing to a subsequence, we may assume that weakly in , strongly in , and a.e. in . Therefore we have in and on , so . From Lemma 7,Thus we have . Hence strongly in . From (19), we see thatas . This contradicts .
Proposition 10. Let . Then the minimization problemis achieved and
Proof. By Lemma 2, we can see thatSince , it is clear thatOn the other hand, for any , we can write , where , and . Hence we haveThus (52) holds. We show that the right-hand side of (52) has a minimizer. Let be a minimizing sequence. ThenBy (1), we haveThus, by Lemma 9, is bounded in . Passing to a subsequence, we may assume that weakly in , strongly in , and a.e. in . Therefore we have on . Sinceit follows from Lemma 7 thatTherefore . It suffices to prove thatIn fact, we can choose a subsequence of so thatSince weakly in , it follows from the Mazur theorem that there exist such that convex hull of and strongly in . Hence we can choose a subsequence of so that strongly in and a.e. in . By the Fatou lemma, we haveSince is a convex function with respect to , we haveTherefore we have This completes the proof.
Lemma 11. Let be a minimizer of . Then is a weak solution of the following system:
Proof. If is a minimizer of , then we can see that, for any , we haveThus we have for all . We claim that In fact, since it is clear that , we haveConversely let . Choose to be a solution ofBy the elliptic regularity theorem, we see that . Define . Then , in , and on . Therefore and .
Hence (67) holds for any . Since , it follows from (67) that is a weak solution of (65).
Remark 12. The system (65) is so called the -curl system. When is a bounded, simply connected domain in without holes and with boundary for some . If , then  showed that the weak solution of system (65) satisfies the fact that for some and there exists a constant depending only on such that .
Proof. Since for any , we see thatThus is a minimizer of (17). On the other hand, for any minimizer of (17), define . Then . From (67), we haveTherefore,By the structure condition (2), we have in , so .
If is a minimizer of (52), we can write , where . If follows from Lemma 6 that we see that . Thus the minimizer of (52) in is unique.
In order to show the estimate in Theorem 1, it suffices to prove the following proposition.
Proposition 14. There exists a constant independent of such that
Proof. Assume that the conclusion is false. Then there exists a sequence such that andFor brevity of notation, we write