A new compact class of open sets under Hausdorff distance and shape optimization
aa r X i v : . [ m a t h . GN ] A ug A complete class of domains underHausdorff-Pompeiu distance
Donghui YangDepartment of Mathematics, Central China Normal University,Wuhan, Hubei, 430079, P.R.of China [email protected]
Abstract
In this paper we obtain a new class of domains and we prove the class iscomplete under the Hausdorff-Pompeiu distance.
Key words. class of domains, Hausdorff-Pompeiu distance
AMS subject classifications. There are many complete classes of domains under Hausdorff-Pompeiu distancehave been found(See [1], [2], [4], [5], [7], [10], [11] etc.) We also can see more casesin the books [3], [6], [8], [9] etc. In this paper, we define a new class of domains andshow it complete under the Hausdorff-Pompeiu distance.We call a domain (connected open set) Ω ⊂ R k satisfies property ( C M ), if forany x, y ∈ Ω there exists a connected compact set K such that x, y ∈ K and [ z ∈ K B ( z, d ∗ M ) ⊂ Ω, here d ∗ = min { dist( x, ∂ Ω) , dist( y, ∂ Ω) } and M > B ( z, r ) an open ball with center z and radius r , dist( · , · ) denotes the Euclidean metric in R k .For given R > C M,R = { Ω ⊂ R k ; B ( x Ω , R ) ⊂ Ω ⊂⊂ B ∗ , Ω is a domain and satisfy the property ( C M ) } , where B ∗ is a bounded domain and x Ω is some point in Ω. The authors were supported by NSFC 10901069. he topology on C M,R is induced by the Hausdorff-Pompeiu distance between thecomplementary sets, i.e., for any Ω , Ω ∈ C M,R , ρ (Ω , Ω ) = max { sup x ∈ B ∗ \ Ω dist( x, B ∗ \ Ω ) , sup y ∈ B ∗ \ Ω dist( B ∗ \ Ω , y ) } . (1.1)We denote by Hlim, the limit in the sense of (1.1).In this work, we obtain the following main result about the family C M,ρ : Theorem 2.1 If { Ω m } ∞ m =1 ⊂ C M,R , then there exists a subsequence { Ω m k } ∞ k =1 of { Ω m } ∞ m =1 such that Hlim k →∞ Ω m k = Ω and Ω ∈ C M,R . i.e., ( C M,R , ρ ) is a compact metric space.
We shall use the following notations: δ ( K , K ) = max { sup x ∈ K d ( x, K ) , sup y ∈ K d ( K , y ) } , where K and K are compact subsets in R k . Following from the definitions of ρ, δ ,we obtain that ρ (Ω , Ω ) = δ ( B ∗ \ Ω , B ∗ \ Ω ) for any open sets Ω , Ω ⊂ B ∗ , hencewe also call δ the Hausdorff-Pompeiu distance. Remark 2.1 If A, A n , n = 1 , , · · · , be compact subsets in R k and δ ( A n , A ) → n → ∞ ), then by the definition of δ we obtain that for any ǫ > N ( ǫ ) > m ≥ N ( ǫ ) we have A ⊂ [ x ∈ A m B ( x, ǫ ) and A m ⊂ [ x ∈ A B ( x, ǫ ).The following Lemma 2.1-Lemma 2.4 were given and proved in [3], [6], [8], [9].Which will be used in this paper. Lemma 2.1
Let
A, A n , n = 1 , , · · · , be compact subsets in R k such that δ ( A n , A ) →
0, then A is the set of all accumulation points of the sequences { x n } such that x n ∈ A n for each n . Lemma 2.2
Let O = { C ⊂ R k ; C is compact } . Then ( O , δ ) is a locally compactand complete metric space. Lemma 2.3
Let A, ˜ A, A n , ˜ A n , n = 1 , , · · · , be compact subsets in R k such that δ ( A n , A ) → δ ( ˜ A n , ˜ A ) →
0. Suppose that A n ⊂ ˜ A n for each n , Then A ⊂ ˜ A .2 emma 2.4 (Γ − property for C M,R ) Assume that { Ω n } ∞ n =1 ⊂ C M,R , Ω ∈ C M,R and Ω = HlimΩ n . Then for each open subset K satisfying K ⊂ Ω , there exists apositive integer n K (depending on K ) such that K ⊂ Ω n for all n ≥ n K .The next Lemma 2.5 is clear, but we show it in the following. Lemma 2.5
Let { x n } ⊂ R k and { r n } ⊂ R be such that r n ≥ r > B ( x n , r n ) = D ⊂ R k . Then there exists x ∈ R k such that B ( x , r ) ⊂ D and x n → x ( n → ∞ ). Proof
Since B ( x n , r n ) ⊂ B ∗ and B ∗ is a bounded subset in R k , there exists asubsequence of { x n } , still denoted by itself, such that x n → x (2.1)for some x ∈ B ∗ .We claim that B ( x , r ) ⊂ D. (2.2)By contradiction, we assume that there did exist y ∈ B ( x , r ) and y D . Since D = Hlim B ( x n , r n ), i.e., δ ( B ∗ \ B ( x n , r n ) , B ∗ \ D ) →
0. It follows from Lemma 2.1that there exists a sequence { y n } satisfying y n ∈ B ∗ \ B ( x n , r n ) and y n → y . (2.3)Hence d ( y n , x n ) ≥ r n ≥ r . By (2.1) and (2.3) we may pass to the limit for n → ∞ to get d ( y , x ) ≥ r , which leads to a contradiction and implies (2.2) as desired. This completes the proof. (cid:3) Lemma 2.6
Let { K n } ∞ n =1 be a sequence of connected compact sets and Hlim K n = K , where K n ⊂ B ∗ for all n ∈ Z + , then K is also connected compact set. Proof
Following from Lemma 2.2 we obtain K is a compact set.Now we show that K is a connected set. Otherwise, there exist at least twocomponents in K , denote one of these by K , and K = K \ K , then K , K arecompact, which shows that l ≡ dist( K , K ) >
0. Since K n ⊂ [ z ∈ K B ( z, l [ z ∈ K B ( z, l (cid:3) ∪ (cid:2) [ z ∈ K B ( z, l (cid:3) for n large enough according to Remark 2.1, followingfrom (cid:2) [ z ∈ K B ( z, l (cid:3) ∩ (cid:2) [ z ∈ K B ( z, l (cid:3) = ∅ and connectedness of K n we obtain thatthe contradiction. (cid:3) Lemma 2.7
Let Ω be a bounded domain and d ( K ) ≡ dist( K, ∂
Ω), where K ⊂ Ωis a nonempty compact sets. Then for all nonempty compact sets K , K ⊂ Ω, wehave | d ( K ) − d ( K ) | ≤ δ ( K , K ) . In other words, d ( K ) is continuous at any compact set K ( K ⊂ Ω) under theHausdorff-Pompeiu distance.
Proof
We assume d ( K ) ≤ d ( K ), then we only need to show d ( K ) ≤ d ( K ) + δ ( K , K ). Since K , ∂ Ω are compact, there exist x ∈ K , y ∈ ∂ Ω such that d ( K ) = | x − y | . Note that | x − x | = dist( x , K ) ≤ sup z ∈ K dist( z, K ) ≤ δ ( K , K ) since K is compact, where x ∈ K . Hence we have d ( K ) = dist( K , ∂ Ω) ≤ | x − y | ≤| x − x | + | x − y | ≤ dist( K , x ) + | x − y | ≤ δ ( K , K ) + d ( K ). (cid:3) The main result in this paper is as follows.
Theorem 2.1 If { Ω m } ∞ m =1 ⊂ C M,R , then there exists a subsequence { Ω m k } ∞ k =1 of { Ω m } ∞ m =1 such that Hlim k →∞ Ω m k = Ω and Ω ∈ C M,R . i.e., ( C M,R , ρ ) is a compact metric space.
Proof ◦ . Ω is not empty.By the definition of C M,R , there exists B ( x m , R ) ⊂ Ω m for every m ∈ Z + . Now weconsider the sequence of domains { B ( x m , R ) } ∞ n =1 , by Lemma 2.5 we obtain that thereexists x ∈ Ω such that B ( x , R ) ⊂ Hlim B ( x n , R ) = D ⊂ Ω and x n → x ( n → ∞ ).Let ˜ A m = B ∗ \ B ( x m , R ) , ˜ A = B ∗ \ D ; A m = B ∗ \ Ω m , A = B ∗ \ Ω. Then δ ( A m , A ) → δ ( ˜ A m , ˜ A ) →
0, by Lemma 2.3 we get B ( x , R ) ⊂ D ⊂ Ω and which show that Ωis not empty.2 ◦ . Ω is a domain.Otherwise, there exists at least two components in Ω, we denote one of componentsof Ω by Ω and Ω = Ω \ Ω . Obviously, Ω ∩ Ω = ∅ and Ω , Ω are open sets as well.There exist two points x , x and d ∗ > B ( x , d ∗ ) ⊂ Ω and B ( x , d ∗ ) ⊂ Ω ,here d ∗ = min { dist( x , ∂ Ω ) , dist( x , ∂ Ω ) } . For any fixed 0 < r < d ∗ , we have B ( x , r ) ∪ B ( x , r ) ⊂ B ( x , d ∗ ) ∪ B ( x , d ∗ ) ⊂ Ω ∪ Ω = Ω, by Lemma 2.4 there4xists N > m ≥ N we have B ( x , r ) ∪ B ( x , r ) ⊂ Ω m , especially, B ( x , r ) ∪ B ( x , r ) ⊂ Ω m . By the definition of C M,R , there exists a connectedcompact set K m with x , x ∈ K m and d ∗ m = min { dist( x , ∂ Ω m ) , dist( x , ∂ Ω m ) } > r and [ z ∈ K m B ( z, d ∗ m M ) ⊂ Ω m for every m ≥ N . Since K m Ω ∪ Ω by Ω ∩ Ω = ∅ ,there exists some y m ∈ K m \ Ω for every m ≥ N , then we obtain a sequence of points { y m } ∞ m = N and { y m } ∞ m = N ∩ Ω = ∅ . Since { y m } ∞ m = N ⊂ B ∗ and B ∗ is bounded, thereexists a subsequence of { y m } ∞ m = N (still denoted { y m } ∞ m = N ) and y Ω such that y m → y as m → ∞ .Now we consider the ball B ( y , r M ). Since y m → y as m → ∞ , there ex-ists N > m ≥ N we have | y m − y | < r M . According to [ z ∈ K m B ( z, r M ) ⊂ Ω m we get B ( y m , r M ) ⊂ Ω m , furthermore, B ( y , r M ) ⊂ Ω m for all m ≥ N = max { N , N } . By the same argument with step 1 ◦ we get B ( y , r M ) ⊂ Ω,which contradict to y Ω and we prove that Ω is a domain.3 ◦ . Ω satisfies the property ( C M ).For any x, y ∈ Ω, we set d ≡ sup { d ( K ); K is a connected compact set, x, y ∈ K and [ z ∈ K B ( z, d ) ⊂ Ω } . Now let { K m } ∞ m =1 be a sequence such that d ( K m ) → d as m → ∞ . Since K m ⊂ B ∗ ( m ∈ Z + ) there exist a set K and a subsequence of { K m } ∞ m =1 (still denoted { K m } ∞ m =1 ) such that δ ( K m , K ) →
0. Following from Lemma 2.6 we get K is aconnected compact set, and by Lemma 2.1 we obtain that x, y ∈ K . Finally, byLemma 2.7 we get d ( K ) = d . Claim 1 : [ z ∈ K B ( z, d ) ⊂ Ω. Proof of Claim 1 : Fix ǫ >
0, there exists N > m ≥ N we have K ⊂ [ z ∈ K m B ( z, ǫ d ( K m ) → d ( m → ∞ ),there exists N > m ≥ N we have d − d ( K m ) < ǫ [ z ∈ K B ( z, d − ǫ ) ⊂ [ z ∈ K m B ( z, d ( K m )) ⊂ Ω for all m ≥ max { N , N } , and we obtainthe Claim 1 by letting ǫ → d ≥ d ∗ M , where d ∗ = min { dist( x, ∂ Ω) , dist( y, ∂ Ω) } .Since B ( x, r ) , B ( y, r ) ⊂ Ω for any given 0 < r < d ∗ , there exists N > m ≥ N we have B ( x, r ) , B ( y, r ) ⊂ Ω m . Hence there exists Q m ⊂ Ω m such5hat x, y ∈ Q m and [ z ∈ Q m B ( z, rM ) ⊂ Ω m for every m ≥ N ( r ) by Ω m ∈ C M,R , here Q m is a connected compact set. By Lemma 2.2 we obtain that there exist a set Q and a subsequence of { Q m } ∞ m = N ( r ) (still denoted { Q m } ) such that δ ( Q m , Q ) → n → ∞ . By Lemma 2.6 we know that Q is also a connected compact set, and byLemma 2.1 we obtain that x, y ∈ Q . claim 2 : [ z ∈ Q B ( z, rM ) ⊂ Ω. Proof of claim 2 : Otherwise, we take x ∗ ∈ (cid:2) [ z ∈ Q B ( z, rM ) (cid:3) \ Ω, then there exists z ∗ ∈ Q such that | x ∗ − z ∗ | = dist( x, Q ). Denote ǫ ∗ = | x ∗ − z ∗ | , then B ( x ∗ , ǫ ∗ ⊂ [ z ∈ Q B ( z, rM − ǫ ∗ Q ⊂ [ z ∈ Q m B ( z, ǫ ∗ m ≥ N ≡ max { N , N ( ǫ ∗ ) } , and so we get B ( x ∗ , ǫ ∗ ⊂ [ z ∈ Q m B ( z, rM ) ⊂ Ω m for all m ≥ N ∗ , bythe same argument with step 1 ◦ we obtain B ( x ∗ , ǫ ∗ ⊂ Ω. Which is a contradictionsince x ∗ Ω.Since 0 < r < d ∗ is arbitrary, letting r → d ∗ we get [ z ∈ Q B ( z, d ∗ M ) ⊂ Ω. By thedefinition of d we obtain d ≥ d ∗ M . Which implies Ω satisfies ( C M ).4 ◦ . Following from 1 ◦ , ◦ , ◦ , we have proved the Theorem 2.1. (cid:3) Remarks 2.2 (i) In the Theorem 2.1, the assumption of the ball B ( x Ω , R ) ⊂ Ω ∈ C M,R is justfor preserving the nonempty limit under Hausdorff-Pompeiu distance.(ii) The property ( C M ) make the domains converge to the domain, which is afundamental property of a class of domains.(iii) We also can discuss γ -convergence on C M,R if we add more properties to thisclass. ( γ -convergence can be found in the references [2]-[11].) But we can not discussthe γ -convergence here, since some domains in C M,R may be very ”bad”. For example,there are standard cusps domains (see [1]) in C M,R .(iv) There is an interesting question we comment here. Replace the property ( C M )by ( C ′ M ): for any x, y ∈ Ω there exists a continuous curve c : [0 , → Ω such that x = c (0) , y = c (1) and [ z ∈ c ([0 , B ( z, d ∗ M ) ⊂ Ω, here d ∗ = min { dist( x, ∂ Ω) , dist( y, ∂ Ω) } and M > C M,R a compact metric space as well?6 eferences [1] A. Adams, Sobolev Spaces, Academic Press, New York, 2003.[2] D. Bucur and J. P. Zol´ e sio, N-dimensiona shape optimization under capaci-tary constraints, J. Differential Equations, 123(2) (1995), 504-522.[3] D. Bucur and G. Buttazzo, Variational Methods in Shape OptimizationProblems, Birkh¨ a user, Boston, 2005.[4] G. Buttazzo and P. Guasoni, Shape optimization problems over classes ofconvex domains, J. Convex Anal., 4(1997), 343-351.[5] D. Chenais, On the existence of a solution in a domain identification problem,J. Math. Anal. Appl., 52 (1975), 189-219.[6] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,Birkh¨ a user, Boston, 2006.[7] G. Dal Maso and U. Mosco, Wiener’s criterion and Γ-convergence, Appl.Math. Optim., 15 (1987), 15-63.[8] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag,Berlin, 1984.[9] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Press Syndi-cate of the University of Cambridge, 1993.[10] V. ˘ S ver´ a k, On optimal shape design, J. Math. Pures Appl., 72(1993), 537-551.[11] Gengsheng Wang and Donghui Yang, Decomposition of vector-valued diver-gence free Sobolev functions and shape optimization for stationary Navier-Stokes equations, Comm. Partial Differential Equations 33 (2008), no. 1-3,429–449. 7 r X i v : . [ m a t h . GN ] A ug A new compact class of open sets under Hausdorffdistance and shape optimization
Donghui YangSchool of Mathematics and Statistics, Central China Normal University,430079, Wuhan, People’s Republic of China [email protected]
Abstract
In this paper we obtain a new class of open sets, and we prove the class iscompact under the Hausdorff distance, then we prove the existence of solutionsof some shape optimization for elliptic equations. There are many papers concerning existence theory for shape optimization prob-lems. There are several types of results: using regularity assumptions for the bound-ary of the unknown open sets (see [4], [5], [9], [11]), using certain capacitary con-straints (see [3], [7], [10]) or using the notion of a generalization perimeter and con-straints or penalty terms constructed with it (see [1], [2], [3], [7]). In general, for theshape optimization problems, we must give a class of open sets and prove the classis complete under the Hausdorff distance in the first place, and then we should provethat the shape optimization problems have at least one solutions.In this work, we give a new class of open sets, and we prove the class is compactunder the Hausdorff distance, then we prove the existence of solutions of some shapeoptimization for elliptic equations.Let B ∗ ⊂ R k be a bounded domain. On the space H ( B ∗ ) we consider thenorm k u k H ( B ∗ ) = (cid:16) Z B ∗ |∇ u | dx (cid:17) . Let be given a smooth symmetric matrix A ∈ M k × k ( C ( B ∗ ) , A = A T (where A T represent the transformation of the matrix A ), and α | ξ | ≤ h A.ξ, ξ i , (1) The author was supported by NSFC 10901069. here 0 < α is a constant. We define the associated operator A : H ( B ∗ ) → H − ( B ∗ ): A = div( A. ∇ ) . (2)For any open set Ω ∈ C M,R ( C M,R will define later), we consider the Dirichletproblem in Ω: u ∈ H (Ω) , − A u Ω = f (3)in the variational sense, i.e., Z Ω h A. ∇ u Ω , ∇ φ i dx = h f | Ω , φ i H − (Ω) × H (Ω) ∀ φ ∈ C ∞ (Ω) (4)with f ∈ H − ( B ∗ ) and f | Ω denoting the restriction of the distribution f to the openset Ω.Because H (Ω) = cl H ( B ∗ ) ( C ∞ (Ω)), then (4) has a unique solution u Ω ∈ H (Ω),which we can extend with zero on B ∗ − Ω, to u , and u ∈ H ( B ∗ ), k u k H ( B ∗ ) = k u Ω k H (Ω) .When we consider the solution of (3), we will implicitly take its extension u . Weshall study the following shape optimization probleminf Ω ∈ C M,R J (Ω) ≡ inf Ω ∈ C M,R n Z B ∗ | u Ω − g | dx o . ( P )In the following, we give the definition of C M,R .We call a open set Ω ⊂ R k with property ( C M ), if for any x, y ∈ Ω, there existsa connected compact set K with x, y ∈ K , such that K ⊂ Ω and [ z ∈ K B ( z, d ∗ M ) ⊂ Ω,here d ∗ = min { dist( x, ∂ Ω) , dist( y, ∂ Ω) } and M > B ( z, r ) an open ball with center z and radius r .For given R > C M,R = { Ω ⊂ R k ; B ( x Ω , R ) ⊂ Ω ⊂ B ∗ , Ω is a open set and satisfy the property ( C M ) } , where B ∗ is a bounded domain and x Ω is some point in Ω.The topology on C M,R is induced by the Hausdorff distance between the comple-mentary sets, i.e., for any Ω , Ω ∈ C M,R , ρ (Ω , Ω ) = max { sup x ∈ B ∗ \ Ω dist( x, B ∗ \ Ω ) , sup y ∈ B ∗ \ Ω dist( B ∗ \ Ω , y ) } , (5)2here dist( · , · ) denotes the Euclidean metric in R N . We denote by Hlim, the limit inthe sense of (5).In this work, we obtain the following main result about the family C M,ρ : Theorem 2.1 If { Ω m } ∞ m =1 ⊂ C M,R , then there exists a subsequence { Ω m k } ∞ k =1 of { Ω m } ∞ m =1 such that Hlim k →∞ Ω m k = Ω and Ω ∈ C M,R . i.e., ( C M,R , ρ ) is a compact metric space.We note that there are many compact classes of open sets under Hausdorff distancehave been found (See [1], [2], [4], [5], [6], [8], [10], [11], [12] etc.). We also can seemore cases in the books [3], [7], [9] etc. But the class of the open sets C M,R has neverappeared in any other place.By Theorem 2.1, we obtain the existence of the optimal solutions for problem ( P ): Theorem 3.1
The shape optimization problem ( P ) has at least one solution. C M,R
We shall use the following notations: δ ( K , K ) = max { sup x ∈ K dist( x, K ) , sup y ∈ K dist( K , y ) } , where K and K are compact subsets in R k . Following from the definitions of ρ, δ ,we obtain that ρ (Ω , Ω ) = δ ( B ∗ \ Ω , B ∗ \ Ω ) for any open sets Ω , Ω ⊂ B ∗ , hencewe also call δ the Hausdorff distance.The following results were given and proved in [3], [7], [9]. Which will be used inthis paper. Lemma 2.1
Let
A, A n , n = 1 , , · · · , be compact subsets in R k such that δ ( A n , A ) →
0, then A is the set of all accumulation points of the sequences { x n } such that x n ∈ A n for each n . Remark 2.1
Following from Lemma 2.1 and the definition of δ , we obtain that forany ǫ > N ( ǫ ) > m ≥ N ( ǫ ) we have A ⊂ [ x ∈ A m B ( x, ǫ )and A m ⊂ [ x ∈ A B ( x, ǫ ). Lemma 2.2
Let A, ˜ A, A n , ˜ A n , n = 1 , , · · · , be compact subsets in R k such that δ ( A n , A ) → δ ( ˜ A n , ˜ A ) →
0. Suppose that A n ⊂ ˜ A n for each n , Then A ⊂ ˜ A .3 emma 2.3 (Γ − property for C M,R ) Assume that { Ω n } ∞ n =1 ⊂ C M,R , Ω ∈ C M,R and Ω = HlimΩ n . Then for each open subset K satisfying K ⊂ Ω , there exists apositive integer n K (depending on K ) such that K ⊂ Ω n for all n ≥ n K . Lemma 2.4
If Ω n ⊂ B ∗ , n ∈ N , are open bounded sets, there exists Ω ⊂ B ∗ ,open, such that Ω = HlimΩ n , on a subsequence. In particular, let O = { C ⊂ B ∗ ; C = ∅ , C is compact } . Then ( O , δ ) is a compact metric space.The next Lemma 2.5 is clearly, but we show it in the following. Lemma 2.5
Let { x n } ⊂ R k and { r n } ⊂ R be such that r n ≥ r > B ( x n , r n ) = D ⊂ R k . Then there exists x ∈ R k such that B ( x , r ) ⊂ D and x n → x ( n → ∞ ). Proof
Since B ( x n , r n ) ⊂ B ∗ and B ∗ is a bounded subset in R k , there exists asubsequence of { x n } , still denoted by itself, such that x n → x (6)for some x ∈ B ∗ .We claim that B ( x , r ) ⊂ D. (7)By contradiction, we assume that there did exist y ∈ B ( x , r ) and y D . Since D = Hlim B ( x n , r n ), i.e., δ ( B ∗ \ B ( x n , r n ) , B ∗ \ D ) →
0. It follows from Lemma 2.1that there exists a sequence { y n } satisfying y n ∈ B ∗ \ B ( x n , r n ) and y n → y . (8)Hence d ( y n , x n ) ≥ r n ≥ r . By (6) and (8) we may pass to the limit for n → ∞ to get d ( y , x ) ≥ r , which leads to a contradiction and implies (7) as desired. This completes the proof. (cid:3) Lemma 2.6
Let { K n } ∞ n =1 be a sequence of connected compact sets and Hlim K n = K , where K n ⊂ B ∗ for all n ∈ Z + , then K is also connected compact set. Proof
By Lemma 2.4, we obtain that K is compact.4ow we show that K is a connected set. Otherwise, there exist at least twocomponents in K , denote one of these by K , and K = K \ K , then K , K arecompact, which shows that l ≡ dist( K , K ) >
0. Since K n ⊂ [ z ∈ K B ( z, l (cid:2) [ z ∈ K B ( z, l (cid:3) ∪ (cid:2) [ z ∈ K B ( z, l (cid:3) for n large enough, following from (cid:2) [ z ∈ K B ( z, l (cid:3) ∩ (cid:2) [ z ∈ K B ( z, l (cid:3) = ∅ and connectedness of K n we obtain the contradiction. (cid:3) Lemma 2.7
Let Ω be a bounded open set and d ( K ) ≡ dist( K, ∂
Ω), where K ⊂ Ω be compact sets. Then for all compact sets K , K ⊂ Ω, we have | d ( K ) − d ( K ) | ≤ δ ( K , K ) . Proof
We assume d ( K ) ≤ d ( K ), then we only need to show d ( K ) ≤ d ( K ) + δ ( K , K ). Since K , ∂ Ω are compact, there exist x ∈ K , y ∈ ∂ Ω such that d ( K ) = | x − y | . Note that dist( x , K ) ≤ sup z ∈ K dist( z, K ) ≤ δ ( K , K ), we obtain d ( K ) =dist( K , ∂ Ω) ≤ dist( K , y ) ≤ dist( K , x ) + | x − y | ≤ d ( K ) + δ ( K , K ). (cid:3) The main result in this paper is as follows.
Theorem 2.1 If { Ω m } ∞ m =1 ⊂ C M,R , then there exists a subsequence { Ω m k } ∞ k =1 of { Ω m } ∞ m =1 such that Hlim k →∞ Ω m k = Ω and Ω ∈ C M,R . i.e., ( C M,R , ρ ) is a compact metric space.
Proof
By Lemma 2.4 we know that there exists Ω ⊂ B ∗ and a subsequence { Ω m k } ∞ k =1 of { Ω m } ∞ m =1 such that Hlim k →∞ Ω m k = Ω, without loss of generality, we stilldenote H lim m →∞ Ω m = Ω. So we only need to prove that Ω ∈ C M,R .In order to prove that Ω ∈ C M,R , we separate it to the following steps.1 ◦ . Ω is not empty.By the definition of C M,R , there exists B ( x m , R ) ⊂ Ω m for every m ∈ Z + . Nowwe consider the sequence of open sets { B ( x m , R ) } ∞ n =1 , by Lemma 2.4 and Lemma 2.5we obtain that there exists x ∈ Ω such that B ( x , R ) ⊂ Hlim B ( x n , R ) = D ⊂ Ω and x n → x ( n → ∞ ). Let ˜ A m = B ∗ \ B ( x m , R ) , ˜ A = B ∗ \ D ; A m = B ∗ \ Ω m , A = B ∗ \ Ω.Then δ ( A m , A ) → δ ( ˜ A m , ˜ A ) →
0, by Lemma 2.2 we get B ( x , R ) ⊂ D ⊂ Ω andwhich show that Ω is not empty.2 ◦ . Ω is connected.Otherwise, there exists at least two components in Ω, we denote two of components5f Ω by Ω , Ω . Obviously, Ω ∩ Ω = ∅ . Since Ω , Ω are all open sets, there exist twopoints x ∈ Ω , x ∈ Ω and d > B ( x , d ) ⊂ Ω and B ( x , d ) ⊂ Ω , here d ∗ = min { dist( x , ∂ Ω ) , dist( x , ∂ Ω ) } . For any 0 < r < d ∗ , by Lemma 2.3 thereexists N > m ≥ N we have B ( x , r ) ∪ B ( x , r ) ⊂ Ω m , further-more, B ( x , r ) ∪ B ( x , r ) ⊂ Ω m . By the definition of C M,R , there exists a connectedcompact set K m for every m ≥ N such that x , x ∈ K m and [ z ∈ K m B ( z, r M ) ⊂ Ω m .Since K m Ω ∪ Ω by Ω ∩ Ω = ∅ , there exists some y m ∈ K m \ Ω for every m ≥ N , then we obtain a sequence of points { y m } ∞ m = N and { y m } ∞ m = N ∩ Ω = ∅ .Since { y m } ∞ m = N ⊂ B ∗ and B ∗ is bounded, there exists a subsequence of { y m } ∞ m = N ,denote by itself, and y Ω such that y m → y as m → ∞ .Now we consider the ball B ( y , r M ). Since y m → y as m → ∞ , there exists N > m ≥ N we have | y m − y | < r M . On the other hand,by [ z ∈ K m B ( z, r M ) ⊂ Ω m we get B ( y m , r M ) ⊂ Ω m , hence, B ( y , r M ) ⊂ Ω m for all m ≥ N = max { N , N } . By the same argument with step 1 ◦ or by Lemma 2.2 weget B ( y , r M ) ⊂ Ω, hence B ( y , d ∗ M ) ⊂ Ω by letting r → d ∗ , which contradict to y Ω and we prove that Ω is connected.3 ◦ . Ω ∈ C M,R .We only need to prove that Ω satisfies the property ( C M ).(1). For any x, y ∈ Ω, we set d ≡ d ( x, y ) ≡ sup { d ( K ); K is connected compact set and x, y ∈ K, [ z ∈ K B ( z, d ) ⊂ Ω } . We note that Ω is a local path connected set (since Ω is an open set) and connected(by step 2 ◦ ), then Ω is a path connected set by the point topological theory. Hence forany x, y ∈ Ω there exist at least one path f : [0 , → Ω such that f (0) = x, f (1) = y .Which implies d = d ( x, y ) exists.Let { K m } ∞ m =1 be a sequence such that d ( K m ) → d as m → ∞ by Lemma 2.4.Since K m ⊂ B ∗ , m ∈ Z + , there exist a set K and a subsequence of { K m } ∞ m =1 , stilldenote by itself, such that δ ( K m , K ) →
0. By Lemma 2.7 we know that d ( K ) is acontinuous function about K under Hausdorff distance, hence d ( K ) = d .Following from Lemma 2.6 we get K is a connected compact set, and by Lemma2.1 we obtain that x, y ∈ K .(2). We claim that [ z ∈ K B ( z, d ) ⊂ Ω. 6f the Claim is false, there exists a point x ∈ (cid:2) [ z ∈ K B ( z, d ) (cid:3) \ Ω. We assumedist( x , K ) = | x − z | , here z ∈ K , and denote ǫ = d − | x − z | , then B ( x , ǫ ⊂ [ z ∈ K B ( z, d − ǫ K ⊂ [ z ∈ K m B ( z, ǫ m large enough (see Remark 2.1),we get B ( x , ǫ ⊂ [ z ∈ K m B ( z, d ( K m )) ⊂ Ω m for d − d ( K m ) < ǫ ◦ we obtain B ( x , ǫ ⊂ Ω. Which is a contradiction since x Ω.(3). We show that d ≥ d ∗ M , where d ∗ = min { dist( x, ∂ Ω) , dist( y, ∂ Ω) } .Since B ( x, r ) , B ( y, r ) ⊂ Ω for any given 0 < r < d ∗ , there exists N ( r ) > m ≥ N ( r ) we have B ( x, r ) , B ( y, r ) ⊂ Ω m . Hence there exists Q m ⊂ Ω m such that [ z ∈ Q m B ( z, rM ) ⊂ Ω m for every m ≥ N ( r ) by Ω m ∈ C M,R , here Q m isa connected compact set. By Lemma 2.4 we obtain that there exist a set Q anda subsequence of { Q m } ∞ m = N ( r ) , still denote by itself, such that δ ( Q m , Q ) → n → ∞ . By Lemma 2.6 we know that Q is also a connected compact set.Now claim: [ z ∈ Q B ( z, rM ) ⊂ Ω.Otherwise, we take x ∗ ∈ (cid:2) [ z ∈ Q B ( z, rM ) (cid:3) \ Ω, then there exists z ∗ ∈ Q such that | x ∗ − z ∗ | = dist( x, Q ). Denote ǫ ∗ = rM −| x ∗ − z ∗ | , then B ( x ∗ , ǫ ∗ ⊂ [ z ∈ Q B ( z, rM − ǫ ∗ Q ⊂ [ z ∈ Q m B ( z, ǫ ∗ m ≥ N ∗ = max { N ( r ) , N ( ǫ ∗ ) } ,and we get B ( x ∗ , ǫ ∗ ⊂ [ z ∈ Q m B ( z, rM ) ⊂ Ω m for all m ≥ N ∗ , by the same argumentwith step 1 ◦ we obtain B ( x ∗ , ǫ ∗ ⊂ Ω. Which is a contradiction since x ∗ Ω.Since 0 < r < d ∗ is arbitrary, letting r → d ∗ we get [ z ∈ Q B ( z, d ∗ M ) ⊂ Ω. By thedefinition of d we obtain d ≥ d ∗ M . Which implies Ω ∈ C M,R .4 ◦ . Following from 1 ◦ , ◦ , ◦ , we have proved the Theorem 2.1. (cid:3) Existence of shape optimization
In this section, we shall prove the existence of problems ( P ). Theorem 3.1.
The shape optimization problem ( P ) has at least one solution. Proof.
Throughout the proof of Theorem 3.1, we shall use spt ( u ) to denote thesupport of u .Let d = inf Ω ∈ C M,R J (Ω) = inf Ω ∈ C M,R n Z B ∗ | u Ω − g | dx o . It is obvious that d > −∞ .Then there exists a sequence { Ω m } ∞ m =1 ⊂ C M,R such that d = lim m →∞ Z B ∗ | u m − g | dx, (9)where u m ≡ u Ω m is the weak solution of (3). By Theorem 2.1, there exist a subse-quence of { Ω n } ∞ n =1 , still denoted by itself, and Ω ∗ ∈ C M,R such that Ω ∗ = HlimΩ n .By taking u = u m , Ω = Ω m in (4), we get Z Ω m h A. ∇ u m , ∇ φ i dx = h f | Ω m , φ i H − (Ω m ) × H (Ω m ) ∀ φ ∈ C ∞ (Ω m )Since u m ∈ H (Ω m ), we have α Z Ω m |∇ u m | dx ≤ Z Ω m h A. ∇ u m , ∇ u m i dx = h f | Ω m , u m i H − (Ω m ) × H (Ω m ) ≤ k f k H − ( B ∗ ) · k u m k H (Ω m ) (10)by (1), which implies that Z Ω m |∇ u m | dx ≤ C, here and throughout the proof of Theorem 3.1, C denotes several positive constantsindependent of m . Let ˆ u m ( x ) = ( u m ( x ) in Ω m , B ∗ − Ω m , (11)then { ˆ u m } ∞ m =1 is bounded in H ( B ∗ ). Hence there exists subsequence of { ˆ u m } , stilldenoted by itself, such thatˆ u m → ˆ u weakly in H ( B ∗ ) and strongly in L ( B ∗ ) (12)for some ˆ u ∈ H ( B ∗ ). 8ow we claim that ˆ u ( x ) ∈ H (Ω ∗ ) . (13)In fact, we only need to show thatˆ u ( x ) = 0 a.e. in B ∗ \ Ω ∗ . (14)Indeed, for any open subset K satisfying K ⊂ B ∗ − Ω ∗ , denote l d = dist( K, Ω ∗ ), sinceΩ ∗ = HlimΩ n , i.e., δ ( B ∗ \ Ω m , B ∗ \ Ω ∗ ) →
0, which implies that for every 0. < l d thereexists an integer m > n ≥ n we have δ ( B ∗ \ Ω m , B ∗ \ Ω ∗ ) < l d δ ( · , · ) we havesup x ∈ B ∗ \ Ω d ( x, B ∗ \ Ω m ) < l d , ∀ m ≥ m . i.e., B ∗ \ Ω ∗ ⊂ { y ∈ R k ; d ( y, B ∗ \ Ω m < l d ) } . So we obtain { y ∈ Ω m ; d ( y, ∂ Ω m ) > l d } ⊂ Ω ∗ , ∀ m ≥ m . Furthermore, we have { y ∈ R k ; d ( y, ∂ Ω ∗ ) < l d } , ∀ m ≥ m . Hence K ⊂ B ∗ \ Ω m , ∀ m ≥ m . Thus Z K | ˆ u ( x ) | dx = lim m →∞ Z K | ˆ u m ( x ) | dx ≤ lim m →∞ Z B ∗ \ Ω m | ˆ u m ( x ) | dx = 0 , which implies that ˆ u ( x ) = 0 a.e. in K . Since K ⊂ K ⊂ B ∗ \ Ω ∗ is arbitrary, (14) andconsequently (13) follow.Then we claim that Z Ω ∗ h A. ∇ ˆ u, ∇ φ i dx = h f | Ω ∗ , φ i H − (Ω ∗ ) × H (Ω ∗ ) ∀ φ ∈ C ∞ (Ω ∗ ) (15)i.e., Z spt ( φ ) h A. ∇ ˆ u, ∇ φ i dx = h f | spt ( φ ) , φ i H − (Ω ∗ ) × H (Ω ∗ ) for each φ ∈ C ∞ (Ω ∗ ). 9et ˆ φ = ( φ in Ω ∗ , B ∗ − Ω ∗ , (16)by Lemma 2.3, there exists a positive integer m ( φ ), such that spt ( ˆ φ ) = spt ( φ ) ⊂ Ω m , for all m ≥ m ( φ ) . Then for each m ≥ m ( φ ) , ˆ φ ∈ C ∞ (Ω m ). So by (4), we have Z Ω m h A. ∇ ˆ u m , ∇ ˆ φ i dx = h f | Ω m , ˆ φ i H − (Ω m ) × H (Ω m ) which, together with (16), implies Z Ω m h A. ∇ ˆ u m , ∇ φ i dx = h f | Ω m , φ i H − (Ω m ) × H (Ω m ) passing to the limit for m → ∞ and using (12) we get (15).Finally, we claim that d ≥ Z B ∗ | ˆ u − g | dx. (17)We notice that ˆ u m → ˆ u strongly in L ( B ∗ ) . Hence we have 12 Z B ∗ | ˆ u m − g | → Z B ∗ | ˆ u − g | . and by (9) we get d = 12 Z B ∗ | ˆ u − g | . i.e. (17) holds.Following from (13), (15) and (17) we obtain that Ω ∗ is a solution of problem ( P ).This completes the proof. (cid:3) eferences [1] D. Bucur and J. P. Zol´ e sio, Wiener’s criterion and shape continuity for theDirichlet problem, Boll. Un. Mat. Ital., B 11(4)(1997), 757-771.[2] D. Bucur and J. P. Zol´ e sio, N-dimensiona shape optimization under capaci-tary constraints, J. Differential Equations, 123(2) (1995), 504-522.[3] D. Bucur and G. Buttazzo, Variational Methods in Shape OptimizationProblems, Birkh¨ a user, Boston, 2005.[4] G. Buttazzo and P. Guasoni, Shape optimization problems over classes ofconvex domains, J. Convex Anal., 4(1997), 343-351.[5] D. Chenais, On the existence of a solution in a domain identification problem,J. Math. Anal. Appl., 52 (1975), 189-219.[6] D. Chenais, Hom´ e omorphisme entre ouverts lipschitziens, Ann. Mat. PuraAppl., 118(4) (1978), 343-398.[7] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,Birkh¨ a user, Boston, 2006.[8] G. Dal Maso and U. Mosco, Wiener’s criterion and Γ-convergence, Appl.Math. Optim., 15 (1987), 15-63.[9] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag,Berlin, 1984.[10] V. ˘ S ver´ aa