Spectral boundary value problems for Laplace--Beltrami operator: moduli of continuity of eigenvalues under domain deformation
aa r X i v : . [ m a t h . A P ] M a y Spectral boundary value problems forLaplace–Beltrami operator: moduli of continuity ofeigenvalues under domain deformation
A.M. Stepin, I.V. TsylinOctober 27, 2018
Abstract
The paper is pertaining to the spectral theory of operators and boundary valueproblems for differential equations on manifolds. Eigenvalues of such problems arestudied as functionals on the space of domains. Resolvent continuity of the corre-sponding operators is established under domain deformation and estimates of conti-nuity moduli of their eigenvalues / eigenfunctions are obtained provided the boundaryof nonperturbed domain is locally represented as a graph of some continuous functionand domain deformation is measured with respect to the Hausdorff–Pompeiu metric. Let M be a smooth connected compact orientable Riemannian manifold (possiblywith boundary), A — elliptic differential operator of second order on M . For an opensubset Ω ( M , ¯Ω ∩ ∂M = ∅ , the following eigenvalue problem A u = λu, u ∈ ◦ H (Ω) , (1)is considered; here solutions are understood in the weak (variational) sense.Our aim is to estimate moduli of continuity for eigenvalues { λ k } of (1) withmultiplicities taken into account and considered as functions of rough domain Ω ;among surveys on this and related topics we notice [5, 12, 15].The problem can be considered in the context of spectral stability of inverse op-erator A − under small perturbations of Ω . In the framework of this approach it issufficient to obtain estimate of convergence rate for corresponding inverse operatorswith respect to uniform topology (resolvent convergence [16]).A convenient instrument for dealing with the resolvent convergence is (equivalentto it) convergence in the sense of Mosco (see [17, 20]) of the spaces ◦ H (Ω) . In thecase when M ⊂ R d is compact, Frehse [11] obtained capacitive conditions equivalentto Mosco’s convergence. n nineties conditions imposed on variations of plane domains (w.r.t. Hausdorffmetric) were investigated so that to ensure the uniform resolvent convergence (see[5]). However the obtained results claimed only the fact of convergence withoutquantitative estimates.Beginning with 2002 some progress was achieved in proving such estimates incase of plane domains, mainly, due to Burenkov and Davies’ approach [6], that wasfurther developed in a series of papers by Burenkov, Lamberti, Lanza de Cristoforis(see a survey in [7]). Method of these authors made it possible to estimate theupper semicontinuity in Ω for eigenvalues of the problem (1) if variation of domainsis restricted to some technical class.In the present paper resolvent continuity of the boundary value problem ( A u = f , u ∈ ◦ H (Ω) ) with respect to domain perturbation (see th. 4.2) is established andusing this fact estimates for moduli of continuity are obtained for eigenvalues andeigenfunctions of the problem (1) provided that 1) the boundary of nonperturbeddomain can be locally represented as a graph of some continuous function and 2) theperturbation of domains is measured by Hausdorff–Pompeiu metric d HP (see sections2.2, 3.1).To formulate our result about eigenvalues we write ∂ Ω ∈ C ,ω if there exists anatlas for the manifold M such that the intersection of ∂ Ω with each of its charts eitheris empty or can be represented in local coordinates as the graph of a function withmodulus of continuity not exceeding C · ω , where ω is a nondecreasing semi-additivefunction such that ω (0) = 0 and C is a positive constant. Theorem 1.1.
Let A be a strongly elliptic operator with Lipschitz coefficients onthe Riemannian manifold ( M, g ) , Ω be a domain in M , ¯Ω ∩ ∂M = ∅ , ∂ Ω ∈ C ,ω , where ω is some modulus of continuity. Then there exist positive constants C n = C n (Ω , A , M ) , δ = δ (Ω , ω, M ) such that for any domain Ω ( M , satisfyingcondition ǫ = d HP (Ω , Ω ) ≤ δ the following estimate holds: | λ n (Ω ) − λ n (Ω ) | ≤ C n · ( ω ( ǫ ) + ǫ ) . Besides, a generalization of Burenkov–Lamberti theorem [8] to the case of domainson manifold (cor. 4.3) is obtained. These results were announced in [19].
Everywhere below we assume that the Riemannian manifold ( M, g ) is C , –smoothconnected orientable compact (possibly with boundary); coordinate homeomorphismsmap in R d ( d = dim M ) endowed with the standard Euclidean norm | · | : | x | = X i ( x i ) , x = ( x , . . . , x d ) ∈ R d , e d = (0 , . . . , , . in the sense of generalized angle (see, for instance, [13], IV.2) .1 Conditions imposed on operator A . Let A ′ be a differential operation on ( M, g ) locally representable in the form − √ det g ∂ i (cid:16) a ij p det g ∂ j u (cid:17) . Assume that coefficients a ij define a continuous symmetric section A of T M . Denote G the section of T M associated with the Riemannian structure g ; let bilinear forms A x and G x in T ∗ x M × T ∗ x M be values of the sections A and G . We assume that thefollowing conditions are fulfilled. A1 There is a positive constant α such that ∀ x ∈ M ∀ ξ ∈ T ∗ x M ⇒ α G x ( ξ, ξ ) ≤ A x ( ξ, ξ ); A2 A sections A belongs to the space C , ( M ) ; a norm in C , ( M ) is defined byfixing some finite subatlas { ( U, κ U ) } U ∈U k A k U C , ( M ) def = max x ∈ M max ξ ∈ T ∗ x M,ξ =0 A x ( ξ, ξ ) G x ( ξ, ξ ) + X U ∈U max ij h a ijU i C , ( κ U ( U )) , [ v ] C , ( κ U ( U )) = sup x,y ∈ κ U ( U ) ,x = y | v ( x ) − v ( y ) || x − y | , v : κ U ( U ) → R , where a ijU are the coordinates of sections A with respect to mapping κ U .All the norms k · k U C , ( M ) are equivalent as regards their dependence on the choice offinite subatlas.As the scalar products in ◦ H (Ω) and L (Ω) , Ω ( M , we choose: ( u, v ) ◦ H (Ω) = Z Ω G ( ∇ u, ∇ v ) dµ, ( u, v ) L (Ω) = Z Ω uvdµ, where measure µ is associated with the Riemann metric g . Let Φ be continuous in ◦ H ( M ) bilinear form defined by the differential operation A ′ , Φ( u, v ) = Z M A ( ∇ u, ∇ v ) dµ. Since the form Φ is positive and bounded in ◦ H (Ω) , operator A associated with Φ isuniquely defined (see, for example, [13]); a function u ∈ ◦ H (Ω) is the weak solutionof the boundary value problem A u = f, f ∈ L (Ω) (2)if and only if ∀ v ∈ ◦ H (Ω) Φ( u, v ) = Z M f vdµ. (3) .2 Domains with boundaries of the class C ,ω .The ω –cusp condition. For a mapping ω : R + → R + such that ω ( r ) − ω (0) is nonnegative and semi-additivewe set ψ ( r ) = p r + ω ( r ) , φ ( r ) = r + ω ( r ) . Let B ρ ( x ) ⊂ R d be the ball with center x and radius ρ in the norm | · | ; we will often use the notion B r ( X ) def = ∪ x ∈ X B r ( x ) fora set X ⊂ R d . Definition 2.1.
We say that an open set Ω ⊂ R d satisfies the uniform ω –cusp con-dition with parameter r at a point x if there exists ξ x ∈ R d , | ξ x | = 1 , such that W1 (cid:2)(cid:0) B ψ ( r ) ( x ) ∩ Ω (cid:1) − C ω,r ( ξ x ) (cid:3) ∩ B ψ ( r ) ( x ) ⊂ Ω ;where C ω,r ( ξ x ) is obtained from C ω,r ( e d ) def = S ω,r ( e d ) ∪ F ω,r ( e d ) by a rotation su-perposing e d on ξ x where F ω,r ( e d ) = (cid:8) z = (˜ z, z d ) ∈ R d : | z | < ψ ( r ) , z d ≥ ω ( r ) (cid:9) , S ω,r ( e d ) = (cid:8) (˜ z, z d ) ∈ R d : ω ( | ˜ z | ) < z d < ω ( r ) , | ˜ z | < r (cid:9) , ˜ z ∈ R d − . Note that the condition (W1) is equivalent to the following condition W2 (cid:2)(cid:0) B ψ ( r ) ( x ) \ Ω (cid:1) + C ω,r ( ξ x ) (cid:3) ∩ (cid:0) B ψ ( r ) ( x ) ∩ Ω (cid:1) = ∅ .In fact, by virtue of symmetric of (W1) and (W2) with respect to changing Ω byits complement it is sufficient to check the implication (W2) ⇒ (W1) . Otherwise ∃ y ∈ (cid:0) B ψ ( r ) ( x ) ∩ Ω (cid:1) : B ψ ( r ) ∩ ( y − C ω,r ( ξ x )) Ω and hence there exists a point z ∈ ∂ Ω ∩ ( y − C ω,r ( ξ x )) . Equivalently this means that y ∈ z + C ω,r ( ξ x ) ; by (W2) this inclusion leads to a contradiction: y ∈ M \ Ω . Remark 2.2.
In the definition above we do not assume that ω (0) = 0 . All thepropositions below are valid without this assumption if it is not stated explicitly. For a matrix A ∈ C , ( ¯Ω) , Ω ⊂ R d define the norm: k A k C , (¯Ω) def = k| A | k L ∞ (Ω) + (cid:13)(cid:13)(cid:13)(cid:13) max j | ∂ j A | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Ω) , | A | = p ˜ r ( A t A ) , where ˜ r is the spectral radius. Additionally we denote B ρ ( y ) def = χ − y ( B ρ ( χ y ( y )) .We say that an atlas W = { ( W y , χ y ) } y ∈ M is ( ρ, ϑ ) –technical, ρ, ϑ > , if W3 ∀ y ∈ M ⇒ B ρ ( y ) ⊂ W y , W4 For every chart ( W, χ ) ∈ W , C , -norm G ◦ χ − does not exceed ϑ , and L ∞ -norm of | G ◦ χ − | can be estimated from below by ϑ − . W5 There exists a finite subatlas U of the atlas for M such that for all the transitionfunctions from U ∈ U into W ∈ W and their inverses C , -norm of the Jacobimatrix (cid:16) ∂x i ′ ∂x i (cid:17) does not exceed ϑ . Definition 2.3.
An open subset Ω ⊂ M satisfies the uniform ω –cusp condition withparameters ( r, ϑ ) if there exists ( ψ ( r ) , ϑ ) –technical atlas W = { ( W y , χ y ) } y ∈ M suchthat for arbitrary y ∈ M the open set χ y (Ω ∩ B ψ ( r ) ( y )) satisfies the ω –cusp conditionwith parameter r at the point χ y ( y ) . This class will be denoted W ωr,ϑ . efinition 2.4. Boundary ∂ Ω of a domain Ω ⊂ M is of class C ,ω if there existssuch a subatlas U = { ( U, κ U ) } for M that nonempty κ U ( ∂ Ω ∩ U ) can be representedby a graph of continuous function g U with modulus of continuity not exceeding C U ω , C U ∈ R + , ω (0) = 0 , where the intersection of κ U (Ω ∩ U ) and off-graph is empty. Domains Ω ⊂ M with boundaries locally representable as graphs of continuousfunctions we call domains of C –class. In the case of compact M by virtue of Cantortheorem one has: for any domain Ω ⊂ M there exists such a positive semi-additivefunction ω Ω , ω Ω (0) = 0 , that ∂ Ω ∈ C ,ω Ω . Proposition 2.5.
For arbitrary domain Ω ⊂ M with boundary of the class C ,ω Ω , ω Ω (0) = 0 , there exists a class W Cω Ω r,ϑ ∋ Ω , where C ≡ const > .Proof. Since M is compact we may assume that for a fixed domain Ω ⊂ M withboundary of the class C ,ω Ω , an atlas U (in the def 2.4) is finite, sets κ U ( U ) ⊂ R d are bounded, and the mappings κ U are defined on ˜ U ⊃ ¯ U . We select a new atlas { ( U ′ , κ U ) } with the property U ′ ⋐ U and set r = ψ − (cid:2) min U dist( ∂κ U U, κ U U ′ ) (cid:3) ,where dist( A, B ) = inf a ∈ A,b ∈ B | a − b | . For every y ∈ M we choose U ′ ∈ U , y ∈ U ′ , and set ( W y , χ y ) = ( U ′ , κ U ) . Thenthere exists such a number ϑ > , that W = { ( W y , χ y ) } y ∈ M is a ( ψ ( r ) , ϑ ) –technicalatlas. In fact, condition (W3) is satisfied according to construction while (W4) and (W5) are fulfilled in view of all the mappings χ y are obtained as restrictions of finitenumber of coordinate diffeomorphisms.Condition (W2) is also valid for ω = Cω Ω , C = max U C U , since images of Ω under coordinate diffeomorphism χ y are located no one side with respect to graph of g y . Similar claim in the case of Lipschitz boundary can be found in [9]. It shouldbe noted that ω ≡ if ω ( h ) = o ( h ) . Nevertheless in the case of manifolds the class C , · ) can be nonempty. Let
X, Y be an arbitrary subsets of a connected metric compact space ( M, d ) . Set d ( x, Y ) = inf y ∈ Y d ( x, y ) and consider the function e ( X, Y ) = sup x ∈ X d ( x, Y ) . If X \ Y = ∅ then e ( X, Y ) = sup x ∈ X \ Y d ( x, ∂Y ) = e ( X \ Y, ∂Y ) . Next introduce ˇ e ( X, Y ) = e ( M \ Y, M \ X ) . In view of ( M \ Y ) \ ( M \ X ) = X \ Y and ∂X = ∂ ( M \ X ) one has ˇ e ( X, Y ) = sup y ∈ M \ Y d ( y, M \ X ) = sup x ∈ X \ Y d ( x, ∂X ) = e ( X \ Y, ∂X ) . (4) n terms of blowing X ε = { x ∈ M | d ( x, X ) < ε } , ε > , and contraction X − ε = { x ∈ X | ∀ z ∈ M : d ( x, z ) < ε ⇒ z ∈ X } the basic functions e and ˇ e can be describedas follows e ( X, Y ) = inf { ε > | X ⊂ Y ε } , ˇ e ( X, Y ) = inf (cid:8) ε > (cid:12)(cid:12) X − ε ⊂ Y (cid:9) . It follows that Hausdorff distance functions can be introduced by the formulas: d H ( X, Y ) = max { e ( X, Y ) , e ( Y, X ) } ; (5) d H ( X, Y ) = max { ˇ e ( X, Y ) , ˇ e ( Y, X ) } . (6)These functions are metrics on the families of closed and open set respectively. Itis useful to consider stronger version of Hausdorff metric, namely upper Hausdorff–Pompeiu distance d HP ( X, Y ) = max (cid:8) d H ( X, Y ) , d H ( X, Y ) (cid:9) . (7) Remark 3.1.
Notice fundamental difference between (7) and (6). The family ofall open subsets in a fixed metric compact space K is compact w.r.t. (6) accordingto Blaschke theorem (see [14]) but w.r.t. (7) this family loses compactness propertythough completeness remains. To see the latter it is sufficient to consider subgraphsof the functions (2 + sin nx ) considered on the closed interval [0 , π ] . For our purposes (see th. 4.2) the following minimum of all the distances of theHausdorff type will be necessary: d HS ( X, Y ) = min (cid:8) e ( X ∆ Y, ∂Y ) , e ( X ∆ Y, ∂X ) , d H ( X, Y ) , d H ( X, Y ) (cid:9) Quantities e ( X, Y ) and ˇ e ( X, Y ) give us four nonequivalent ways to measure dis-tances, thus d HS is the weakest quantity defining convergence of sets among thosethat can be constructed by means of e ( X, Y ) , ˇ e ( X, Y ) , e ( Y, X ) , ˇ e ( Y, X ) . For a Hilbert space V and its closed subspaces V and V . Consider the problems: u i ∈ V i , Φ( u i , v i ) = h f, v i i ∀ v i ∈ V i , where f ∈ V ′ , and Φ is a bilinear continuous function on V possessing positive α, β ∈ R such that α k u k V ≤ Φ( u, u ) ≤ β k u k V ∀ u ∈ V. By Lax–Milgram lemma solutions u i = G ( f ; V i ) exist and are unique. All the state-ments of this subsections can be found in [18]. To formulate the following lemma andits corollary we need the standard notation d V ( v, A ) for the distance between v ∈ V and a subset A ⊂ V . emma 3.2. For solutions u and u the following inequality holds: k u − u k V ≤ r βα ( d V ( u , V ∩ V ) + d V ( u , V ∩ V )) . (8) Moreover, if V , is a closed subspace, containing V ∩ V then for u , = G ( f ; V , ) the inequality k u − u k V ≤ r βα (cid:0) d V ( u , , V ) + d V ( u , , V ) (cid:1) (9) takes place. Corollary 3.3.
The estimate (9) takes the form k u − u k V ≤ βα (cid:0) d V ( u , , V ) + d V ( u , , V ) (cid:1) where u , = G ( f ; V , ) and V , contains V ∪ V . Let O ( ( M \ ∂M ) be open non-void and M O def = M \ O . The assumption Ω ⊂ M O will be assumed for every domain considered below. Let p = p M O denote Friedrichsconstant (i.e. inf n p (cid:12)(cid:12)(cid:12) k u k L ( M O ) ≤ p k u k ◦ H M O o ) and k u k V Ω def = Z Ω A ( ∇ u, ∇ u ) dµ, k u k L Ω def = p Z Ω | u | dµ, (10)be the norms in the spaces V Ω = ◦ H (Ω) and L Ω = L (Ω) respectively with thestandard norms in these spaces defined by the formulae: k u k ◦ H (Ω) def = Z Ω G ( ∇ u, ∇ u ) dµ ; k u k L (Ω) def = Z Ω | u | dµ. (11)In addition, we denote V = V M O , L = L M O .Again by Lax-Milgram lemma it follows that problem (2) possesses unique colution u f def = G ( f ; V Ω ) ∈ V Ω for arbitrary f ∈ V ′ ; thus kG ( f ; V Ω ) k V ≤ α − k f k V ′ ∀ f ∈ V ′ . (12)Let pairs ( u (1) n , λ (1) n ) , ( u (2) n , λ (2) n ) be solutions to the eigenvalue problem (1) fordomains Ω and Ω respectively; enumeration { λ ( i ) n } in ascending order is meant.Denote by P Ω : V → V Ω the orthogonal projection V onto V Ω and set S (1) n =span( u (1)1 , . . . , u (1) n ) . Lemma 3.4 (cf. [3]) . Fix n ∈ N and assume there are positive numbers A n and B n < p such that for every u ∈ S (1) n the inequalities k P Ω u − u k V ≤ A n k u k L , k P Ω u − u k L ≤ B n k u k L , (13) hold; then λ (1) n ≥ λ (2) n − A n ( √ p − √ B n ) p . .4 Local estimate necessary for resolvent convergence Arguments in this section are based on a generalization of the technique proposedin [18]. We start with the following
Lemma 3.5 (cf. [10] p. 24, [1] p. 49) . Let M be a compact manifold, d a met-ric associated with a Riemannian structure on M , F a family of opened balls with inf B ∈F diam B > and A the set of its centers. Then there exists a finite subfamily { B r j ( a j ) } Jj =1 ⊂ F such that A ⊂ ∪ Jj =1 B r j ( a j ) , with { B r j / ( a j ) } Jj =1 being disjoint. Proposition 3.6.
Let
X, Y ⊂ M O be open, atlas W be ( ρ, ϑ ) –technical and v ∈ V Y .Assume that for every y ∈ Λ = Y \ X there exists a vector ν ( y ) , such that x ∈ χ y ( B ρ ( y ) \ X ) ⇒ ( x + ν ( y )) / ∈ χ y ( Y ∩ B ρ ( y )) , and a function H ∈ L ( M ) satisfies the inequality k v yν ( y ) − v k V B ρ ( y ) ≤ k H k L ( B ρ ( y )) , for every y ∈ Λ ρ = ∪ y ∈ Λ B ρ ( y ) , where v yh = v ◦ χ − y ◦ ( x + h ) ◦ χ y . Then there existsa function w ∈ V X (independent on the choice of H ) such that k w − v k V ≤ C k H k L (Λ ρ ) , where C = C ( M O , ϑ, ρ, A ) ≡ const .Proof. Let O R ( y ) = { z ∈ M | δ ( z, y ) < R } be the geodesic ball with respect to themetric associated with the Riemannian structure g . Applying lemma 3.5 to the family F = {O ρ ϑ ( y ) } y ∈ Λ and the set A = Λ , we get the finite set { x j } Jj =1 such that i = j ⇒ O ρ ϑ ( x j ) ∩ O ρ ϑ ( x i ) = ∅ , Λ ⊂ ∪ j ∈ J O ρ ϑ ( x j ) ⊂ ∪ j ∈ J B ρ ( x j ) , B ρ ϑ ( x i ) ∩ B ρ ϑ ( x j ) = ∅ . Hence J ≤ µ ( M )min j µ ( O ρ ϑ ( x j )) ≤ µ ( M )inf z ∈ M µ ( O ρ ϑ ( z )) , (14)and by virtue of M compact and measure µ absolutely continuous there exists a point z ∈ M where (positive) inf in (14) is reached; thus J < C ( M, ϑ, ρ ) .The required function w will be built by means of a certain partition of unity. Toconstruct it we introduce the functions ϕ j ( x ) = min { , (3 − | χ x j ( x ) − χ x j ( x j ) | /ρ ) + } , ϕ ( x ) = min { , δ ( x, Λ) /ρ } . and g ( x, ξ ) = G x ( ξ, ξ ) . Then one has ≤ ϕ j ( x ) ≤ and supp( ϕ j ) ⊂ B ρ ( x j ) , ϕ j | B ρ/ ( x j ) ≡ , g ( x, ∇ φ j ) / ≤ C ( M ) ρ − B ρ ( x j ) ( x ) , j ∈ Z + ∩ [0 , J ] , here A ( x ) = 1 is the indicator of A ⊂ M ; thus ≤ P Jj =0 ϕ j ≤ J . The function κ j = ϕ j P Jk =0 ϕ k possess the following properties: J X j =0 κ j ≡ , ≤ κ j ≤ , g ( x, ∇ κ j ) / ≤ C ( M, ϑ, ρ ) /ρ = C ( M, ϑ, ρ ) C ( M ) ρ . If w = P Jj =0 κ j ˜ v j , ˜ v j = B ρ ( x j ) v x j ν ( x j ) , then k v − w k L ≤ p Z M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J X j =0 κ j ( v − ˜ v j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ≤ J X j =0 p Z B ρ ( x j ) | v − v x j ν ( x j ) | dµ = J X j =0 k v − v x j ν ( x j ) k L B ρ ( xj ) ≤ J X j =0 k v − v x j ν ( x j ) k V B ρ ( xj ) ≤ C ( M, ϑ, ρ ) Z Λ ρ H dµ,
Using the symbol a ( x, ξ ) = A x ( ξ, ξ ) of operator A one finally has k v − w k V = Z M a x, ∇ J X j =0 κ j ( v − ˜ v j ) dµ ≤ C ( M, ϑ, ρ ) J X j =0 Z M a ( x, ∇ κ j ) | v − ˜ v j | dµ + 2 J X j =0 Z M κ j a ( x, ∇ v − ∇ ˜ v j ) dµ ≤≤ C ( M, ϑ, ρ ) J X j =0 Z B ρ ( x j ) (cid:18) C ( A )p ρ H + H (cid:19) dµ ≤ C k H k L (Λ ρ ) . W ωr,ϑ . Lemma 3.7 (see [4]) . If v ∈ H ( R d ) , x ∈ R d , the for any h ∈ R d , | h | < ρ, Z B ρ ( x ) | v ( x + h ) − v ( x ) | dx ≤ | h | Z B ρ ( x ) |∇ v ( x ) | dx The following claim allows one to find a function H satisfying condition in Propo-sition 3.6. Theorem 3.8.
Let Z ⊂ M , Z ∈ W ωr,ϑ , f ∈ L ( M ) , u = G ( f ; V Z ) , y ∈ M , h = | h | ξ y , | h | < ψ ( r ) . Then for every positive ρ < ψ ( r ) there exists such a number ˜ C = ˜ C ( M O , r, ϑ, A ) that k u − u yh k V B ρ ( y ) ≤ ˜ C k u k V B ρ ( y ) h k u k V B ρ ( y ) + k f k L Z ∩B ρ ( y ) i · | h | . roof. We need the function κ y = min ( , (cid:18) − | x − y | ρ (cid:19) + ) ◦ χ y , it possesses the following properties: ≤ κ y ≤ , g ( x, ∇ κ y ) ≤ C ( M ) ρ − , κ y | B ρ ( y ) ≡ . For v ∈ H ( B ρ ( y )) we introduce notation T y,h v = (1 − κ y ) v + κ y v yh , and remark that T y,h v − v = κ y (cid:0) v yh − v (cid:1) . Since Z ∈ W ωr,ϑ , one has v ∈ V Z ⇒ T y,h v ∈ V Z ⊂ V, supp( v ) , supp( T y,h v ) ⊂ ¯ Z, (15)and hence k u yh − u k V B ψ ( r )( y ) = kT y,h u − u k V B ψ ( r )( y ) ≤ Φ( T y,h u, T y,h u ) − Φ( u, u ) + 2 h f, u − T y,h u i . (16)The last summand can be estimated by lemma 3.7: h f, u − T y,h u i ≤ ϑ | h | k u k H ( B ρ ) k f k L ( Z ∩B ρ ( y )) ≤ ϑ p − α − | h |k u k V B ρ k f k L Z ∩B ρ ( y ) . To estimate the remaining term in (16) we use the formula ∇ ( T y,h v ) = κ y ∇ ( v yh ) + (1 − κ y ) ∇ v + ∇ κ y ( v yh − v ) = T y,h ∇ v + ∇ κ y ( v yh − v ) , and property (15) implies that Φ( T y,h u, T y,h u ) − Φ( u, u ) does not exceed Z Z a ( y, T y,h ∇ u + ∇ κ y ( u yh − u )) dµ − Z Z a ( y, T y,h ∇ u ) dµ + (17) Z Z a ( y, T y,h ∇ u ) dµ − Z Z a ( y, ∇ u ) dµ. (18)Since for ξ, η ∈ R d the following inequality a ( x, ξ + η ) − a ( x, ξ ) ≤ ( a ( x, η ) a ( x, ξ + η )) / ≤ a ( x, η ) / (2 a ( x, ξ ) / + a ( x, η ) / ) , holds, we can apply it for ξ = T y,h ∇ u , η = ∇ κ y ( u yh − u )) , hence integrals in line (17)can be estimated as follows Z Z a ( y, T y,h ∇ u + ∇ κ y ( u yh − u )) dµ − Z Z a ( y, T y,h ∇ u ) dµ ≤ Z Z a ( y, T y,h ∇ u + ∇ κ y ( u yh − u )) dµ − Z Z ∩B ρ ( y ) a ( y, T y,h ∇ u ) dµ ≤ Z Z a (cid:0) y, ∇ κ y ( u yh − u ) (cid:1) / h(cid:0) a ( y, ∇ κ y ( u yh − u )) (cid:1) / + 2 ( a ( y, T y,h ∇ u )) / i dµ ≤ (cid:18)Z Z a ( y, η ) dµ (cid:19) / · "(cid:18)Z Z a ( y, η ) dµ (cid:19) / + 2 (cid:18)Z Z a ( y, ξ ) dµ (cid:19) / ≤ ˜ C ( M, ϑ, A ) k u − u yh k L B ρ ( y ) (cid:16) ˜ C ( M, ϑ, A ) k u − u yh k L B ρ ( y ) + 2 k a ( x, T y,h ∇ u ) k L B ρ ( y ) (cid:17) . t follows from definition of T y,h that k a ( x, T y,h ∇ u ) k L ( B ρ ( y )) ≤ C ( M, ρ ) k u k V B ρ ( y ) . Therefore applying lemma 3.7 one has Z Z a ( y, T y,h ∇ u + ∇ κ y ( u yh − u )) dµ − Z Z a ( y, T y,h ∇ u ) dµ ≤ C ( M, ϑ, A ) (cid:16) ˜ C ( M, ϑ, A ) + ˜ C ( M, ρ ) (cid:17) k u k V B ρ ( y ) | h | = ˜ C ( M, ϑ, ρ, A ) k u k V B ρ ( y ) | h | . The integrals in (18) can be estimated with regards for convexity of a : a ( x, T y,h ∇ v ) − a ( x, ∇ v ) ≤ (1 − κ y ) a ( x, ∇ v ) + κ y a ( x, ∇ ( v yh )) − a ( x, ∇ v ) = κ y (cid:2) a ( x, ∇ v yh ) − a ( x, ∇ v ) (cid:3) , where x ∈ supp κ y . Hence we get Z Z a ( y, T y,h ∇ u ) dµ − Z Z a ( y, ∇ u ) dµ ≤ Z Z ∩B ρ ( y ) κ y (cid:2) a ( x, ∇ ( v yh )) − a ( x, ∇ v ) (cid:3) dµ = Z Z ∩ χ − y ( B ρ ( y )+ h ) ( κ y ) y − h a ( x − h, ∇ v ) d ( µ y − h ) − Z Z ∩B ρ ( y ) κ y a ( x, ∇ v ) dµ ≤ Z Z ∩B ρ ( y ) ( κ y ) y − h ( a ( x − h, ∇ v ) − κ y a ( x − h, ∇ v )) d ( µ y − h )++ Z Z ∩B ρ ( y ) κ y a ( x − h, ∇ v ))( dµ y − h − dµ ) + Z Z ∩B ρ ( y ) κ y ( a ( x − h, ∇ v ) − a ( x, ∇ v )) dµ ≤ ˜ C ( M O , ϑ, ρ, A ) | h | · k u k V B ρ ( y ) . ω -cusp con-dition In the proposition of this subsection Ω ⊂ M ⊂ R d denotes a domain of the class W ωr, ;notation Ω ε see in subsection 3.1. Lemma 3.9.
Let Ω satisfy uniform ω -cusp condition with parameter r at point x .Then for any postitve ε ≤ ρ = ψ ( r ) the set Ω ε satisfies uniform ω -cusp condition withparameter r = ψ − ( ψ ( r ) / at the point x .Proof. Choose h ∈ C ω,r ( ξ x ) and y ∈ B ρ/ ( x ) ; if y ∈ Ω ε then there exists z ∈ Ω suchthat | z − y | < ε ≤ ρ . Since | z − x | < ρ + ρ < ρ then from the local definition 2.1one has z − h ∈ Ω , dist( y − h, Ω) ≤ | y − h − ( z − h ) | = | y − z | < ε, that is y − h ∈ Ω ε . emark 3.10. By simple geometric reason it is easy to see that < ε ≤ φ − (cid:18) ψ ( r )2 (cid:19) ⇒ B ε ( φ ( ε ) ξ ) ⊂ C ω,r ( ξ ) . In fact, for sufficiently small ε the inclusion B ε (( ω ( ε ) + ε ) ξ ) ⊂ S ω,r ( ξ ) ∪ F ω,r ( ξ ) . holds. As regards sufficiency it is ensured by the inequality φ ( ε ) < ψ ( r ) . Lemma 3.11.
Let Z satisfies uniform ω -cusp condition with parameter R at the point y , ρ = ψ ( R ) , ω (0) = 0 , and − φ − (cid:18) ψ ( R )4 (cid:19) ≤ η ≤ ≤ ε ≤ φ − (cid:18) ψ ( R )4 (cid:19) Then x ∈ B ρ ( y ) \ Z η ⇒ x + [ φ ( ε ) + φ ( − η )] ξ y / ∈ Z ε . Proof.
Let us denote z = x + [ φ ( ε ) + φ ( − η )] ξ y , t = x + φ ( − η ) ξ y ; assuming t ∈ Z because of inclusion t ∈ B ρ ( y ) one has by Remark 3.10 that B − η ( x ) = B − η ( t − φ ( − η ) ξ y ) ⊂ t − C ω,r ( ξ y ) ⊂ Z, that is x ∈ Z η . This contradiction means that t Z . Applying Remark 3.10 againone has B ε ( z ) = B ε ( t + φ ( ε ) ξ y ) ⊂ t + C ω,r ( ξ y ) ⊂ R d \ Z, hence dist( z, Z ) ≥ ε . Lemma 3.12.
Let Ω , Ω be open subsets of R d , Ω ∈ W ωr,ϑ , ρ = ψ ( r ) , ω (0) = 0 , e (Ω , Ω ) ≤ φ − (cid:16) ψ ( r )2 (cid:17) . Then ˇ e (Ω , Ω ) ≤ φ ( e (Ω , Ω )) .If in addition Ω ∈ W ωr,ϑ and d H (Ω , Ω ) ≤ φ − (cid:16) ψ ( r )2 (cid:17) one has d HP (Ω , Ω ) ≤ φ ( d HS (Ω , Ω )) . Proof.
It is sufficient to prove the first statement; the second follows from definitionof ˇ e ( X, Y ) and the fact that locally M \ ¯Ω satisfies ω -cusp condition if Ω satisfies thiscondition.If λ = e (Ω , Ω ) then Ω ⊂ Ω λ . For y ∈ Ω \ Ω one has y + φ ( λ ) ξ y ∈ R d \ Ω λ ⊂ R d \ Ω . by lemma 3.11 and inequality λ ≤ φ − (cid:16) ψ ( r )2 (cid:17) . It follows that dist( y, R d \ Ω ) ≤ φ ( λ ) and hence ˇ e (Ω , Ω ) = sup y ∈ Ω \ Ω d ( y, R d \ Ω ) ≤ φ ( λ ) . .6.2 Global analogs of lemmas 3.9 and 3.11. Corollary 3.13. If Ω ∈ W ωr,ϑ and ω (0) = 0 then for each positive ε not exceeding min n φ − (cid:16) ψ ( r ) − ω ( r )2( ϑ +1) (cid:17) , ψ ( r ) ϑ o the following holds: O ε (Ω) ∈ W ω +( ϑ +1) φ ( ε ) r ,ϑ , where r = ψ − (cid:16) ψ ( r )2 (cid:17) .Proof. Fix y ∈ M and set Y = χ y (cid:0) Ω ∩ B ψ ( r ) ( y ) (cid:1) . Since Y satisfies uniform ω -cuspcondition with parameter r at the point χ y ( y ) , then according to lemma 3.9 for eachpositive ε ≤ ψ ( r ) ϑ the set B ε/ϑ also satisfies uniform ω -cusp condition with parameter r = ψ − ( ψ ( r ) / at the point χ y ( y ) . Setting Z = Y ε/ϑ , R = r , ρ = ψ ( R ) = ψ ( r ) / one obtains from lemma 3.11 that x ∈ B ρ ( y ) \ Z ⇒ x + φ (cid:18)(cid:18) ϑ − ϑ (cid:19) ε (cid:19) ξ y / ∈ Z ( ϑ − ϑ ) ε = (cid:16) Y ε/ϑ (cid:17) ( ϑ − ϑ ) ε = Y εϑ + ( ϑ − ϑ ) ε = Y ϑε . now because of lemma 3.9 for each positive ε < ψ ( r ) /ϑ the set Y ϑε satisfies uniform ω -cusp condition with parameter r = ψ − ( ψ ( r ) / at the point χ y ( y ) . Thus x ∈ χ y (cid:2) B ψ ( r ) ( y ) \O ε (Ω) (cid:3) ⊂ B ρ ( y ) \ Z ⇒ x + φ (( ϑ − /ϑ ) ε ) ξ y / ∈ Y ϑε ⇒ x + φ (( ϑ − /ϑ ) ε ) ξ y + C ω,r ( ξ y ) ⊂ R d \ Y ϑε ⇒ x + (2 ϑ + 1) φ ( ε ) ξ y + C ω,r ( ξ y ) ⊂ R d \ Y ϑε ⇒ x + C ω +(2 ϑ +1) φ ( ε ) ,r/ ( ξ y ) ⊂ R d \ Y ϑε ⇒ x + C ω +(2 ϑ +1) φ ( ε ) ,r/ ( ξ y ) ⊂ χ y (cid:0) B ψ ( r ) ( y ) \O ε (Ω) (cid:1) , if ε ≤ φ − (cid:16) ψ ( r ) − ω ( r )2( ϑ +1) (cid:17) .For negative η introduce O η (Ω) def = { y ∈ Ω | O | η | ( y ) ∩ ∂ Ω = ∅ } . Corollary 3.14. If ω (0) = 0 and < − η, ε < φ − (cid:16) ψ ( R )4( ϑ +1) (cid:17) then x ∈ χ y (cid:0) B ψ ( r ) ( y ) \O η (Ω) (cid:1) ⇒ x + ( ϑ + 1)( φ ( ε ) + φ ( − η )) ξ y / ∈ χ y (cid:0) O ε (Ω) ∩ B ψ ( r ) ( y ) (cid:1) . Similarly to checking of the previous corollary by setting Z = χ y (cid:0) Ω ∩ B ψ ( r ) ( y ) (cid:1) , R = r , ρ = ψ ( r ) and applying lemma 3.11 one has x ∈ χ y (cid:0) O η (Ω) ∩ B ψ ( r ) ( y ) (cid:1) ⊂ B ψ ( r ) ( y ) \ Y ϑη ⇒ x + [ φ ( − ϑη ) + φ ( ϑε )] ξ y / ∈ Y ϑε , it remains to note that Y ϑε ⊂ χ y (cid:0) O ε (Ω) ∩ B ψ ( r ) ( y ) (cid:1) . Lemma 4.1. If Ω ∈ W ωr,ϑ , ω (0) = 0 , < − η, ε < φ − (cid:16) ψ ( R )4( ϑ +1) (cid:17) , ε ≤ dist(Ω , ∂M O ) .Then for arbitrary f ∈ L ( M ) and Λ = O ε (Ω) \O η (Ω) there exists w η ∈ V O η (Ω) and ¯ C = ¯ C ( M O , ϑ, r, A ) such that k u ε − w η k V ≤ ¯ C · ( φ ( ε ) + φ ( − η )) k u ε k V O ϑψ ( r )(Λ) (cid:16) k u ε k V O ϑψ ( r )(Λ) + k f k L O ϑψ ( r )(Λ) (cid:17) , here u ε = G ( f ; V O ε (Ω) ) .Proof. Since Ω ∈ W ωr,ϑ it follows from corollary3.13 that O ε (Ω) ∈ W ω +( ϑ +1) φ ( ε ) r ,ϑ , r = ψ − (cid:16) ψ ( r )2 (cid:17) . Applying theorem 3.8 for Z = O ε (Ω) one can find ˜ C = ˜ C ( M O , r, ϑ, A ) ,such that the following estimate holds k v − v yν ( y ) k V B ρ ( y ) ≤ ˜ C k v k V B ρ ( y ) h k v k V B ρ ( y ) + k f k L Z ∩B ρ ( y ) i · ( ϑ + 1)( φ ( ε ) + φ ( − η )) =˜ C · ( ϑ + 1)( φ ( ε ) + φ ( − η )) Z Z A ( ∇ v, ∇ v ) dµ + Z B ρ ( y ) A ( ∇ v, ∇ v ) dµ Z Z ∩B ρ ( y ) f dµ ! / ≤√ C · ( ϑ + 1)( φ ( ε ) + φ ( − η )) " (1 + κ ) Z Z A ( ∇ v, ∇ v ) dµ + 1 κ Z B ρ ( y ) f dµ , where v = u ε , ν ( y ) = ( ϑ + 1)( φ ( ε ) + φ ( − η )) ξ y . Thus, it follows from 3.14 that onecan use proposition 3.6 for Y = O ε (Ω) , X = O η (Ω) , v = u ε , ν ( y ) = ( ϑ + 1)( φ ( ε ) + φ ( − η )) ξ y H = √ C · ( ϑ + 1)( φ ( ε ) + φ ( − η )) (cid:20) (1 + κ ) A ( ∇ v, ∇ v ) + 1 κ f (cid:21) , and get that there exists a function w η ∈ V O η (Ω) such that k u ε − w η k V ≤ √ C · ( ϑ + 1)( φ ( ε ) + φ ( − η )) · (cid:16) (1 + κ ) k u ε k V Λ3 ψ ( r ) + κ − k f k L Λ3 ψ ( r ) (cid:17) . Choosing κ = k f k L Λ3 ψ ( r ) k u ε k V Λ3 ψ ( r ) we obtain the required inequality.Now the following resolvent continuity will be shown with respect to domain per-turbation. Theorem 4.2.
Let Ω ∈ W ωr,ϑ , ω (0) = 0 , f ∈ L ( M ) , u i = G ( f ; V Ω i ) , i = 1 , , thenfor sufficiently small ǫ = e (Ω ∆Ω , ∂ Ω ) there exists a constant Γ = Γ( M O , W ωr,ϑ , A ) such that k u − u k V ≤ Γ · φ ( ǫ ) k f k L ′ k f k V ′ . (19) If in addition Ω ∈ W ωr,ϑ and d HS (Ω , Ω ) is sufficiently small then k u − u k V ≤ Γ · φ ( d HS (Ω , Ω )) k f k L ′ k f k V ′ . Proof.
Set ε = e (Ω , Ω ) , ˇ ε = ˇ e (Ω , Ω ) , η = e (Ω , Ω ) , ˇ η = ˇ e (Ω , Ω ) . To establish (19) we use inequality (9) of lemma 3.2, where V = V Ω , V = V Ω , V , = V O ε (Ω ) , α = β = 1 . s V O − ˇ η (Ω ) ⊂ V ∩ V and d ( u , , V ) ≤ d ( u , , V O − ˇ η (Ω ) ) one has that k u − u k V ≤ (cid:0) d V ( u , , V Ω ) + d V ( u , , V Ω ) (cid:1) ≤ d V (cid:0) u , , V O − ˇ η (Ω ) (cid:1) . Now to get estimate (19) it is sufficient to use lemma 4.1 and inequality (12).If in addition Ω ∈ W ωr,ϑ then interchanging Ω and Ω in the above argumentsone obtains the estimate k u − u k V ≤ Γ · φ (min { e (Ω ∆Ω , ∂ Ω ) , e (Ω ∆Ω , ∂ Ω ) } ) k f k L ′ k f k V ′ . We notice incidentally that 3.12 enables one to obtain sufficient smallness of d HP (Ω , Ω ) .Hence it sufficient to establish the following estimates k u − u k V ≤ Γ · φ (cid:0) d H (Ω , Ω ) (cid:1) k f k L ′ k f k V ′ , (20) k u − u k V ≤ Γ · φ ( d H (Ω , Ω )) k f k L ′ k f k V ′ . (21)Now, the first one follows from lemma 4.1, inequality (12) and estimate (8) in lemma3.2: V O − ˇ η (Ω ) ⊂ V ∩ V , d V ( u , V ∩ V ) ≤ d V ( u , V O − ˇ η (Ω ) ) ≤ Γ · φ (ˇ η ) k f k L ′ k f k V ′ ,V O − ˇ ε (Ω ) ⊂ V ∩ V , d V ( u , V ∩ V ) ≤ d V ( u , V O − ˇ ε (Ω ) ) ≤ Γ · φ (ˇ ε ) k f k L ′ k f k V ′ . As regards (21) it follows again from lemma 4.1, inequality (12) and corollary 3.3where V , = V O ε (Ω ) , V , = V O η (Ω ) . Resolvent continuity in Ω established above enables one to estimate distance be-tween eigenspaces of the spectral boundary value problems for A with close domains Ω and Ω . Namely, let ∂ Ω ∈ C ,ω , ν k be k -th (in decreasing order) eigenvalue ofthe operator G ( · , V Ω ) without taking multiplicity into account, E k (Ω ) be the corre-sponding eigenspace, number r > be such that B r ( ν k ) ∩ spec( G ( · , V Ω )) = { ν k } .Then there exists δ = δ ( r, ν k ) > so that as soon as ǫ = e (Ω ∆Ω , ∂ Ω ) < δ for thegeneralized angle (see, for instance, [13]) ˜ δ V ( A, B ) = max ( sup u ∈ A, k u k V =1 d V ( u, B ) , sup u ∈ B, k u k V =1 d V ( u, A ) ) between subspaces A, B ⊂ V the following estimate takes place ˜ δ V ( E k (Ω ) , E k (Ω )) ≤ Γ · C ( ν k , r ) · φ ( ǫ ) / , where E k (Ω ) is the range of the operator πi Z | ν k − ξ | = r ( G ( · , V Ω ) − ξ ) − dξ. .2 Proof of the Theorem 1.1 By means of Proposition 2.5 for domain Ω with the boundary of class C ,ω one hasa class W Cωr,ϑ (without loss of generality one can assume that C = 1 ). Let u ( i ) n ∈ V Ω , i = 1 , , be eigenfunctions associated with eigenvalues λ ( i ) n of the problem (1)considered in domains Ω i and ¯ u (1) n ∈ ◦ H (Ω ) be weak solution of the equation A u = λ (1) n u (1) n , u ∈ ◦ H (Ω ) . Taking advantage of the Theorem 4.2 one has the estimate k u (1) n − P Ω u (1) n k V ≤ k u (1) n − ¯ u (1) n k V ≤ Γ · φ ( e (Ω ∆Ω , ∂ Ω )) k λ (1) n u (1) n k L ′ k λ (1) n u (1) n k V ′ , besides the following inequality holds: k λ ( i ) n u ( i ) n k H − ( M O ) ≤ C ( n, A , p) q λ ( i ) n k u ( i ) n k L ( M O ) . Since λ (1) n ≤ γ n does not exceed the n -th eigenvalue of the problem (1), with Ω beinga ball contained in Ω ∩ Ω , one has k u (1) j − P Ω u (1) j k V ≤ C j φ ( e (Ω ∆Ω , ∂ Ω )) , j = 1 , . . . , n Now applying lemma 3.4 one derives the estimate from below for λ (1) n . To obtainsimilar estimate for λ (2) n we write k u (2) n − P Ω u (2) n k V ≤ k u (2) n − ¯ u (2) n k V ≤ Γ · φ ( e (Ω ∆Ω , ∂ Ω )) k λ (2) n u (2) n k L ′ k λ (2) n u (2) n k V ′ , where ¯ u (2) n = G ( λ (2) n u (2) n ; V Ω ) . Carrying on the above arguments we obtain the re-quired conclusionIf in addition Ω ∈ W ωr,ϑ then k u (1) n − P Ω u (1) n k V ≤ Γ · φ ( d HS (Ω , Ω )) k λ (1) n u (1) n k L ′ k λ (1) n u (1) n k V ′ Corollary 4.3 (Manifold version of Burenkov–Lamberti theorem) . Let operator A on the manifold ( M, g ) satisfies conditions ( A1 ) – ( A2 ), Ω , Ω ∈ W ωr,ϑ , ω (0) = 0 .Then there exists constants C n = C n ( M, ω, r, ϑ, A ) > , δ = δ ( M O , ω, r, ϑ ) > suchthat conditions Ω ⋐ M O , d HS (Ω , Ω ) ≤ δ imply inequality | λ (1) n − λ (2) n | ≤ C n ( ω ( d HS (Ω , Ω )) + d HS (Ω , Ω )) , where { λ ( i ) n } are eigenvalues of the problem (1) for domains Ω i indexed in ascendingorder with multiplicities taken into account. eferences [1] Ambrosio L., Fusco N., Pallara D.
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