1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25

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1 .. IV ( ) / 25

2 1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25

3 1. Ω ε B ε t u ε u ε = 0 in (0, T) Ω ε (P ε ) Bu = 0 on (0, T) Ω ε u ε (0) = u ε in Ω 0 ε u ε (t, x) lim ε 0 u ε lim σ( B ), ε 0 Ω ε lim ( B z) 1 ε 0 Ω ε ( ) / 25

4 1. Quantum Graph 1930 ( ) / 25

5 1. [ 92 Hale and Raugel] G 1 G Y ( ) / 25

6 ( ) / 25

7 2.1 G = (V, E) V = {v i } i I ( I < + ) E = {e j } j J ( J < + ) l j (0, + ) e j (i.e. e j [0, l j ] = {s R 0 s l j }) ( ) / 25

8 2.1 G = (V, E) V = {v i } i I ( I < + ) E = {e j } j J ( J < + ) l j (0, + ) e j (i.e. e j [0, l j ] = {s R 0 s l j }) L 2 (G) := {ψ : G C ψ j := ψ e j L 2 (e j ) ( j J)} l j ψ j L 2 (e j ) ψ j 2 := ψ L 2 j (s) 2 ds < + (e j ) ψ, ϕ L 2 (G) := l j j J 0 0 ψ j (s) ϕ j (s) ds ( ) / 25

9 2.2 G H = d2 ds 2 with boundary conditions on each vertex 1 (Dirichlet condition) D(H) = {ψ H 2 (G) ψ(v) = 0 (v V)} 2 (Neumann condition) D(H) = {ψ H 2 (G) ψ (v) = 0 (v V)} ( ) / 25

10 (Kirchhoff condition) ψ C(G) N dψ j ds (0) = 0 j=1 dψ j ds (l j) = 0 ( j = 1,..., N) O G ( ) / 25

11 2.3 3 (Kirchhoff condition) ψ C(G) N dψ j ds (0) = 0 j=1 dψ j ds (l j) = 0 ( j = 1,..., N) 4 (δ-type condition) ψ C(G) N dψ j (0) = αψ(0) ds j=1 dψ j ds (l j) = 0 ( j = 1,..., N) α R G O ( ) / 25

12 3.1 Ω Q[u, w] = u w dx Ω ( ) / 25

13 3.1 Ω Q[u, w] = u w dx Ω u C 2 (Ω) u ν = 0 on Ω Q[u, w] = Ω ( u ν ) w ds x u w dx = u w dx Ω Ω Q[u, w] = u, w L 2 ( ) / 25

14 3.1 H = L 2 (Ω) Q[u, w] = u w dx Ω D( N ) := {u Ω H2 (Ω) u ν = 0 u, w D(Q) = H 1 (Ω) on Ω} Q[u, w] = u, w H (u D( N Ω ), w H1 (Ω)) ( ) / 25

15 3.1 H = L 2 (Ω) Q[u, w] = u w dx Ω D( N ) := {u Ω H2 (Ω) u ν = 0 u, w D(Q) = H 1 (Ω) on Ω} Q[u, w] = u, w H (u D( N Ω ), w H1 (Ω)) φ(u) = Q[u, u] φ Ω Neumann Laplacian N Ω φ N Ω ( ) / 25

16 3.2 Kirchhoff Laplacian Q 0 [ψ, ϕ] = N j=1 l j 0 ψ (s) j ϕ (s) ds j ψ, ϕ H 1 (G) = { ψ C(G) ψ j H 1 (e j )} ψ Kirchhoff N { Q 0 [ψ, ϕ] = ψ (s) ϕ s=l j l j } j j(s) ψ (s) ϕ j j(s) ds = j=1 l j N j=1 0 s=0 ψ j (s) ϕ j(s) ds 0 Q 0 d2 ds 2 = ψ, ϕ L 2 (G) with ψ C(G), e j E v dψ j (v) = 0 (v V) ds ( ) / 25

17 . Definition 3.3 Mosco [cf: 69 Mosco]. H 2 Φ ε, Φ : H (, + ] Φ ε Φ Mosco 2 1. u ε u in H weakly = Φ(u) lim inf Φ ε (u ε ). ε u H, u ε H s.t. u ε u strongly, Φ(u) = lim Φ ε (u ε ) ε +0 y H = R, Φ ε (x) = Φ(x) = x x 2 + ε 2 Φ ε ε Φ 0 x ( ) / 25

18 . Theorem 3.3 Mosco [cf: 69 Mosco]. 2 Φ ε A ε 1. Φ ε Φ Mosco 2. e t A ε e t A strongly 3. (z A ε ) 1 (z A) 1 strongly (Im z 0) e t A A (1- ) (z A) 1 A. ( ) / 25

19 4.1 Ω ε Neumann Laplacian dµ ε = 1 ωε n 1 dx φ ε : H ε = L 2 (Ω ε, dµ ε ) [0, + ] φ ε (u) = Q ε [u, u] u 2 dµ ε if u H 1 (Ω ε, dµ ε ) = Ω ε + otherwise φ ε ε +0 ωε n 1 ε n 1 Ω ε ( ) / 25

20 4.2 Gromov-Hausdorff [cf: 03 Kuwae-Shioya] f ε : Ω ε G Gromov-Hausdorff ( Ωε, O, dµ ε = dx/(ωε n 1 ) ) (G, O, ds) (ε +0) lim ψ f ε dµ ε = ψ ds ε +0 Ω ε G = ψ C 0 (G) N j=1 l j 0 ψ j (s) ds O O Ω ε G ( ) / 25

21 . 4.3 Mosco [cf: 03 Kuwae-Shioya] Definition. (X ε, dm ε ) (X, dm) Gromov-Hausdorff H ε = L 2 (X ε, dm ε ) H = L 2 (X, dm) 2 Φ ε : H ε (, + ] Φ : H (, + ] Φ ε Φ Mosco 2 1. H ε u ε u H weakly = Φ(u) lim inf Φ ε (u ε ). ε u H, u ε H ε s.t. u ε u strongly, Φ(u) = lim Φ ε (u ε ) ε +0 u ε u strongly lim u ε u f ε 2 dm ε = 0 ε +0 X ε f ε : X ε X ( ) / 25

22 4.4 Mosco [cf: 03 Kuwae-Shioya]. Theorem. 2 Φ ε A ε 1. Φ ε Φ Mosco 2. e t A ε e t A strongly 3. (z A ε ) 1 (z A) 1 strongly (Im z 0) e t A A (1- ) (z A) 1 A. ( ) / 25

23 4.5 φ ε : L 2 (Ω ε, dµ ε ) [0, + ] φ : L 2 (G) [0, + ] φ ε (u) = u 2 dµ ε if u H 1 (Ω ε, dµ ε ) Ω ε φ(ψ) = N l j j=1 0 ψ j (s) 2 ds if ψ H 1 (G) H 1 (G) = { ψ C(G) ψ j H 1 (e j ) ( j = 1,..., N)} O O Ω ε G ( ) / 25

24 4.6. Theorem. φ ε φ ε +0 Mosco. Ω ε Neumann. Laplacian G Kirchhoff Laplacian φ ε (u) = u 2 dµ ε if u H 1 (Ω ε, dµ ε ) Ω ε N l j φ(ψ) = ψ j (s) 2 ds if ψ H 1 (G) j=1 0 H 1 (G) = { ψ C(G) ψ j H 1 (e j ) ( j = 1,..., N)} O O Ω ε G ( ) / 25

25 4.6. Theorem. φ ε φ ε +0 Mosco. Ω ε Neumann. Laplacian G Kirchhoff Laplacian V C 0 (R n ), V 0, V ε (x) = (1/ε)V(x/ε) C V V mass φ ε (u) = u 2 dµ ε + Ω ε V ε u 2 dµ ε Ω ε φ(ψ) = N l j j=1 0 φ ε φ Mosco ψ j (s) 2 ds + C V ψ(o) 2 ( ) / 25

26 Kirchhoff B.C. Neumann B.C. ( ) / 25

27 {u ε } ε>0 sup {Q ε [u ε ] + u ε 2 } < + ε>0 L 2 (Ω E,dµ ε ) u ε k ψ D j,ε D 1,ε Ω ε O J ε D 3,ε D 2,ε w ε j (y) := uε D j,ε (y 1, εy ) y = (y 1, y ) (0, l j ) B 1 sup ε>0 w ε j H1 < +, lim y wε ε 0 j L 2 = 0 w ε k j ψ j ( ) / 25

28 ψ j G ψ ψ D(Q 0 ) = {ψ C(G) ψ j H 1 (e j )} J ε v ε (z) := u ε Jε (εz) (z J = ε 1 J ε ) lim z v ε L 2 = 0 v ε C ε : const ε 0 D j,ε J ε D 1,ε Ω ε O J ε D 3,ε D 2,ε ψ j (0) = lim k C ε k ψ C(G) ( ) / 25

29 1. H. Attouch, Variational Convergence for Functions and Operators, G. Dal Maso, An Introduction to Γ-Convergence, N. Kenmochi, Monotonicity and compactness Methods for Nonlinear Variational Inequalities, Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 4, ed. M. Chiopt, Chapter 4, , North Holland, Amsterdam, K. Kuwae and T. Shioya, Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom., 11 (2003), U. Mosco, Convergence of convex sets and of solutions variational inequalities, Advances Math., 3(1969), ( ) / 25

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i 1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,