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

.. IV 2012 10 4 ( ) 2012 10 4 1 / 25

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 ( ) 2012 10 4 2 / 25

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 ( ) 2012 10 4 3 / 25

1. Quantum Graph 1930 ( ) 2012 10 4 4 / 25

1. [ 92 Hale and Raugel] G 1 G Y ( ) 2012 10 4 5 / 25

1. 2. 3.. 4 ( ) 2012 10 4 6 / 25

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 }) ( ) 2012 10 4 7 / 25

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 ( ) 2012 10 4 7 / 25

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)} ( ) 2012 10 4 8 / 25

2.3 1 3 (Kirchhoff condition) ψ C(G) N dψ j ds (0) = 0 j=1 dψ j ds (l j) = 0 ( j = 1,..., N) O G ( ) 2012 10 4 9 / 25

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 ( ) 2012 10 4 10 / 25

3.1 Ω Q[u, w] = u w dx Ω ( ) 2012 10 4 11 / 25

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 ( ) 2012 10 4 11 / 25

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)) ( ) 2012 10 4 12 / 25

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 ( ) 2012 10 4 12 / 25

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 ( ) 2012 10 4 13 / 25

. 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 +0 2. 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 ( ) 2012 10 4 14 / 25

. 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. ( ) 2012 10 4 15 / 25

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 ( ) 2012 10 4 16 / 25

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 ( ) 2012 10 4 17 / 25

. 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 +0 2. 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 ( ) 2012 10 4 18 / 25

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. ( ) 2012 10 4 19 / 25

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 ( ) 2012 10 4 20 / 25

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 ( ) 2012 10 4 21 / 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 ( ) 2012 10 4 21 / 25

Kirchhoff B.C. Neumann B.C. ( ) 2012 10 4 22 / 25

{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 ( ) 2012 10 4 23 / 25

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) ( ) 2012 10 4 24 / 25

1. H. Attouch, Variational Convergence for Functions and Operators, 1984. 2. G. Dal Maso, An Introduction to Γ-Convergence, 1993. 3. 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, 203-298, North Holland, Amsterdam, 2007. 4. K. Kuwae and T. Shioya, Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom., 11 (2003), 599 673. 5. U. Mosco, Convergence of convex sets and of solutions variational inequalities, Advances Math., 3(1969), 510-585. ( ) 2012 10 4 25 / 25