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On vector-valued singular perturbation problems involving potentials - - PowerPoint PPT Presentation

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On vector-valued singular perturbation problems involving potentials vanishing on curves Nelly Andr e and Itai Shafrir Univ. Tours, Technion Itai Shafrir mass constraint A phase transition problem (Cahn-Hilliard energy: Modica, Sternberg)

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On vector-valued singular perturbation problems involving potentials vanishing on curves

Nelly Andr´ e and Itai Shafrir

  • Univ. Tours, Technion

Itai Shafrir mass constraint

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A phase transition problem (Cahn-Hilliard energy: Modica, Sternberg)

Let W : R → [0, ∞) be a smooth double-well potential: W (t) ≥ 0, ∀t, W (t) = 0 ⇐ ⇒ t ∈ {a1, a2} .

Itai Shafrir mass constraint

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A phase transition problem (Cahn-Hilliard energy: Modica, Sternberg)

Let W : R → [0, ∞) be a smooth double-well potential: W (t) ≥ 0, ∀t, W (t) = 0 ⇐ ⇒ t ∈ {a1, a2} .

W

Itai Shafrir mass constraint

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A phase transition problem (Cahn-Hilliard energy: Modica, Sternberg)

Let W : R → [0, ∞) be a smooth double-well potential: W (t) ≥ 0, ∀t, W (t) = 0 ⇐ ⇒ t ∈ {a1, a2} .

W

For simplicity, take the potential W (t) = t2(1 − t)2.

Itai Shafrir mass constraint

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A phase transition problem (Cahn-Hilliard energy: Modica, Sternberg)

Let W : R → [0, ∞) be a smooth double-well potential: W (t) ≥ 0, ∀t, W (t) = 0 ⇐ ⇒ t ∈ {a1, a2} .

W

For simplicity, take the potential W (t) = t2(1 − t)2. Let G ⊂ RN be a bounded domain with µ(G) = 1.

Itai Shafrir mass constraint

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A phase transition problem (Cahn-Hilliard energy: Modica, Sternberg)

Let W : R → [0, ∞) be a smooth double-well potential: W (t) ≥ 0, ∀t, W (t) = 0 ⇐ ⇒ t ∈ {a1, a2} .

W

For simplicity, take the potential W (t) = t2(1 − t)2. Let G ⊂ RN be a bounded domain with µ(G) = 1. Let c ∈ (0, 1) be given.

Itai Shafrir mass constraint

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A phase transition problem (Cahn-Hilliard energy: Modica, Sternberg)

Let W : R → [0, ∞) be a smooth double-well potential: W (t) ≥ 0, ∀t, W (t) = 0 ⇐ ⇒ t ∈ {a1, a2} .

W

For simplicity, take the potential W (t) = t2(1 − t)2. Let G ⊂ RN be a bounded domain with µ(G) = 1. Let c ∈ (0, 1) be given. For each ε > 0 let uε be a minimizer for Eε(u) = ˆ

G

|∇u|2 + W (u) ε2 , u ∈ H1(G), ˆ

G

u = c .

Itai Shafrir mass constraint

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Theorem (Modica, Sternberg 87)

  • uεn → u0 = χG1 in L1(G) with µ(G1) = c.

Itai Shafrir mass constraint

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Theorem (Modica, Sternberg 87)

  • uεn → u0 = χG1 in L1(G) with µ(G1) = c.
  • The surface area of ∂G1 ∩ G is minimal, i.e.,

PerG G1 = min{PerG A : A ⊂ G, |A| = c} .

Itai Shafrir mass constraint

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Theorem (Modica, Sternberg 87)

  • uεn → u0 = χG1 in L1(G) with µ(G1) = c.
  • The surface area of ∂G1 ∩ G is minimal, i.e.,

PerG G1 = min{PerG A : A ⊂ G, |A| = c} .

  • lim

ε→0 εEε(uε) = 2D PerG{u0 = 1}, where

Itai Shafrir mass constraint

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Theorem (Modica, Sternberg 87)

  • uεn → u0 = χG1 in L1(G) with µ(G1) = c.
  • The surface area of ∂G1 ∩ G is minimal, i.e.,

PerG G1 = min{PerG A : A ⊂ G, |A| = c} .

  • lim

ε→0 εEε(uε) = 2D PerG{u0 = 1}, where

D := ˆ 1

  • W (s) ds .

Itai Shafrir mass constraint

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Two scenarios

Itai Shafrir mass constraint

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Two scenarios

  • 1. G convex:

G G1 G2 u ~1 u ~0

ε ε

Itai Shafrir mass constraint

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Two scenarios

  • 1. G convex:

G G1 G2 u ~1 u ~0

ε ε

  • 2. G non-convex:

G1 G2 u ~1 u ~0

ε ε

G

Itai Shafrir mass constraint

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Two scenarios

  • 1. G convex:

G G1 G2 u ~1 u ~0

ε ε

  • 2. G non-convex:

G1 G2 u ~1 u ~0

ε ε

G

Or

G G

1 u ~1 ε u ~0 ε

G2

Itai Shafrir mass constraint

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The isoperimetric profile

The isoperimetric profile of G is the function I : (0, µ(G)) → R+:

Itai Shafrir mass constraint

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The isoperimetric profile

The isoperimetric profile of G is the function I : (0, µ(G)) → R+: I(t) = min{PerG V : V ⊂ G, µ(V ) = t} .

Itai Shafrir mass constraint

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The isoperimetric profile

The isoperimetric profile of G is the function I : (0, µ(G)) → R+: I(t) = min{PerG V : V ⊂ G, µ(V ) = t} . Clearly I is symmetric around µ(G)/2.

Itai Shafrir mass constraint

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The isoperimetric profile

The isoperimetric profile of G is the function I : (0, µ(G)) → R+: I(t) = min{PerG V : V ⊂ G, µ(V ) = t} . Clearly I is symmetric around µ(G)/2.

I(t)

, , t

0.5 1 | | β1 β2

G convex

Itai Shafrir mass constraint

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The isoperimetric profile

The isoperimetric profile of G is the function I : (0, µ(G)) → R+: I(t) = min{PerG V : V ⊂ G, µ(V ) = t} . Clearly I is symmetric around µ(G)/2.

I(t)

, , t

0.5 1 | | β1 β2

G convex

G nonconvex

, ,

0.5 1 | β1 β2

| α

I(t)

t

|

Itai Shafrir mass constraint

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The isoperimetric profile

The isoperimetric profile of G is the function I : (0, µ(G)) → R+: I(t) = min{PerG V : V ⊂ G, µ(V ) = t} . Clearly I is symmetric around µ(G)/2.

I(t)

, , t

0.5 1 | | β1 β2

G convex

G nonconvex

, ,

0.5 1 | β1 β2

| α

I(t)

t

|

Note: When G is convex, I(t) is concave.

Itai Shafrir mass constraint

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The vector-valued problem

Consider W : R2 → [0, ∞) vanishing on two closed curves Γ1, Γ2.

Itai Shafrir mass constraint

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The vector-valued problem

Consider W : R2 → [0, ∞) vanishing on two closed curves Γ1, Γ2.

  • 0 is inside Γ1 which lies inside Γ2.

Itai Shafrir mass constraint

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The vector-valued problem

Consider W : R2 → [0, ∞) vanishing on two closed curves Γ1, Γ2.

  • 0 is inside Γ1 which lies inside Γ2.
  • Wnn > 0 on Γ1 ∪ Γ2.

Itai Shafrir mass constraint

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The vector-valued problem

Consider W : R2 → [0, ∞) vanishing on two closed curves Γ1, Γ2.

  • 0 is inside Γ1 which lies inside Γ2.
  • Wnn > 0 on Γ1 ∪ Γ2.
  • W satisfies a coercivity condition at infinity.

Itai Shafrir mass constraint

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The vector-valued problem

Consider W : R2 → [0, ∞) vanishing on two closed curves Γ1, Γ2.

  • 0 is inside Γ1 which lies inside Γ2.
  • Wnn > 0 on Γ1 ∪ Γ2.
  • W satisfies a coercivity condition at infinity.

. 0

Γ

2

M2 M1 Γ

1

m1 m2 Rc

c

r

m M m M 1 1 2 2

.

R

Itai Shafrir mass constraint

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Our Problem: Let G ⊂ RN be a smooth bounded domain with µ(G) = 1.

Itai Shafrir mass constraint

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Our Problem: Let G ⊂ RN be a smooth bounded domain with µ(G) = 1. Given Rc ∈ (M1, m2), study the limit as ε → 0 of the minimizers {uε} for min{Eε(u) = ˆ

G

|∇u|2 + W (u) ε2 : u ∈ H1(G, R2), ˆ

G

|u| = Rc}.

Itai Shafrir mass constraint

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Our Problem: Let G ⊂ RN be a smooth bounded domain with µ(G) = 1. Given Rc ∈ (M1, m2), study the limit as ε → 0 of the minimizers {uε} for min{Eε(u) = ˆ

G

|∇u|2 + W (u) ε2 : u ∈ H1(G, R2), ˆ

G

|u| = Rc}. We expect uεn → u0 with ´

G |u0| = Rc,

Itai Shafrir mass constraint

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Our Problem: Let G ⊂ RN be a smooth bounded domain with µ(G) = 1. Given Rc ∈ (M1, m2), study the limit as ε → 0 of the minimizers {uε} for min{Eε(u) = ˆ

G

|∇u|2 + W (u) ε2 : u ∈ H1(G, R2), ˆ

G

|u| = Rc}. We expect uεn → u0 with ´

G |u0| = Rc,

  • u0(x) ∈ Γ1, x ∈ G1,

u0(x) ∈ Γ2, x ∈ G2.

Itai Shafrir mass constraint

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Our Problem: Let G ⊂ RN be a smooth bounded domain with µ(G) = 1. Given Rc ∈ (M1, m2), study the limit as ε → 0 of the minimizers {uε} for min{Eε(u) = ˆ

G

|∇u|2 + W (u) ε2 : u ∈ H1(G, R2), ˆ

G

|u| = Rc}. We expect uεn → u0 with ´

G |u0| = Rc,

  • u0(x) ∈ Γ1, x ∈ G1,

u0(x) ∈ Γ2, x ∈ G2.

c

r

m M m M 1 1 2 2

.

R

Itai Shafrir mass constraint

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Our Problem: Let G ⊂ RN be a smooth bounded domain with µ(G) = 1. Given Rc ∈ (M1, m2), study the limit as ε → 0 of the minimizers {uε} for min{Eε(u) = ˆ

G

|∇u|2 + W (u) ε2 : u ∈ H1(G, R2), ˆ

G

|u| = Rc}. We expect uεn → u0 with ´

G |u0| = Rc,

  • u0(x) ∈ Γ1, x ∈ G1,

u0(x) ∈ Γ2, x ∈ G2.

c

r

m M m M 1 1 2 2

.

R

Hence, µ(G1) ∈ [β1, β2] = I0 := [ m2−Rc

m2−m1 , M2−Rc M2−M1 ].

Itai Shafrir mass constraint

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Our Problem: Let G ⊂ RN be a smooth bounded domain with µ(G) = 1. Given Rc ∈ (M1, m2), study the limit as ε → 0 of the minimizers {uε} for min{Eε(u) = ˆ

G

|∇u|2 + W (u) ε2 : u ∈ H1(G, R2), ˆ

G

|u| = Rc}. We expect uεn → u0 with ´

G |u0| = Rc,

  • u0(x) ∈ Γ1, x ∈ G1,

u0(x) ∈ Γ2, x ∈ G2.

c

r

m M m M 1 1 2 2

.

R

Hence, µ(G1) ∈ [β1, β2] = I0 := [ m2−Rc

m2−m1 , M2−Rc M2−M1 ].

µ(G1) := α satisfies I(α) = mint∈I0 I(t).

Itai Shafrir mass constraint

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The results

  • I. The “convex case”

Itai Shafrir mass constraint

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The results

  • I. The “convex case”

I(t)

, , t

0.5 1 | | β1 β2

G convex

Theorem (Andr´ e-Sh 2011) If α ∈ {β1, β2},

Itai Shafrir mass constraint

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The results

  • I. The “convex case”

I(t)

, , t

0.5 1 | | β1 β2

G convex

Theorem (Andr´ e-Sh 2011) If α ∈ {β1, β2},then uεn → u0 = χG1x1 + χG2x2 in L1(G),

Itai Shafrir mass constraint

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The results

  • I. The “convex case”

I(t)

, , t

0.5 1 | | β1 β2

G convex

Theorem (Andr´ e-Sh 2011) If α ∈ {β1, β2},then uεn → u0 = χG1x1 + χG2x2 in L1(G), with xj ∈ Γj,

Itai Shafrir mass constraint

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The results

  • I. The “convex case”

I(t)

, , t

0.5 1 | | β1 β2

G convex

Theorem (Andr´ e-Sh 2011) If α ∈ {β1, β2},then uεn → u0 = χG1x1 + χG2x2 in L1(G), with xj ∈ Γj,|xj| = mj (j = 1, 2) or |xj| = Mj (j = 1, 2).

G G1 G2

u ~x u ~x

ε ε 1 2

Itai Shafrir mass constraint

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x1 and x2 are not necessarily the “closest” pair p1, p2

Itai Shafrir mass constraint

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x1 and x2 are not necessarily the “closest” pair p1, p2 Passing from pj to xj costs (presumably) ∼ ε−1/2

Itai Shafrir mass constraint

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x1 and x2 are not necessarily the “closest” pair p1, p2 Passing from pj to xj costs (presumably) ∼ ε−1/2 We proved Eε(uε) = 2DI(α)

ε

+ O(ε−1/2).

Itai Shafrir mass constraint

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x1 and x2 are not necessarily the “closest” pair p1, p2 Passing from pj to xj costs (presumably) ∼ ε−1/2 We proved Eε(uε) = 2DI(α)

ε

+ O(ε−1/2). In the case G = S2 we have (Andr´ e-Sh 08):

Itai Shafrir mass constraint

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x1 and x2 are not necessarily the “closest” pair p1, p2 Passing from pj to xj costs (presumably) ∼ ε−1/2 We proved Eε(uε) = 2DI(α)

ε

+ O(ε−1/2). In the case G = S2 we have (Andr´ e-Sh 08): Eε(uε) = 2DI(α) ε + c0 ε1/2 + o(ε−1/2)

Itai Shafrir mass constraint

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x1 and x2 are not necessarily the “closest” pair p1, p2 Passing from pj to xj costs (presumably) ∼ ε−1/2 We proved Eε(uε) = 2DI(α)

ε

+ O(ε−1/2). In the case G = S2 we have (Andr´ e-Sh 08): Eε(uε) = 2DI(α) ε + c0 ε1/2 + o(ε−1/2) Using c0 we can know which xj is selected among several candidates.

Itai Shafrir mass constraint

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  • II. The nonconvex case

Itai Shafrir mass constraint

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  • II. The nonconvex case

G nonconvex

, ,

0.5 1 | β1 β2

| α

I(t)

t

|

Itai Shafrir mass constraint

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  • II. The nonconvex case

G nonconvex

, ,

0.5 1 | β1 β2

| α

I(t)

t

|

Assume α ∈ (β1, β2).

Itai Shafrir mass constraint

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  • II. The nonconvex case

G nonconvex

, ,

0.5 1 | β1 β2

| α

I(t)

t

|

Assume α ∈ (β1, β2). Let G ⊂ R2(technical??).

Itai Shafrir mass constraint

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  • II. The nonconvex case

G nonconvex

, ,

0.5 1 | β1 β2

| α

I(t)

t

|

Assume α ∈ (β1, β2). Let G ⊂ R2(technical??). Assume ∃! geodesic ¯ γ realizing D = inf

γ(0)∈Γ1 γ(1)∈Γ2

ˆ 1

  • W (γ(t))

1/2 |γ′(t)| dt.

. 0

Γ

2

M2 M1 Γ

1 1

m2 Rc p2 − γ p1 m

Itai Shafrir mass constraint

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Let G1 ⊂ G s.t. µ(G1) = α and PerG G1 = I(α).

Itai Shafrir mass constraint

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Let G1 ⊂ G s.t. µ(G1) = α and PerG G1 = I(α). Theorem (Andr´ e-Sh 2014) uεn → u0 in H1

loc(G \ Σ) ∩ Cloc(G \ Σ) where Σ = ∂G1 ∩ ∂G2

Itai Shafrir mass constraint

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Let G1 ⊂ G s.t. µ(G1) = α and PerG G1 = I(α). Theorem (Andr´ e-Sh 2014) uεn → u0 in H1

loc(G \ Σ) ∩ Cloc(G \ Σ) where Σ = ∂G1 ∩ ∂G2

u0 = (U1, U2) ∈ H1(G1, Γ1) × H1(G2, Γ2) is a minimizer for

Itai Shafrir mass constraint

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Let G1 ⊂ G s.t. µ(G1) = α and PerG G1 = I(α). Theorem (Andr´ e-Sh 2014) uεn → u0 in H1

loc(G \ Σ) ∩ Cloc(G \ Σ) where Σ = ∂G1 ∩ ∂G2

u0 = (U1, U2) ∈ H1(G1, Γ1) × H1(G2, Γ2) is a minimizer for E0 := min ˆ

G1

|∇v1|2 + ˆ

G2

|∇v2|2 : vj ∈ H1(Gj, Γj) , Tr vj

  • ∂Gj∩G = pj , j = 1, 2 ,

ˆ

G1

|v1| + ˆ

G2

|v2| = Rc

  • .

Itai Shafrir mass constraint

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Let G1 ⊂ G s.t. µ(G1) = α and PerG G1 = I(α). Theorem (Andr´ e-Sh 2014) uεn → u0 in H1

loc(G \ Σ) ∩ Cloc(G \ Σ) where Σ = ∂G1 ∩ ∂G2

u0 = (U1, U2) ∈ H1(G1, Γ1) × H1(G2, Γ2) is a minimizer for E0 := min ˆ

G1

|∇v1|2 + ˆ

G2

|∇v2|2 : vj ∈ H1(Gj, Γj) , Tr vj

  • ∂Gj∩G = pj , j = 1, 2 ,

ˆ

G1

|v1| + ˆ

G2

|v2| = Rc

  • .

Moreover, Eε(uε) = 2D

ε I(α) + E0 + o(1).

Itai Shafrir mass constraint

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Let G1 ⊂ G s.t. µ(G1) = α and PerG G1 = I(α). Theorem (Andr´ e-Sh 2014) uεn → u0 in H1

loc(G \ Σ) ∩ Cloc(G \ Σ) where Σ = ∂G1 ∩ ∂G2

u0 = (U1, U2) ∈ H1(G1, Γ1) × H1(G2, Γ2) is a minimizer for E0 := min ˆ

G1

|∇v1|2 + ˆ

G2

|∇v2|2 : vj ∈ H1(Gj, Γj) , Tr vj

  • ∂Gj∩G = pj , j = 1, 2 ,

ˆ

G1

|v1| + ˆ

G2

|v2| = Rc

  • .

Moreover, Eε(uε) = 2D

ε I(α) + E0 + o(1).

G1 G2

G

U2 = p2 U1 = p1

Σ

∂U1 ∂ν = 0 ∂U2 ∂ν = 0

Itai Shafrir mass constraint

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Remarks

Itai Shafrir mass constraint

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Remarks

Sternberg (87) introduced the problem and proved a Γ-convergence result.

Itai Shafrir mass constraint

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Remarks

Sternberg (87) introduced the problem and proved a Γ-convergence result. Rubinstein, Sternberg and Keller (89) studied a related reaction-diffusion equation.

Itai Shafrir mass constraint

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Remarks

Sternberg (87) introduced the problem and proved a Γ-convergence result. Rubinstein, Sternberg and Keller (89) studied a related reaction-diffusion equation. F.H. Lin, X.B. Pan and C. Wang (2012) found the asymptotic behavior of the minimizers with (“well prepared”) Dirichlet b.c.:

Itai Shafrir mass constraint

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Remarks

Sternberg (87) introduced the problem and proved a Γ-convergence result. Rubinstein, Sternberg and Keller (89) studied a related reaction-diffusion equation. F.H. Lin, X.B. Pan and C. Wang (2012) found the asymptotic behavior of the minimizers with (“well prepared”) Dirichlet b.c.:

  • Arbitrary dimension

Itai Shafrir mass constraint

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Remarks

Sternberg (87) introduced the problem and proved a Γ-convergence result. Rubinstein, Sternberg and Keller (89) studied a related reaction-diffusion equation. F.H. Lin, X.B. Pan and C. Wang (2012) found the asymptotic behavior of the minimizers with (“well prepared”) Dirichlet b.c.:

  • Arbitrary dimension
  • L1-convergence.

Itai Shafrir mass constraint