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

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) Let W : R → [0 , ∞ ) be a smooth double-well potential: W ( t ) ≥ 0 , ∀ t , W ( t ) = 0 ⇐ ⇒ t ∈ { a 1 , a 2 } . Itai Shafrir mass constraint

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 ∈ { a 1 , a 2 } . W Itai Shafrir mass constraint

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 ∈ { a 1 , a 2 } . W For simplicity, take the potential W ( t ) = t 2 (1 − t ) 2 . Itai Shafrir mass constraint

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 ∈ { a 1 , a 2 } . W For simplicity, take the potential W ( t ) = t 2 (1 − t ) 2 . Let G ⊂ R N be a bounded domain with µ ( G ) = 1. Itai Shafrir mass constraint

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 ∈ { a 1 , a 2 } . W For simplicity, take the potential W ( t ) = t 2 (1 − t ) 2 . Let G ⊂ R N be a bounded domain with µ ( G ) = 1. Let c ∈ (0 , 1) be given. Itai Shafrir mass constraint

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 ∈ { a 1 , a 2 } . W For simplicity, take the potential W ( t ) = t 2 (1 − t ) 2 . Let G ⊂ R N be a bounded domain with µ ( G ) = 1. Let c ∈ (0 , 1) be given. For each ε > 0 let u ε be a minimizer for |∇ u | 2 + W ( u ) ˆ ˆ , u ∈ H 1 ( G ) , E ε ( u ) = u = c . ε 2 G G Itai Shafrir mass constraint

Theorem (Modica, Sternberg 87) • u ε n → u 0 = χ G 1 in L 1 ( G ) with µ ( G 1 ) = c. Itai Shafrir mass constraint

Theorem (Modica, Sternberg 87) • u ε n → u 0 = χ G 1 in L 1 ( G ) with µ ( G 1 ) = c. • The surface area of ∂ G 1 ∩ G is minimal, i.e., Per G G 1 = min { Per G A : A ⊂ G , | A | = c } . Itai Shafrir mass constraint

Theorem (Modica, Sternberg 87) • u ε n → u 0 = χ G 1 in L 1 ( G ) with µ ( G 1 ) = c. • The surface area of ∂ G 1 ∩ G is minimal, i.e., Per G G 1 = min { Per G A : A ⊂ G , | A | = c } . • lim ε → 0 ε E ε ( u ε ) = 2 D Per G { u 0 = 1 } , where Itai Shafrir mass constraint

Theorem (Modica, Sternberg 87) • u ε n → u 0 = χ G 1 in L 1 ( G ) with µ ( G 1 ) = c. • The surface area of ∂ G 1 ∩ G is minimal, i.e., Per G G 1 = min { Per G A : A ⊂ G , | A | = c } . • lim ε → 0 ε E ε ( u ε ) = 2 D Per G { u 0 = 1 } , where ˆ 1 � D := W ( s ) ds . 0 Itai Shafrir mass constraint

Two scenarios Itai Shafrir mass constraint

Two scenarios 1. G convex: G G 2 G 1 u ~0 u ~1 ε ε Itai Shafrir mass constraint

Two scenarios 1. G convex: G G 2 G 1 u ~0 u ~1 ε ε 2. G non-convex: G G 2 G 1 u ~0 u ~1 ε ε Itai Shafrir mass constraint

Two scenarios 1. G convex: G G 2 G 1 u ~0 u ~1 ε ε 2. G non-convex: G G 2 G 1 u ~0 u ~1 ε ε Or G G 2 G 1 u ~0 u ~1 ε ε Itai Shafrir mass constraint

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

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

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

The isoperimetric profile The isoperimetric profile of G is the function I : (0 , µ ( G )) → R + : I ( t ) = min { Per G V : V ⊂ G , µ ( V ) = t } . Clearly I is symmetric around µ ( G ) / 2. I(t) G convex , , | | t 0.5 1 0 β 1 β 2 Itai Shafrir mass constraint

The isoperimetric profile The isoperimetric profile of G is the function I : (0 , µ ( G )) → R + : I ( t ) = min { Per G V : V ⊂ G , µ ( V ) = t } . Clearly I is symmetric around µ ( G ) / 2. I(t) G convex I(t) G nonconvex t , , | | | , , | | t 0.5 1 0.5 1 0 β 1 α β 2 0 β 1 β 2 Itai Shafrir mass constraint

The isoperimetric profile The isoperimetric profile of G is the function I : (0 , µ ( G )) → R + : I ( t ) = min { Per G V : V ⊂ G , µ ( V ) = t } . Clearly I is symmetric around µ ( G ) / 2. I(t) G convex I(t) G nonconvex t , , | | | , , | | t 0.5 1 0.5 1 0 β 1 α β 2 0 β 1 β 2 Note: When G is convex, I ( t ) is concave. Itai Shafrir mass constraint

The vector-valued problem Consider W : R 2 → [0 , ∞ ) vanishing on two closed curves Γ 1 , Γ 2 . Itai Shafrir mass constraint

The vector-valued problem Consider W : R 2 → [0 , ∞ ) vanishing on two closed curves Γ 1 , Γ 2 . • 0 is inside Γ 1 which lies inside Γ 2 . Itai Shafrir mass constraint

The vector-valued problem Consider W : R 2 → [0 , ∞ ) vanishing on two closed curves Γ 1 , Γ 2 . • 0 is inside Γ 1 which lies inside Γ 2 . • W nn > 0 on Γ 1 ∪ Γ 2 . Itai Shafrir mass constraint

The vector-valued problem Consider W : R 2 → [0 , ∞ ) vanishing on two closed curves Γ 1 , Γ 2 . • 0 is inside Γ 1 which lies inside Γ 2 . • W nn > 0 on Γ 1 ∪ Γ 2 . • W satisfies a coercivity condition at infinity. Itai Shafrir mass constraint

The vector-valued problem Consider W : R 2 → [0 , ∞ ) vanishing on two closed curves Γ 1 , Γ 2 . • 0 is inside Γ 1 which lies inside Γ 2 . • W nn > 0 on Γ 1 ∪ Γ 2 . • W satisfies a coercivity condition at infinity. Γ 2 M 1 Γ 1 . 0 M 2 m 1 R c m 2 . 0 m M R m M 1 1 c 2 r 2 Itai Shafrir mass constraint

Our Problem: Let G ⊂ R N be a smooth bounded domain with µ ( G ) = 1. Itai Shafrir mass constraint

Our Problem: Let G ⊂ R N be a smooth bounded domain with µ ( G ) = 1. Given R c ∈ ( M 1 , m 2 ), study the limit as ε → 0 of the minimizers { u ε } for ˆ |∇ u | 2 + W ( u ) ˆ : u ∈ H 1 ( G , R 2 ) , min { E ε ( u ) = | u | = R c } . ε 2 G G Itai Shafrir mass constraint

Our Problem: Let G ⊂ R N be a smooth bounded domain with µ ( G ) = 1. Given R c ∈ ( M 1 , m 2 ), study the limit as ε → 0 of the minimizers { u ε } for ˆ |∇ u | 2 + W ( u ) ˆ : u ∈ H 1 ( G , R 2 ) , min { E ε ( u ) = | u | = R c } . ε 2 G G ´ We expect u ε n → u 0 with G | u 0 | = R c , Itai Shafrir mass constraint

Our Problem: Let G ⊂ R N be a smooth bounded domain with µ ( G ) = 1. Given R c ∈ ( M 1 , m 2 ), study the limit as ε → 0 of the minimizers { u ε } for ˆ |∇ u | 2 + W ( u ) ˆ : u ∈ H 1 ( G , R 2 ) , min { E ε ( u ) = | u | = R c } . ε 2 G G ´ We expect u ε n → u 0 with G | u 0 | = R c , � u 0 ( x ) ∈ Γ 1 , x ∈ G 1 , u 0 ( x ) ∈ Γ 2 , x ∈ G 2 . Itai Shafrir mass constraint

Our Problem: Let G ⊂ R N be a smooth bounded domain with µ ( G ) = 1. Given R c ∈ ( M 1 , m 2 ), study the limit as ε → 0 of the minimizers { u ε } for ˆ |∇ u | 2 + W ( u ) ˆ : u ∈ H 1 ( G , R 2 ) , min { E ε ( u ) = | u | = R c } . ε 2 G G ´ We expect u ε n → u 0 with G | u 0 | = R c , � u 0 ( x ) ∈ Γ 1 , x ∈ G 1 , u 0 ( x ) ∈ Γ 2 , x ∈ G 2 . . 0 m M R m c M 1 1 2 2 r Itai Shafrir mass constraint

Our Problem: Let G ⊂ R N be a smooth bounded domain with µ ( G ) = 1. Given R c ∈ ( M 1 , m 2 ), study the limit as ε → 0 of the minimizers { u ε } for ˆ |∇ u | 2 + W ( u ) ˆ : u ∈ H 1 ( G , R 2 ) , min { E ε ( u ) = | u | = R c } . ε 2 G G ´ We expect u ε n → u 0 with G | u 0 | = R c , � u 0 ( x ) ∈ Γ 1 , x ∈ G 1 , u 0 ( x ) ∈ Γ 2 , x ∈ G 2 . . 0 m M R m c M 1 1 2 2 r Hence, µ ( G 1 ) ∈ [ β 1 , β 2 ] = I 0 := [ m 2 − R c m 2 − m 1 , M 2 − R c M 2 − M 1 ] . Itai Shafrir mass constraint

Our Problem: Let G ⊂ R N be a smooth bounded domain with µ ( G ) = 1. Given R c ∈ ( M 1 , m 2 ), study the limit as ε → 0 of the minimizers { u ε } for ˆ |∇ u | 2 + W ( u ) ˆ : u ∈ H 1 ( G , R 2 ) , min { E ε ( u ) = | u | = R c } . ε 2 G G ´ We expect u ε n → u 0 with G | u 0 | = R c , � u 0 ( x ) ∈ Γ 1 , x ∈ G 1 , u 0 ( x ) ∈ Γ 2 , x ∈ G 2 . . 0 m M R m c M 1 1 2 2 r Hence, µ ( G 1 ) ∈ [ β 1 , β 2 ] = I 0 := [ m 2 − R c m 2 − m 1 , M 2 − R c M 2 − M 1 ] . µ ( G 1 ) := α satisfies I ( α ) = min t ∈I 0 I ( t ). Itai Shafrir mass constraint

The results I. The “convex case” Itai Shafrir mass constraint

The results I. The “convex case” I(t) G convex , , | | t 0 0.5 β 2 1 β 1 Theorem (Andr´ e-Sh 2011) If α ∈ { β 1 , β 2 } , Itai Shafrir mass constraint