Asymptotic Development by Gamma Convergence

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Gianni Asymptotic Development Dirichlet–Neumann Problems Singularly Perturbed Problems Gamma Asymptotic Development Modica–Mortola, Cahn–Hilliard Functional

Asymptotic Development by Gamma Convergence

Gianni Dal Maso’s 65th birthday

Giovanni Leoni

Carnegie Mellon University

January 27, 2020

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CMU, October 2002

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CMU, Workshop 2003

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Miami, December 2009

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CMU, May 2016

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2-quasiconvexity. Dal Maso, Fonseca, G. L., & Morini, ARMA, 2004;

Image restoration. Dal Maso, Fonseca, G. L., & Morini, SIMA, 2009;

Counterexample integral representation. Dal Maso, Fonseca,

& G. L., Adv. Calc. Var., 2010;

Singularly perturbed problems. Chermisi, Dal Maso, Fonseca,

& G. L., Indiana Univ. Math. J, 2011;

Singular parabolic equations, Hilbert transform. Dal Maso, Fonseca, & G. L., ARMA, 2014;

Second order Gamma convergence. Dal Maso, Fonseca, & G.

L., Calc. Var. Partial Di¤erential Equations, 2015;

Nonlocal functionals. Dal Maso, Fonseca, & G. L., Trans.

Amer. Math. Soc., 2018;

Minimizing movements, elliptic systems in domains with

corners. Dal Maso, Fonseca, & G. L., in preparation.

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Outline of the Talk:

Gamma-Asymptotic Development of Order k

Singularly perturbed Neumann-Robin boundary problems

G.Gravina and G. L., 2019, SIAM J. Math. Anal.,

Asymptotic development of order 2 for Cahn–Hilliard functional

G.L. and R. Murray, 2016, ARMA, & 2019, Proc. Amer. Math. Soc.

G. Dal Maso, I. Fonseca and G.L., 2015, Calc. Var. Partial

Di¤erential Equations.

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Outline of the Talk:

Gamma-Asymptotic Development of Order k

Singularly perturbed Neumann-Robin boundary problems

G.Gravina and G. L., 2019, SIAM J. Math. Anal., Asymptotic development of order 2 for Cahn–Hilliard functional

G.L. and R. Murray, 2016, ARMA, & 2019, Proc. Amer. Math. Soc.

G. Dal Maso, I. Fonseca and G.L., 2015, Calc. Var. Partial

Di¤erential Equations.

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Outline of the Talk:

Gamma-Asymptotic Development of Order k

Singularly perturbed Neumann-Robin boundary problems G.Gravina and G. L., 2019, SIAM J. Math. Anal.,

Asymptotic development of order 2 for Cahn–Hilliard functional

G.L. and R. Murray, 2016, ARMA, & 2019, Proc. Amer. Math. Soc.

G. Dal Maso, I. Fonseca and G.L., 2015, Calc. Var. Partial

Di¤erential Equations.

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Outline of the Talk:

Gamma-Asymptotic Development of Order k

Singularly perturbed Neumann-Robin boundary problems G.Gravina and G. L., 2019, SIAM J. Math. Anal., Asymptotic development of order 2 for Cahn–Hilliard functional

G.L. and R. Murray, 2016, ARMA, & 2019, Proc. Amer. Math. Soc.

G. Dal Maso, I. Fonseca and G.L., 2015, Calc. Var. Partial

Di¤erential Equations.

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Outline of the Talk:

Gamma-Asymptotic Development of Order k

Singularly perturbed Neumann-Robin boundary problems G.Gravina and G. L., 2019, SIAM J. Math. Anal., Asymptotic development of order 2 for Cahn–Hilliard functional

G.L. and R. Murray, 2016, ARMA, & 2019, Proc. Amer.

Math. Soc.

G. Dal Maso, I. Fonseca and G.L., 2015, Calc. Var. Partial

Di¤erential Equations.

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Outline of the Talk:

Gamma-Asymptotic Development of Order k

Singularly perturbed Neumann-Robin boundary problems G.Gravina and G. L., 2019, SIAM J. Math. Anal., Asymptotic development of order 2 for Cahn–Hilliard functional

G.L. and R. Murray, 2016, ARMA, & 2019, Proc. Amer.

Math. Soc.

G. Dal Maso, I. Fonseca and G.L., 2015, Calc. Var. Partial

Di¤erential Equations.

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Gamma-Asymptotic Development of Order k

Gamma-Convergence

De Giorgi (1975), De Giorgi and Franzoni (1975)

De…nition

X metric space, F ε : X ! [ ∞, ∞ ] , ε > 0, Γ-converges if there exists F ⁽ ⁰ ⁾ ^{: X} ! [ ∞, ∞ ] such that

for every x 2 X and every x _ε ! ^x, lim inf

ε ! ⁰

⁺

F ε ( x _ε ) F ⁽ ⁰ ⁾ ( x ) ,

for every x 2 X there exists x _ε ! x such that lim

ε ! ⁰

⁺

F ε ( x _ε ) F ⁽ ⁰ ⁾ ( x )

We write F ε

! F Γ ⁽ ⁰ ⁾ ^.

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Gamma-Convergence

De Giorgi (1975), De Giorgi and Franzoni (1975)

De…nition

X metric space, F ε : X ! [ ∞, ∞ ] , ε > 0, Γ-converges if there exists F ⁽ ⁰ ⁾ ^{: X} ! [ ∞, ∞ ] such that

for every x 2 X and every x _ε ! ^x, lim inf

ε ! ⁰

⁺

F ε ( x _ε ) F ⁽ ⁰ ⁾ ( x ) , for every x 2 X there exists x _ε ! x such that

lim

ε ! ⁰

⁺

F ε ( x _ε ) F ⁽ ⁰ ⁾ ( x ) We write F ε

! F Γ ⁽ ⁰ ⁾ ^.

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Gamma-Asymptotic Developments

Anzellotti and Baldo (1993)

De…nition

X metric space, F ε : X ! ( ∞, ∞ ] has a Γ-asymptotic development of order k,

F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , if there exist F ⁽ ⁱ ⁾ ^{: X} ! [ ^{∞, ∞} ] , i = 1, . . . , k, such that

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^{, where} F ε ⁽ ⁰ ⁾ : = F ε ,

F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

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Gamma-Asymptotic Developments

Anzellotti and Baldo (1993)

De…nition

X metric space, F ε : X ! ( ∞, ∞ ] has a Γ-asymptotic development of order k,

F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , if there exist F ⁽ ⁱ ⁾ ^{: X} ! [ ^{∞, ∞} ] , i = 1, . . . , k, such that

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^{, where} F ε ⁽ ⁰ ⁾ : = F ε , F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

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Gianni Asymptotic Development Dirichlet–Neumann Problems Singularly Perturbed Problems Gamma Asymptotic Development Modica–Mortola, Cahn–Hilliard Functional Gamma-Asymptotic Development of Order k

F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g

F ⁽ ⁱ ⁾ ^{∞ in X} n U ⁱ ¹

let ε m ! ⁰ ⁺ and assume that there is a minimizer x _m 2 ^{X of} F ε

m

and x _m ! ^x

f limits of minimizers of F ε

m

g U ^k U ⁰ ^,

inf F ^ε

^m

= inf F ⁽ ⁰ ⁾ + ε m inf F ⁽ ¹ ⁾ + + ε ^k _m inf F ⁽ ^k ⁾ + o ( ε ^k _m ) ,

Goal/Hope …nd k so that f limits of minimizers of F ε

m

g = U k .

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F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g F ⁽ ⁱ ⁾ ^{∞ in X} n U ⁱ ¹

let ε m ! ⁰ ⁺ and assume that there is a minimizer x _m 2 ^{X of} F ε

m

and x _m ! ^x

f limits of minimizers of F ε

m

g U ^k U ⁰ ^,

inf F ^ε

^m

= inf F ⁽ ⁰ ⁾ + ε m inf F ⁽ ¹ ⁾ + + ε ^k _m inf F ⁽ ^k ⁾ + o ( ε ^k _m ) ,

Goal/Hope …nd k so that f limits of minimizers of F ε

m

g = U k .

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F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g F ⁽ ⁱ ⁾ ^{∞ in X} n U ⁱ ¹

let ε m ! ⁰ ⁺ and assume that there is a minimizer x _m 2 ^{X of} F ε

m

and x _m ! ^x

f limits of minimizers of F ε

m

g U ^k U ⁰ ^,

inf F ^ε

^m

= inf F ⁽ ⁰ ⁾ + ε m inf F ⁽ ¹ ⁾ + + ε ^k _m inf F ⁽ ^k ⁾ + o ( ε ^k _m ) ,

Goal/Hope …nd k so that f limits of minimizers of F ε

m

g = U k .

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F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g F ⁽ ⁱ ⁾ ^{∞ in X} n U ⁱ ¹

let ε m ! ⁰ ⁺ and assume that there is a minimizer x _m 2 ^{X of} F ε

m

and x _m ! ^x

f limits of minimizers of F ε

m

g U ^k U ⁰ ^,

inf F ^ε

^m

= inf F ⁽ ⁰ ⁾ + ε m inf F ⁽ ¹ ⁾ + + ε ^k _m inf F ⁽ ^k ⁾ + o ( ε ^k _m ) ,

Goal/Hope …nd k so that f limits of minimizers of F ε

m

g = U k .

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F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g F ⁽ ⁱ ⁾ ^{∞ in X} n U ⁱ ¹

let ε m ! ⁰ ⁺ and assume that there is a minimizer x _m 2 ^{X of} F ε

m

and x _m ! ^x

f limits of minimizers of F ε

m

g U ^k U ⁰ ^,

inf F ^ε

^m

= inf F ⁽ ⁰ ⁾ + ε m inf F ⁽ ¹ ⁾ + + ε ^k _m inf F ⁽ ^k ⁾ + o ( ε ^k _m ) ,

Goal/Hope …nd k so that f limits of minimizers of F ε

m

g = U k .

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F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ε F ⁽ ¹ ⁾ + + ε ^k F ⁽ ^k ⁾ + o ( ε ^k ) , F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g F ⁽ ⁱ ⁾ ^{∞ in X} n U ⁱ ¹

let ε m ! ⁰ ⁺ and assume that there is a minimizer x _m 2 ^{X of} F ε

m

and x _m ! ^x

f limits of minimizers of F ε

m

g U ^k U ⁰ ^,

inf F ^ε

^m

= inf F ⁽ ⁰ ⁾ + ε m inf F ⁽ ¹ ⁾ + + ε ^k _m inf F ⁽ ^k ⁾ + o ( ε ^k _m ) ,

Goal/Hope …nd k so that f limits of minimizers of F ε

m

g = U k .

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = ε ^k j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R,

F ε ⁽ ⁱ ⁾ ( x ) = ε ^k ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i = 1, . . . , k 1, U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

F ε ⁽ ^k ⁾ ( x ) = j ^x j ^,

f limits of minimizers of F ε

m

g = U ^k U ^k ¹ = = U ⁰ ^.

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = ε ^k j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ε ^k ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i = 1, . . . , k 1,

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R, F ε ⁽ ^k ⁾ ( x ) = j ^x j ^,

f limits of minimizers of F ε

m

g = U ^k U ^k ¹ = = U ⁰ ^.

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = ε ^k j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ε ^k ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i = 1, . . . , k 1, U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

F ε ⁽ ^k ⁾ ( x ) = j ^x j ^,

f limits of minimizers of F ε

m

g = U ^k U ^k ¹ = = U ⁰ ^.

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = ε ^k j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ε ^k ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i = 1, . . . , k 1, U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

F ε ⁽ ^k ⁾ ( x ) = j ^x j ^,

f limits of minimizers of F ε

m

g = U ^k U ^k ¹ = = U ⁰ ^.

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = ε ^k j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ε ^k ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i = 1, . . . , k 1, U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

F ε ⁽ ^k ⁾ ( x ) = j ^x j ^,

f limits of minimizers of F ε

m

g = U ^k U ^k ¹ = = U ⁰ ^.

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example

Let X = [ 0, 1 ] and F ^ε ( x ) = ^ε

n j ^x j ^{if x} 2 ( ² ⁿ ^{, 2} ⁿ ⁺ ¹ ] , n 2 N, 0 if x = 0.

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = [ ^{0, 1} ] ,

F ε ⁽ ^k ⁾ ( x ) = ^ε

n k j ^x j ^{if x} 2 ( ² ⁿ ^{, 2} ⁿ ⁺ ¹ ] , n 2 ^N,

0 if x = 0.

U ^k ^: = f minimizers of F ⁽ ^k ⁾ g = [ ^{0, 2} ^k ] ,

f limits of minimizers of F ε

m

g = ^T k U k .

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example

Let X = [ 0, 1 ] and F ^ε ( x ) = ^ε

n j ^x j ^{if x} 2 ( ² ⁿ ^{, 2} ⁿ ⁺ ¹ ] , n 2 N, 0 if x = 0.

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = [ ^{0, 1} ] ,

F ε ⁽ ^k ⁾ ( x ) = ^ε

n k j ^x j ^{if x} 2 ( ² ⁿ ^{, 2} ⁿ ⁺ ¹ ] , n 2 ^N,

0 if x = 0.

U ^k ^: = f minimizers of F ⁽ ^k ⁾ g = [ ^{0, 2} ^k ] ,

f limits of minimizers of F ε

m

g = ^T k U k .

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example

Let X = [ 0, 1 ] and F ^ε ( x ) = ^ε

n j ^x j ^{if x} 2 ( ² ⁿ ^{, 2} ⁿ ⁺ ¹ ] , n 2 N, 0 if x = 0.

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^0, U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = [ ^{0, 1} ] ,

F ε ⁽ ^k ⁾ ( x ) = ^ε

n k j ^x j ^{if x} 2 ( ² ⁿ ^{, 2} ⁿ ⁺ ¹ ] , n 2 ^N,

0 if x = 0.

U ^k ^: = f minimizers of F ⁽ ^k ⁾ g = [ ^{0, 2} ^k ] ,

f limits of minimizers of F ε

m

g = ^T k U k .

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = e ^1/ε j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ⁰

U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ^e

1/ε

ε ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i , U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

f limits of minimizers of F ε

m

g = f ⁰ g U ⁱ = = U ⁰ = _R

for all i

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = e ^1/ε j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ⁰

U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R,

F ε ⁽ ⁱ ⁾ ( x ) = ^e

1/ε

ε ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i , U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

f limits of minimizers of F ε

m

g = f ⁰ g U ⁱ = = U ⁰ = _R

for all i

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = e ^1/ε j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ⁰

U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ^e

1/ε

ε ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i ,

U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

f limits of minimizers of F ε

m

g = f ⁰ g U ⁱ = = U ⁰ = _R

for all i

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = e ^1/ε j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ⁰

U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ^e

1/ε

ε ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i , U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

f limits of minimizers of F ε

m

g = f ⁰ g U ⁱ = = U ⁰ = _R

for all i

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F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ε

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

Example Let X = R and

F ε ( x ) = e ^1/ε j ^x j F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ⁰

U ⁰ ^: = f minimizers of F ⁽ ⁰ ⁾ g = ^R, F ε ⁽ ⁱ ⁾ ( x ) = ^e

1/ε

ε ⁱ j ^x j ^, F ε ⁽ ⁱ ⁾

! F Γ ⁽ ⁱ ⁾ 0 for all i , U ⁱ ^: = f minimizers of F ⁽ ⁱ ⁾ g = R,

f limits of minimizers of F ε

m

g = f ⁰ g U ⁱ = = U ⁰ = _R

for all i

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More generally, De…nition

X metric space, F ε : X ! ( ∞, ∞ ] has a Γ-asymptotic development of order k,

F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ω 1 ( ε )F ⁽ ¹ ⁾ + + ω k ( ε )F ⁽ ^k ⁾ + o ( ε ^k ) , if there exist F ⁽ ⁱ ⁾ ^{: X} ! [ ^{∞, ∞} ] , i = 1, . . . , k, such that

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^{, where} F ε ⁽ ⁰ ⁾ : = F ε ,

ω 0 1, ω i > 0, ω i ( ε ) ! ^{0, ω} ⁱ ( ε ) /ω i 1 ( ε ) ! ^{0 as ε} ! ⁰ ⁺ ^, F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ω i ( ε ) /ω i 1 ( ε )

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

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More generally, De…nition

X metric space, F ε : X ! ( ∞, ∞ ] has a Γ-asymptotic development of order k,

F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ω 1 ( ε )F ⁽ ¹ ⁾ + + ω k ( ε )F ⁽ ^k ⁾ + o ( ε ^k ) , if there exist F ⁽ ⁱ ⁾ ^{: X} ! [ ^{∞, ∞} ] , i = 1, . . . , k, such that

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^{, where} F ε ⁽ ⁰ ⁾ : = F ε ,

ω 0 1, ω i > 0, ω i ( ε ) ! ^{0, ω} ⁱ ( ε ) /ω i 1 ( ε ) ! ^{0 as ε} ! ⁰ ⁺ ^,

F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾ ω i ( ε ) /ω i 1 ( ε )

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

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More generally, De…nition

X metric space, F ε : X ! ( ∞, ∞ ] has a Γ-asymptotic development of order k,

F ε ⁽ ⁰ ⁾

= F Γ ⁽ ⁰ ⁾ + ω 1 ( ε )F ⁽ ¹ ⁾ + + ω k ( ε )F ⁽ ^k ⁾ + o ( ε ^k ) , if there exist F ⁽ ⁱ ⁾ ^{: X} ! [ ^{∞, ∞} ] , i = 1, . . . , k, such that

F ε ⁽ ⁰ ⁾

! F Γ ⁽ ⁰ ⁾ ^{, where} F ε ⁽ ⁰ ⁾ : = F ε ,

ω 0 1, ω i > 0, ω i ( ε ) ! ^{0, ω} ⁱ ( ε ) /ω i 1 ( ε ) ! ^{0 as ε} ! ⁰ ⁺ ^, F ε ⁽ ⁱ ⁾ : = F ε ⁽ ⁱ ¹ ⁾ inf F ⁽ ⁱ ¹ ⁾

ω i ( ε ) /ω i 1 ( ε )

! F Γ ⁽ ⁱ ⁾ ^{for i} = 1, . . . , k.

(39)

Dirichlet–Neumann Problems

8 <

:

∆u 0 = f in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = g on Γ D ,

Ω R ^N open, bounded, ∂Ω Lipschitz continuous,

ν outward unit normal to ∂ Ω,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = _{∅, Γ} N , Γ D relatively open, nonempty

f 2 ^L ² ( _Ω ) , g 2 ^H ^3/2 ( ∂ Ω )

(40)

Dirichlet–Neumann Problems

8 <

:

∆u 0 = f in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = g on Γ D ,

Ω R ^N open, bounded, ∂Ω Lipschitz continuous, ν outward unit normal to ∂ Ω,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = _{∅, Γ} N , Γ D relatively open, nonempty

f 2 ^L ² ( _Ω ) , g 2 ^H ^3/2 ( ∂ Ω )

(41)

Dirichlet–Neumann Problems

8 <

:

∆u 0 = f in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = g on Γ D ,

Ω R ^N open, bounded, ∂Ω Lipschitz continuous, ν outward unit normal to ∂ Ω,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = _{∅, Γ} N , Γ D relatively open, nonempty

f 2 ^L ² ( _Ω ) , g 2 ^H ^3/2 ( ∂ Ω )

(42)

Dirichlet–Neumann Problems

8 <

:

∆u 0 = f in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = g on Γ D ,

Ω R ^N open, bounded, ∂Ω Lipschitz continuous, ν outward unit normal to ∂ Ω,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = _{∅, Γ} N , Γ D relatively open, nonempty

f 2 ^L ² ( _Ω ) , g 2 ^H ^3/2 ( ∂ Ω )

(43)

8 <

:

∆u 0 = f in Ω,

∂ _ν u 0 = 0 on Γ N , u 0 = g on Γ D ,

Figure: Courtesy of G. Gravina

(44)

Loss of Regularity

8 <

:

∆u 0 = 0 in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = 0 on Γ D , N = 2, Ω = _R ² ₊ = f( ^x ¹ ^{, x} ² ) : x 2 > 0 g ^,

Γ N = ( ∞, 0 ) f ⁰ g ^, ^Γ ^D = ( 0, ∞ ) f ⁰ g ^, in polar coordinates

u 0 ( r , θ ) = r ^1/2 sin ( θ/2 ) ,

u ₀ 2 / ^H ² ( Ω \ ^B (( 0, 0 ) , r )) for every r > 0.

(45)

Loss of Regularity

8 <

:

∆u 0 = 0 in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = 0 on Γ D , N = 2, Ω = _R ² ₊ = f( ^x ¹ ^{, x} ² ) : x 2 > 0 g ^, Γ N = ( ∞, 0 ) f ⁰ g ^, ^Γ ^D = ( 0, ∞ ) f ⁰ g ^,

in polar coordinates

u 0 ( r , θ ) = r ^1/2 sin ( θ/2 ) ,

u ₀ 2 / ^H ² ( Ω \ ^B (( 0, 0 ) , r )) for every r > 0.

(46)

Loss of Regularity

8 <

:

∆u 0 = 0 in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = 0 on Γ D , N = 2, Ω = _R ² ₊ = f( ^x ¹ ^{, x} ² ) : x 2 > 0 g ^, Γ N = ( ∞, 0 ) f ⁰ g ^, ^Γ ^D = ( 0, ∞ ) f ⁰ g ^, in polar coordinates

u 0 ( r , θ ) = r ^1/2 sin ( θ/2 ) ,

u ₀ 2 / ^H ² ( Ω \ ^B (( 0, 0 ) , r )) for every r > 0.

(47)

Loss of Regularity

8 <

:

∆u 0 = 0 in Ω,

∂ ν u ₀ = 0 on Γ N , u ₀ = 0 on Γ D , N = 2, Ω = _R ² ₊ = f( ^x ¹ ^{, x} ² ) : x 2 > 0 g ^, Γ N = ( ∞, 0 ) f ⁰ g ^, ^Γ ^D = ( 0, ∞ ) f ⁰ g ^, in polar coordinates

u 0 ( r , θ ) = r ^1/2 sin ( θ/2 ) ,

u ₀ 2 / ^H ² ( Ω \ ^B (( 0, 0 ) , r )) for every r > 0.

(48)

Main Hypotheses

8 <

:

∆u 0 = f in Ω,

∂ _ν u ₀ = 0 on Γ N , u 0 = g on Γ D ,

Ω R ² open, bounded, connected ∂Ω of class C ^1,1 ,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = f ^P ¹ ^{, P} ² g ^, Γ N , Γ D relatively open, nonempty,

∂ Ω \ ^B ( P _i , δ i ) is a segment, i = 1, 2, δ i > 0 small, ( r _i , θ i ) polar coordinates at P _i , , i = 1, 2,

f 2 ^L ² ( Ω ) , g 2 ^H ^3/2 ( ∂Ω ) .

(49)

Main Hypotheses

8 <

:

∆u 0 = f in Ω,

∂ _ν u ₀ = 0 on Γ N , u 0 = g on Γ D ,

Ω R ² open, bounded, connected ∂Ω of class C ^1,1 ,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = f ^P ¹ ^{, P} ² g ^, Γ N , Γ D relatively open, nonempty,

∂ Ω \ ^B ( P _i , δ i ) is a segment, i = 1, 2, δ i > 0 small, ( r _i , θ i ) polar coordinates at P _i , , i = 1, 2,

f 2 ^L ² ( Ω ) , g 2 ^H ^3/2 ( ∂Ω ) .

(50)

Main Hypotheses

8 <

:

∆u 0 = f in Ω,

∂ _ν u ₀ = 0 on Γ N , u 0 = g on Γ D ,

Ω R ² open, bounded, connected ∂Ω of class C ^1,1 ,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = f ^P ¹ ^{, P} ² g ^, Γ N , Γ D relatively open, nonempty,

∂ Ω \ ^B ( P i , δ i ) is a segment, i = 1, 2, δ i > 0 small,

( r _i , θ i ) polar coordinates at P _i , , i = 1, 2,

f 2 ^L ² ( Ω ) , g 2 ^H ^3/2 ( ∂Ω ) .

(51)

Main Hypotheses

8 <

:

∆u 0 = f in Ω,

∂ _ν u ₀ = 0 on Γ N , u 0 = g on Γ D ,

Ω R ² open, bounded, connected ∂Ω of class C ^1,1 ,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = f ^P ¹ ^{, P} ² g ^, Γ N , Γ D relatively open, nonempty,

∂ Ω \ ^B ( P i , δ i ) is a segment, i = 1, 2, δ i > 0 small, ( r _i , θ i ) polar coordinates at P _i , , i = 1, 2,

f 2 ^L ² ( Ω ) , g 2 ^H ^3/2 ( ∂Ω ) .

(52)

Main Hypotheses

8 <

:

∆u 0 = f in Ω,

∂ _ν u ₀ = 0 on Γ N , u 0 = g on Γ D ,

Ω R ² open, bounded, connected ∂Ω of class C ^1,1 ,

∂ Ω = _Γ N [ Γ D , Γ N \ Γ D = f ^P ¹ ^{, P} ² g ^, Γ N , Γ D relatively open, nonempty,

∂ Ω \ ^B ( P i , δ i ) is a segment, i = 1, 2, δ i > 0 small, ( r _i , θ i ) polar coordinates at P _i , , i = 1, 2,

f 2 ^L ² ( Ω ) , g 2 ^H ^3/2 ( ∂Ω ) .

(53)

8 <

:

∆u 0 = f in Ω,

∂ _ν u 0 = 0 on Γ N , u 0 = g on Γ D , Theorem

u 0 = u reg +

∑ 2 i = 1

c i ϕ i ( r i ) r _i ^1/2 sin ( θ i /2 ) , u _reg 2 ^H ² ( Ω ) , c _i 2 ^{R, with}

k ^u ^reg k H

²

( _Ω ) +

∑ 2 i = 1

j ^c ⁱ j ^c k ^f k L

²

( _Ω ) + c k ^g k H

^3/2

( _Ω ) ,

ϕ i 2 ^C ^∞ ([ 0, ∞ )) , ϕ i = 1 in [ 0, δ i /2 ] and ϕ i = 0 outside [ 0, δ i ] .

Grisvard, Kondratev, Maz’ja, Oleinik, Plamenevski¼¬, etc.

(54)

8 <

:

∆u 0 = f in Ω,

∂ _ν u 0 = 0 on Γ N , u 0 = g on Γ D , Theorem

u 0 = u reg +

∑ 2 i = 1

c i ϕ i ( r i ) r _i ^1/2 sin ( θ i /2 ) , u _reg 2 ^H ² ( Ω ) , c _i 2 ^{R, with}

k ^u ^reg k H

²

( _Ω ) +

∑ 2 i = 1

j ^c ⁱ j ^c k ^f k L

²

( _Ω ) + c k ^g k H

^3/2

( _Ω ) ,

ϕ i 2 ^C ^∞ ([ 0, ∞ )) , ϕ i = 1 in [ 0, δ i /2 ] and ϕ i = 0 outside [ 0, δ i ] .

Grisvard, Kondratev, Maz’ja, Oleinik, Plamenevski¼¬, etc.

(55)

Singularly Perturbed Neumann–Robin

Lions (73), Colli-Franzone (73, 74), Costabel & Dauge (93) 8 <

:

∆u ε = f in Ω,

∂ ν u _ε = 0 on Γ N , ε∂ _ν u _ε + u _ε = g on Γ D . What is the advantage?

u 0 r ^1/2

u _ε r log r

(56)

Singularly Perturbed Neumann–Robin

Lions (73), Colli-Franzone (73, 74), Costabel & Dauge (93) 8 <

:

∆u ε = f in Ω,

∂ ν u _ε = 0 on Γ N , ε∂ _ν u _ε + u _ε = g on Γ D . What is the advantage?

u 0 r ^1/2

u _ε r log r

(57)

Singularly Perturbed Neumann–Robin

Lions (73), Colli-Franzone (73, 74), Costabel & Dauge (93) 8 <

:

∆u ε = f in Ω,

∂ ν u _ε = 0 on Γ N , ε∂ _ν u _ε + u _ε = g on Γ D . What is the advantage?

u 0 r ^1/2

u _ε r log r

(58)

Asymptotics

8 <

:

∆u ε = f in Ω,

∂ _ν u _ε = 0 on Γ N , ε∂ ν u _ε + u _ε = g on Γ D . Theorem (Costabel & Dauge)

Under the main hypotheses, with f = 0, k ^u ^ε ^u ⁰ k L

²

( _Ω ) = O( ^{ε log ε} ) ,

k ^u ε u ₀ k H

¹⁺^s

( _Ω ) = O( ^ε ^{1/2 s} ) , s 2 ( ¹ 2 , ¹ ₂ ) ,

k ^u ε u 0 k L

²

( _Γ

_D

) = O( ^ε j ^{log ε} j ^1/2 ) .

(59)

Asymptotics

8 <

:

∆u ε = f in Ω,

∂ _ν u _ε = 0 on Γ N , ε∂ ν u _ε + u _ε = g on Γ D . Theorem (Costabel & Dauge)

Under the main hypotheses, with f = 0, k ^u ^ε ^u ⁰ k L

²

( _Ω ) = O( ^{ε log ε} ) ,

k ^u ε u ₀ k H

¹⁺^s

( _Ω ) = O( ^ε ^{1/2 s} ) , s 2 ( ¹ 2 , ¹ ₂ ) ,

k ^u ε u 0 k L

²

( _Γ

_D

) = O( ^ε j ^{log ε} j ^1/2 ) .

(60)

Asymptotics

8 <

:

∆u ε = f in Ω,

∂ _ν u _ε = 0 on Γ N , ε∂ ν u _ε + u _ε = g on Γ D . Theorem (Costabel & Dauge)

Under the main hypotheses, with f = 0, k ^u ^ε ^u ⁰ k L

²

( _Ω ) = O( ^{ε log ε} ) ,

k ^u ε u ₀ k H

¹⁺^s

( _Ω ) = O( ^ε ^{1/2 s} ) , s 2 ( ¹ 2 , ¹ ₂ ) ,

k ^u ε u 0 k L

²

( _Γ

_D

) = O( ^ε j ^{log ε} j ^1/2 ) .

(61)

Gamma Asymptotic Development

8 <

:

∆u ε = f in Ω,

∂ ν u _ε = 0 on Γ N , ε∂ _ν u _ε + u _ε = g on Γ D .

(N-R)

Solutions u _ε of (N-R) are critical points of F _ε ( v ) =

Z

Ω 1

2 jr ^v j ² + fv dx + ¹ 2ε

Z

Γ

D

( v g ) ² ds, v 2 ^H ¹ ( _Ω ) .

Study compactness of bounded sequences. Study Gamma asymptotic development.

Extend F _ε to L ² ( _Ω ) via F _ε ( v ) : = _{∞ for v} 2 ^L ² ( _Ω ) n ^H ¹ ( _Ω ) .

(62)

Gamma Asymptotic Development

8 <

:

∆u ε = f in Ω,

∂ ν u _ε = 0 on Γ N , ε∂ _ν u _ε + u _ε = g on Γ D .

(N-R)

Solutions u _ε of (N-R) are critical points of F _ε ( v ) =

Z

Ω 1

2 jr ^v j ² + fv dx + ¹ 2ε

Z

Γ

D

( v g ) ² ds, v 2 ^H ¹ ( _Ω ) . Study compactness of bounded sequences.

Study Gamma asymptotic development.

Extend F _ε to L ² ( _Ω ) via F _ε ( v ) : = _{∞ for v} 2 ^L ² ( _Ω ) n ^H ¹ ( _Ω ) .

(63)

Gamma Asymptotic Development

8 <

:

∆u ε = f in Ω,

∂ ν u _ε = 0 on Γ N , ε∂ _ν u _ε + u _ε = g on Γ D .

(N-R)

Solutions u _ε of (N-R) are critical points of F _ε ( v ) =

Z

Ω 1

2 jr ^v j ² + fv dx + ¹ 2ε

Z

Γ

D

( v g ) ² ds, v 2 ^H ¹ ( _Ω ) . Study compactness of bounded sequences.

Study Gamma asymptotic development.

Extend F _ε to L ² ( _Ω ) via F _ε ( v ) : = _{∞ for v} 2 ^L ² ( _Ω ) n ^H ¹ ( _Ω ) .

(64)

Gamma Asymptotic Development

8 <

:

∆u ε = f in Ω,

∂ ν u _ε = 0 on Γ N , ε∂ _ν u _ε + u _ε = g on Γ D .

(N-R)

Solutions u _ε of (N-R) are critical points of F _ε ( v ) =

Z

Ω 1

2 jr ^v j ² + fv dx + ¹ 2ε

Z

Γ

D

( v g ) ² ds, v 2 ^H ¹ ( _Ω ) . Study compactness of bounded sequences.

Study Gamma asymptotic development.

Extend F _ε to L ² ( _Ω ) via F _ε ( v ) : = _{∞ for v} 2 ^L ² ( _Ω ) n ^H ¹ ( _Ω ) .

(65)

Gamma Asymptotic Development of Order 0

F _ε ( v ) =

Z

Ω 1

2 jr ^v j ² + fv dx + ¹ 2ε

Z

Γ

D

( v g ) ² ds, v 2 ^H ¹ ( Ω ) .

Theorem (G. Gravina & G.L.)

Under the main hypotheses, F _ε ! ^Γ ^F ⁰ ^{in L} ² ( _Ω ) , where F ₀ ( v ) =

Z

Ω 1

2 jr ^v j ² + fv dx

if v 2 ^H ¹ ( Ω ) and v = g on Γ D , F 0 ( v ) = ∞ otherwise in L ² ( Ω ) .

min F 0 = F 0 ( u 0 ) , u 0 solution to Dirichlet–Neumann problem.

(66)

Compactness (of Order 1)

F _ε ( v ) : =

Z

Ω 1

2 jr ^v j ² + fv dx + ¹ 2ε

Z

Γ

D

( v g ) ² ds, v 2 ^H ¹ ( Ω ) .

Theorem (G. Gravina & G.L.)

Under the main hypotheses, let ε n ! ⁰ ⁺ ^{and v} ⁿ 2 ^H ¹ ( _Ω ) be such that

sup

n

F _ε

_n

( v _n ) F ₀ ( u ₀ ) + c ε n j ^{log ε} ⁿ j ^. Then (up to a subsequence)

v _n u ₀

ε ^1/2 n j ^{log ε} ⁿ j ^1/2 * w ₀ in H ¹ ( _Ω ) , v _n u ₀

ε n j ^{log ε} ⁿ j ^1/2 * v ₀ in L ² ( _Γ D ) .

(67)

Hardy’s Inequality p=N

Compactness relies on the Hardy-type inequality Theorem (Machihara, Ozawa, & Wadade, ’12) If N = 2 and u 2 ^H ¹ ( B ( 0, r )) , r > 0, then

Asymptotic Development by Gamma Convergence