Generalized functions in the theory of PDEs and in the ...bottazzi/stuff/padova.pdf · Generalized...

Generalizedfunctions in

the theory ofPDEs and inthe Calculusof Variations

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Coherence withthe distributions

Coherence withthe Youngmeasures

Applications

Calculus ofvariations

Conclusions

Extras

Generalized functions in the theory of PDEsand in the Calculus of Variations

Emanuele Bottazzi, University of Trento

IMSE 2016, July 26, 2016

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Distributions

Definition (Distribution)

A distribution is a linear continuous functional T : D ′ → R.

Definition (Distributional derivative)

The derivative ∂T of T is given by

〈∂T , ϕ〉 = −〈T , ∂ϕ〉.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Young measures

Definition (Young measure)

A Young measure is a measurable function ν : Ω→ P(R),where P(R) is the set of probability measures over R.

Iff ∈ C 0

b (R), then the “composition” f (ν) is defined by

f (ν(x)) =

∫Rf (y)dν(x).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Young measures

Definition (Young measure)

A Young measure is a measurable function ν : Ω→ P(R),where P(R) is the set of probability measures over R. Iff ∈ C 0

b (R), then the “composition” f (ν) is defined by

f (ν(x)) =

∫Rf (y)dν(x).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The main theorem of Young measures

Theorem

Let znn∈N be a bounded sequence of L∞(Ω) functions. Thenthere exists a Young measure ν such that

f (zn)∗ f (ν)

for all f ∈ C 0b (R).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The basic idea of nonstandard analysis

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Nonstandard analysis in a nutshell

Let P(X ) be the power set of X , and define

V(X ) =⋃n∈NPn(X ).

Definition (Nonstandard universe)

A nonstandard universe is a triple (V(R),V(∗R), ∗) such that:∗ : V(R)→ V(∗R);∗ maps R properly into ∗R (i.e. R 6= ∗R);∗ preserves “elementary properties”.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Nonstandard analysis in a nutshell

Let P(X ) be the power set of X , and define

V(X ) =⋃n∈NPn(X ).

Definition (Nonstandard universe)

A nonstandard universe is a triple (V(R),V(∗R), ∗) such that:∗ : V(R)→ V(∗R);∗ maps R properly into ∗R (i.e. R 6= ∗R);∗ preserves “elementary properties”.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The field ∗R

Proposition

∗R is an ordered field that properly extends R. Moreover, thereexists a nonzero infinitesimal x ∈ ∗R.

Proposition

Every finite number x ∈ ∗R can be written in a unique way as

x = x + ε

where x ∈ R and ε = x − x is infinitesimal.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The field ∗R

Proposition

∗R is an ordered field that properly extends R. Moreover, thereexists a nonzero infinitesimal x ∈ ∗R.

Proposition

Every finite number x ∈ ∗R can be written in a unique way as

x = x + ε

where x ∈ R and ε = x − x is infinitesimal.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Generalized functions

Definition (Generalized functions)

Let N ∈ ∗N be infinite and let ε = N−1. We define

X = nε : n ∈ ∗Z and − N2 ≤ n ≤ N2.

The set of generalized functions is the set

∗RX = f : X→ ∗R.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The set of bounded generalized functions

Definition (Inner product)

Let f , g ∈ ∗RX. We define

〈f , g〉X = ε∑x∈X

f (x)g(x).

Definition (Bounded generalized functions)

D ′X = f ∈ ∗RX : 〈f , ∗φ〉X is finite for all φ ∈ D ′(R).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The set of bounded generalized functions

Definition (Inner product)

Let f , g ∈ ∗RX. We define

〈f , g〉X = ε∑x∈X

f (x)g(x).

Definition (Bounded generalized functions)

D ′X = f ∈ ∗RX : 〈f , ∗φ〉X is finite for all φ ∈ D ′(R).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Generalized functions and distributions

Theorem

We define a projection π : D ′X → D ′(R) by

〈π(f ), ψ〉 = 〈f , ∗ψ〉X.

π is linear and surjective.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Generalized derivative

Definition (Generalized derivative)

For a function f ∈ ∗RX, we define the grid derivative df as

df (x) =f (x + ε)− f (x)

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Generalized derivative and distributional derivative

Proposition

The following diagram commutes:

D ′Xd−→ D ′X

π ↓ ↓ πD(R)

∂−→ D(R)

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Generalized functions and Young measures

Theorem

If u ∈ D ′X is finite for all x ∈ X, there exists a Young measureνu : Ω→ M(R) such that

〈∗f (u), ∗ϕ〉 =

(∫Rfdνu(x)

)ϕ(x)dx

for all f ∈ C 0b (R) and for all ϕ ∈ D(Ω).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Periodic generalized functions induce homogeneousYoung measures

Corollary

If u ∈ D ′X is finite and periodic of period Mε with (Mε) = 0,then the Young measure ν associated to u is homogeneous, and∫

Rfdν(x) =

M−1∑i=0

f (u(iε))

for all f ∈ C 0b (R).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Calculus of variations

Consider the problem of minimizing the functional

J(u) =

(∫ x

0u(t)dt

+ (u2(x)− 1)2dx

with u ∈ L2([0, 1]).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Calculus of variations

Letu1 = χ[k,k+1/2) − χ[k+1/2,k+1), k ∈ Z

and let un : [0, 1]→ R be defined by

un(x) = u1(nx).

Then unn∈N is a minimizing sequence for J, but J has nominimum.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The generalized function solution

In the setting of generalized functions, the above problembecomes

J(u) = ε

N∑n=0

(ε n∑i=0

u(iε)

+ (u(nε)2 − 1)2

The functions unn∈N are still a minimizing sequence for J,and it can be proved that the function uN/2 defined by

uN/2(nε) = (−1)n

is a minimizer for J.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The generalized function solution

In the setting of generalized functions, the above problembecomes

J(u) = ε

N∑n=0

(ε n∑i=0

u(iε)

+ (u(nε)2 − 1)2

The functions unn∈N are still a minimizing sequence for J,and it can be proved that the function uN/2 defined by

uN/2(nε) = (−1)n

is a minimizer for J.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

From the generalized function solution to theYoung measure solution

Since uN/2 is finite for all x ∈ [0, 1]X, it has a correspondingYoung measure ν : Ω→ M(R). For all f ∈ C 0

b (R), ν satisfies∫Rf (y)dν(x) =

2(∗f (u(0)) + ∗f (u(ε)))

2(f (1) + f (−1))

That is, ν(x) =1

2(δ1 + δ−1).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

2(∗f (u(0)) + ∗f (u(ε)))

2(f (1) + f (−1))

That is, ν(x) =1

2(δ1 + δ−1).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

2(∗f (u(0)) + ∗f (u(ε)))

2(f (1) + f (−1))

That is, ν(x) =1

2(δ1 + δ−1).

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The Young Measure solution

We can compute

(ν(x)2 − 1)2 =

(τ2 − 1)2dν(x)

2(12 − 1)2 +

2((−1)2 − 1)2 = 0

and ∫ x

0ν(t)dt =

∫Rτdν(t)dt = 0.

Hence, ν is a minimizer for J in the sense of Young measures.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

We can compute

(ν(x)2 − 1)2 =

(τ2 − 1)2dν(x)

2(12 − 1)2 +

2((−1)2 − 1)2 = 0

and ∫ x

0ν(t)dt =

∫Rτdν(t)dt = 0.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

We can compute

(ν(x)2 − 1)2 =

(τ2 − 1)2dν(x)

2(12 − 1)2 +

2((−1)2 − 1)2 = 0

and ∫ x

0ν(t)dt =

∫Rτdν(t)dt = 0.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

We can compute

(ν(x)2 − 1)2 =

(τ2 − 1)2dν(x)

2(12 − 1)2 +

2((−1)2 − 1)2 = 0

and ∫ x

0ν(t)dt =

∫Rτdν(t)dt = 0.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

An ill-posed PDE

Consider the Neumann initial value problem

ut(x , t) = ∆φ(u(x , t)), x ∈ Ω ⊂ Rk , t ≥ 0u(x , 0) = u0(x), x ∈ Ω.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

An ill-posed PDE

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

An ill-posed PDE

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

An ill-posed PDE

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The generalized function formulation

A generalized formulation of the ill-posed PDE is obtained bydiscretizing the Laplacian:

∆Xφ(u(x , t)) =k∑

didi (φ(u(x − εei )))

k∑i=1

φ(u(x + eiε, t))− 2φ(u(xε, t)) + φ(u(x − eiε, t))

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Existence and uniqueness of the solution for thegeneralized problem

The generalized formulation then becomes a system of ODEs:

ut(x , t) = ∆Xφ(u(x , t)), x ∈ ΩX ⊂ Xk , t ≥ 0u(x , 0) = u0(x), x ∈ ΩX,

Theorem

The above problem with Neumann boundary conditions has aunique global solution u.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Existence and uniqueness of the solution for thegeneralized problem

The generalized formulation then becomes a system of ODEs:

ut(x , t) = ∆Xφ(u(x , t)), x ∈ ΩX ⊂ Xk , t ≥ 0u(x , 0) = u0(x), x ∈ ΩX,

Theorem

The above problem with Neumann boundary conditions has aunique global solution u.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Properties of the generalized solution

Theorem

The generalized solution satisfies the following properties:

the mass of the solution is preserved, i.e.∑x∈ΩX

u(x , t) =∑x∈ΩX

u0(x) for all t ≥ 0;

there is an asymptotically stable equilibrium for u.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

Perspectives for future research

Formulation of physical phenomena in ∗RX.

Study of classically ill-posed PDEs and variationalproblems in ∗RX.

Study of the relation between ∗RX, Colombeau’s algebrasand Todorov’s generalized functions.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The Heaviside distribution

The Heaviside distribution H is defined by

〈H, ψ〉 =

∫ +∞

0ψ(x)dx .

If h is the function:

h(x) =

0 if x ≤ 01 if x ≥ 0,

〈H, ψ〉 =

∫Rh(x)ψ(x)dx .

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The Heaviside distribution

The Heaviside distribution H is defined by

〈H, ψ〉 =

∫ +∞

0ψ(x)dx .

If h is the function:

h(x) =

0 if x ≤ 01 if x ≥ 0,

〈H, ψ〉 =

∫Rh(x)ψ(x)dx .

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The derivative of the Heaviside distribution

The derivative of the Heaviside distribution is defined by

〈H ′, ψ〉 = −〈H, ψ′〉 = −∫ +∞

0ψ′(x)dx = ψ(0).

This distribution is called the Dirac distribution centered at 0.

There is no function f : R→ R such that

〈δ0, ψ〉 =

∫Rf (x)ψ(x)dx .

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

The derivative of the Heaviside distribution

The derivative of the Heaviside distribution is defined by

〈H ′, ψ〉 = −〈H, ψ′〉 = −∫ +∞

0ψ′(x)dx = ψ(0).

This distribution is called the Dirac distribution centered at 0.There is no function f : R→ R such that

〈δ0, ψ〉 =

∫Rf (x)ψ(x)dx .

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

A naive calculation

In physics, it would be convenient to perform the calculation∫R

(Hn−Hm)H ′dx

∣∣∣∣+∞−∞− Hm+1

∣∣∣∣+∞−∞

n + 1− 1

m + 1.

However, the above calculation is not acceptable in the senseof distributions.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

A naive calculation

(Hn−Hm)H ′dx =Hn+1

∣∣∣∣+∞−∞− Hm+1

∣∣∣∣+∞−∞

n + 1− 1

m + 1.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

A naive calculation

∣∣∣∣+∞−∞− Hm+1

∣∣∣∣+∞−∞

n + 1− 1

m + 1.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

A naive calculation

∣∣∣∣+∞−∞− Hm+1

∣∣∣∣+∞−∞

n + 1− 1

m + 1.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

A nonstandard Heaviside function

Let M ∈ ∗N be an infinite hypernatural satisfying Mε ≈ 0, anddefine

h(iε) =

0 if i ≤ 0i/M if 0 ≤ i ≤ M1 if M ≤ i .

dh(iε) =

0 if i ≤ 0 or M ≤ i1/Mε 0 ≤ i ≤ M.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

A nonstandard Heaviside function

Let M ∈ ∗N be an infinite hypernatural satisfying Mε ≈ 0, anddefine

h(iε) =

0 if i ≤ 0i/M if 0 ≤ i ≤ M1 if M ≤ i .

dh(iε) =

0 if i ≤ 0 or M ≤ i1/Mε 0 ≤ i ≤ M.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

A less naive calculation

We can perform the calculation

ε∑x∈X

(hn(x)− hm(x))dh(x)

M∑i=0

≈∫ 1

0xn − xmdx =

n + 1+

m + 1.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

ε∑x∈X

(hn(x)− hm(x))dh(x) =1

M∑i=0

≈∫ 1

0xn − xmdx =

n + 1+

m + 1.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

ε∑x∈X

M∑i=0

≈∫ 1

0xn − xmdx

n + 1+

m + 1.

EmanueleBottazzi

Introduction

Distributions

Young measures

Nonstandardanalysis

The mainresults

Applications

Conclusions

Extras

ε∑x∈X

M∑i=0

≈∫ 1

0xn − xmdx =

n + 1+

m + 1.

Generalized functions in the theory of PDEs and in the ...bottazzi/stuff/padova.pdf · Generalized...

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