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
PDEs
Conclusions
Extras
Generalized functions in the theory of PDEsand in the Calculus of Variations
Emanuele Bottazzi, University of Trento
IMSE 2016, July 26, 2016
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
PDEs
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〉.
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
PDEs
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).
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
PDEs
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).
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
PDEs
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).
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
PDEs
Conclusions
Extras
The basic idea of nonstandard analysis
Image from “Elementary Calculus: An Infinitesimal Approach”, c© 2000 by H. Jerome Keisler.
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
PDEs
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”.
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
PDEs
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”.
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
PDEs
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.
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
PDEs
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.
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
PDEs
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.
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
PDEs
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).
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
PDEs
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).
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
PDEs
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.
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
PDEs
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)
ε.
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
PDEs
Conclusions
Extras
Generalized derivative and distributional derivative
Proposition
The following diagram commutes:
D ′Xd−→ D ′X
π ↓ ↓ πD(R)
∂−→ D(R)
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
PDEs
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〉 =
∫Ω
(∫Rfdνu(x)
)ϕ(x)dx
for all f ∈ C 0b (R) and for all ϕ ∈ D(Ω).
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
PDEs
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) =
(1
M
M−1∑i=0
f (u(iε))
).
for all f ∈ C 0b (R).
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
PDEs
Conclusions
Extras
Calculus of variations
Consider the problem of minimizing the functional
J(u) =
∫ 1
0
(∫ x
0u(t)dt
)2
+ (u2(x)− 1)2dx
with u ∈ L2([0, 1]).
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
PDEs
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.
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
PDEs
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)
)2
+ (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.
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
PDEs
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)
)2
+ (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.
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
PDEs
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) =
(1
2(∗f (u(0)) + ∗f (u(ε)))
)
=1
2(f (1) + f (−1))
That is, ν(x) =1
2(δ1 + δ−1).
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
PDEs
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) =
(1
2(∗f (u(0)) + ∗f (u(ε)))
)=
1
2(f (1) + f (−1))
That is, ν(x) =1
2(δ1 + δ−1).
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
PDEs
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) =
(1
2(∗f (u(0)) + ∗f (u(ε)))
)=
1
2(f (1) + f (−1))
That is, ν(x) =1
2(δ1 + δ−1).
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
PDEs
Conclusions
Extras
The Young Measure solution
We can compute
(ν(x)2 − 1)2 =
∫R
(τ2 − 1)2dν(x)
=1
2(12 − 1)2 +
1
2((−1)2 − 1)2 = 0
and ∫ x
0ν(t)dt =
∫ x
0
∫Rτdν(t)dt = 0.
Hence, ν is a minimizer for J in the sense of Young measures.
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
PDEs
Conclusions
Extras
The Young Measure solution
We can compute
(ν(x)2 − 1)2 =
∫R
(τ2 − 1)2dν(x)
=1
2(12 − 1)2 +
1
2((−1)2 − 1)2 = 0
and ∫ x
0ν(t)dt =
∫ x
0
∫Rτdν(t)dt = 0.
Hence, ν is a minimizer for J in the sense of Young measures.
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
PDEs
Conclusions
Extras
The Young Measure solution
We can compute
(ν(x)2 − 1)2 =
∫R
(τ2 − 1)2dν(x)
=1
2(12 − 1)2 +
1
2((−1)2 − 1)2 = 0
and ∫ x
0ν(t)dt =
∫ x
0
∫Rτdν(t)dt = 0.
Hence, ν is a minimizer for J in the sense of Young measures.
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
PDEs
Conclusions
Extras
The Young Measure solution
We can compute
(ν(x)2 − 1)2 =
∫R
(τ2 − 1)2dν(x)
=1
2(12 − 1)2 +
1
2((−1)2 − 1)2 = 0
and ∫ x
0ν(t)dt =
∫ x
0
∫Rτdν(t)dt = 0.
Hence, ν is a minimizer for J in the sense of Young measures.
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
PDEs
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.
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
PDEs
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.
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
PDEs
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.
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
PDEs
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.
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
PDEs
Conclusions
Extras
The generalized function formulation
A generalized formulation of the ill-posed PDE is obtained bydiscretizing the Laplacian:
∆Xφ(u(x , t)) =k∑
i=1
didi (φ(u(x − εei )))
=
k∑i=1
φ(u(x + eiε, t))− 2φ(u(xε, t)) + φ(u(x − eiε, t))
ε2
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
PDEs
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.
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
PDEs
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.
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
PDEs
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.
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
PDEs
Conclusions
Extras
Perspectives for future research
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.
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
PDEs
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,
then
〈H, ψ〉 =
∫Rh(x)ψ(x)dx .
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
PDEs
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,
then
〈H, ψ〉 =
∫Rh(x)ψ(x)dx .
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
PDEs
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 .
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
PDEs
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 .
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
PDEs
Conclusions
Extras
A naive calculation
In physics, it would be convenient to perform the calculation∫R
(Hn−Hm)H ′dx
=Hn+1
n + 1
∣∣∣∣+∞−∞− Hm+1
m + 1
∣∣∣∣+∞−∞
=1
n + 1− 1
m + 1.
However, the above calculation is not acceptable in the senseof distributions.
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
PDEs
Conclusions
Extras
A naive calculation
In physics, it would be convenient to perform the calculation∫R
(Hn−Hm)H ′dx =Hn+1
n + 1
∣∣∣∣+∞−∞− Hm+1
m + 1
∣∣∣∣+∞−∞
=1
n + 1− 1
m + 1.
However, the above calculation is not acceptable in the senseof distributions.
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
PDEs
Conclusions
Extras
A naive calculation
In physics, it would be convenient to perform the calculation∫R
(Hn−Hm)H ′dx =Hn+1
n + 1
∣∣∣∣+∞−∞− Hm+1
m + 1
∣∣∣∣+∞−∞
=1
n + 1− 1
m + 1.
However, the above calculation is not acceptable in the senseof distributions.
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
PDEs
Conclusions
Extras
A naive calculation
In physics, it would be convenient to perform the calculation∫R
(Hn−Hm)H ′dx =Hn+1
n + 1
∣∣∣∣+∞−∞− Hm+1
m + 1
∣∣∣∣+∞−∞
=1
n + 1− 1
m + 1.
However, the above calculation is not acceptable in the senseof distributions.
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
PDEs
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 .
Then
dh(iε) =
0 if i ≤ 0 or M ≤ i1/Mε 0 ≤ i ≤ M.
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
PDEs
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 .
Then
dh(iε) =
0 if i ≤ 0 or M ≤ i1/Mε 0 ≤ i ≤ M.
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
PDEs
Conclusions
Extras
A less naive calculation
We can perform the calculation
ε∑x∈X
(hn(x)− hm(x))dh(x)
=1
M
M∑i=0
(i
M
)n
−(
i
M
)m
≈∫ 1
0xn − xmdx =
1
n + 1+
1
m + 1.
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
PDEs
Conclusions
Extras
A less naive calculation
We can perform the calculation
ε∑x∈X
(hn(x)− hm(x))dh(x) =1
M
M∑i=0
(i
M
)n
−(
i
M
)m
≈∫ 1
0xn − xmdx =
1
n + 1+
1
m + 1.
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
PDEs
Conclusions
Extras
A less naive calculation
We can perform the calculation
ε∑x∈X
(hn(x)− hm(x))dh(x) =1
M
M∑i=0
(i
M
)n
−(
i
M
)m
≈∫ 1
0xn − xmdx
=1
n + 1+
1
m + 1.
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
PDEs
Conclusions
Extras
A less naive calculation
We can perform the calculation
ε∑x∈X
(hn(x)− hm(x))dh(x) =1
M
M∑i=0
(i
M
)n
−(
i
M
)m
≈∫ 1
0xn − xmdx =
1
n + 1+
1
m + 1.
Top Related