Symmetrization methods and applications to PDE’s
Vincenzo Ferone
Università di Napoli “Federico II”
CIMPA-UNESCO-EGYPT School
Recent Developments in the Theory of Elliptic PDE
Alexandria, January 26-February 3, 2009
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
1 / 14
Isoperimetric inequality
Following the tradition, the history of isoperimetric inequality originates
from a question which goes back to Queen Dido.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
2 / 14
Isoperimetric inequality
Following the tradition, the history of isoperimetric inequality originates
from a question which goes back to Queen Dido.
Is it possible to find a curve in the plane with given length such that the
enclosed region has the largest area?
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
2 / 14
Isoperimetric inequality
Following the tradition, the history of isoperimetric inequality originates
from a question which goes back to Queen Dido.
Is it possible to find a curve in the plane with given length such that the
enclosed region has the largest area?
Such a problem has been addressed, for example, by Archimede, Keplero,
Galilei, Bernoulli, Schwarz, Steiner, etc.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
2 / 14
Isoperimetric inequality
Following the tradition, the history of isoperimetric inequality originates
from a question which goes back to Queen Dido.
Is it possible to find a curve in the plane with given length such that the
enclosed region has the largest area?
Such a problem has been addressed, for example, by Archimede, Keplero,
Galilei, Bernoulli, Schwarz, Steiner, etc.
Complete proofs of the fact that the circumference has the required
property have been given only at the beginning of the last century:
Hurwitz, Minkowski, Lebesgue, Blaschke, Bonnesen, etc. (plane case),
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
2 / 14
Isoperimetric inequality
Following the tradition, the history of isoperimetric inequality originates
from a question which goes back to Queen Dido.
Is it possible to find a curve in the plane with given length such that the
enclosed region has the largest area?
Such a problem has been addressed, for example, by Archimede, Keplero,
Galilei, Bernoulli, Schwarz, Steiner, etc.
Complete proofs of the fact that the circumference has the required
property have been given only at the beginning of the last century:
Hurwitz, Minkowski, Lebesgue, Blaschke, Bonnesen, etc. (plane case),
Tonelli, Schmidt, Radò, De Giorgi, etc. (n-dimensional case).
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
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Isoperimetric inequality
Let E ⊂ Rn be a measuble set of finite measure, then:
1/n
nωn |E |1−1/n ≤ Per (E ).
Equality holds true if and only if E is a ball.
(ωn denotes the measure of the unit ball in Rn )
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Isoperimetric inequality
Let E ⊂ Rn be a measuble set of finite measure, then:
1/n
nωn |E |1−1/n ≤ Per (E ).
Equality holds true if and only if E is a ball.
(ωn denotes the measure of the unit ball in Rn )
In other words, it holds:
Per (E ] ) ≤ Per (E ),
where E ] is a ball such that |E ] | = |E |.
[E. De Giorgi, 1954]
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Symmetrization methods (I)
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Isoperimetric inequality
Let E ⊂ Rn be a measuble set of finite measure, then:
1/n
nωn |E |1−1/n ≤ Per (E ).
Equality holds true if and only if E is a ball.
(ωn denotes the measure of the unit ball in Rn )
In other words, it holds:
Per (E ] ) ≤ Per (E ),
where E ] is a ball such that |E ] | = |E |.
[E. De Giorgi, 1954]
When n = 2 it reads as:
2
V. Ferone (Univ. di Napoli “Federico II”)
p
π|E | ≤ Per (E ).
Symmetrization methods (I)
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A delicate question in isoperimetric inequality concerns the equality case
and the uniqueness of the optimal domain (the ball).
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Let E be a domain in the plane with fixed perimeter, which shape does
maximize the width of E , w (E )?
(w (E ) is the smallest distance between two parallel lines such that the
strip made by them contains E )
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
4 / 14
Let E be a domain in the plane with fixed perimeter, which shape does
maximize the width of E , w (E )?
(w (E ) is the smallest distance between two parallel lines such that the
strip made by them contains E )
Per (E ) ≥ πw (E )
Equality holds if E is a set of constant width like the disk and the
“rounded” poligons as in the figure.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
4 / 14
Let E be a domain in the plane with fixed perimeter, which shape does
maximize the width of E , w (E )?
(w (E ) is the smallest distance between two parallel lines such that the
strip made by them contains E )
Per (E ) ≥ πw (E )
Equality holds if E is a set of constant width like the disk and the
“rounded” poligons as in the figure.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
4 / 14
Let E be a domain in the plane with fixed perimeter, which shape does
maximize the width of E , w (E )?
(w (E ) is the smallest distance between two parallel lines such that the
strip made by them contains E )
Per (E ) ≥ πw (E )
Equality holds if E is a set of constant width like the disk and the
“rounded” poligons as in the figure.
All the above sets have the same perimeter (for a fixed width) and the one
bounded by three arcs has minimal area.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
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Isoperimetric inequality
A brief list of approaches used to prove isoperimetric inequality.
When n = 2 one can use a completely analytical approach based on
Wirtinger inequality
Z 2π
Z 2π
2
u (t) dt ≤
(u 0 (t))2 dt
0
0
where u is such that
Z
2π
u(0) = u(2π),
u(t) dt = 0.
0
The inequality holds as an equality if and only if
u(t) = a cos t + b sin t.
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Isoperimetric inequality
A brief list of approaches used to prove isoperimetric inequality.
When n = 2 one can use a completely analytical approach based on
Wirtinger inequality
One can use the notion of Minkowski content and Brunn-Minkowski
inequality.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
5 / 14
Isoperimetric inequality
A brief list of approaches used to prove isoperimetric inequality.
When n = 2 one can use a completely analytical approach based on
Wirtinger inequality
One can use the notion of Minkowski content and Brunn-Minkowski
inequality.
It is possible to use the properties of Steiner symmetrization and the
definition of perimeter in the sense of De Giorgi.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
5 / 14
Isoperimetric inequality
A brief list of approaches used to prove isoperimetric inequality.
When n = 2 one can use a completely analytical approach based on
Wirtinger inequality
One can use the notion of Minkowski content and Brunn-Minkowski
inequality.
It is possible to use the properties of Steiner symmetrization and the
definition of perimeter in the sense of De Giorgi.
It is possible to use techniques from problem of optimal mass
transportation.
...
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
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Isoperimetric inequality
The approach based on symmetrization consists of two steps.
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Symmetrization methods (I)
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Isoperimetric inequality
The approach based on symmetrization consists of two steps.
Step 1. One proves that there exists a sets which minimizes the
perimeter among all the sets with given measure.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
6 / 14
Isoperimetric inequality
The approach based on symmetrization consists of two steps.
Step 1. One proves that there exists a sets which minimizes the
perimeter among all the sets with given measure.
Step 2. One proves that the unique set which realizes the minimum is
the ball.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
6 / 14
Isoperimetric inequality
The approach based on symmetrization consists of two steps.
Step 1. One proves that there exists a sets which minimizes the
perimeter among all the sets with given measure.
Step 2. One proves that the unique set which realizes the minimum is
the ball.
Step 1 is achieved by using the definition of perimeter given by De Giorgi
which allows to successfully use semicontinuity and compactness
arguments.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
6 / 14
Isoperimetric inequality
The approach based on symmetrization consists of two steps.
Step 1. One proves that there exists a sets which minimizes the
perimeter among all the sets with given measure.
Step 2. One proves that the unique set which realizes the minimum is
the ball.
Step 1 is achieved by using the definition of perimeter given by De Giorgi
which allows to successfully use semicontinuity and compactness
arguments.
Step 2 is achieved via Steiner symmetrization.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
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Steiner symmetrization
Steiner symmetrization of a set E ⊂ R2 with respect to the line r consists
e following the following rule.
in transforming it into another set E
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Steiner symmetrization
Steiner symmetrization of a set E ⊂ R2 with respect to the line r consists
e following the following rule.
in transforming it into another set E
For any line s perpendicular to r we have:
e ∩ s is empty if E ∩ s is empty;
E
e ∩ s is a line segment on s centered on r with the same length as
E
E ∩ s (not empty).
E
e
E
s
r
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Steiner symmetrization
Symmetrization has the following properties:
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Steiner symmetrization
Symmetrization has the following properties:
e which is symmetric with respect to r ;
it transforms E in the set E
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Symmetrization methods (I)
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Steiner symmetrization
Symmetrization has the following properties:
e which is symmetric with respect to r ;
it transforms E in the set E
area stays unchanged under symmetrization
e |;
|E | = |E
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
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Steiner symmetrization
Symmetrization has the following properties:
e which is symmetric with respect to r ;
it transforms E in the set E
area stays unchanged under symmetrization
e |;
|E | = |E
perimeter decreases under symmetrization,
e ),
Per (E ) ≥ Per (E
and, for a convex set, if the set is not symmetric the above inequality
is strict.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
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Steiner symmetrization
Symmetrization has the following properties:
e which is symmetric with respect to r ;
it transforms E in the set E
area stays unchanged under symmetrization
e |;
|E | = |E
perimeter decreases under symmetrization,
e ),
Per (E ) ≥ Per (E
and, for a convex set, if the set is not symmetric the above inequality
is strict.
In any dimension n one can use similar arguments symmetrizing with
respect to an hyperplane.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
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Isoperimetric inequalities
The term isoperimetric inequality is used in the literature in order to
identify inequalities which come from maximum or minimum problems in
which not necessarily perimeter and area are involved. For example, they
can be about geometrical quantities as diameter, general notions of
perimeter, weighted measures, or quantities from physics as capacity,
eigenvalues for boundary value problems, torsional rigidity.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
9 / 14
Isoperimetric inequalities
The term isoperimetric inequality is used in the literature in order to
identify inequalities which come from maximum or minimum problems in
which not necessarily perimeter and area are involved. For example, they
can be about geometrical quantities as diameter, general notions of
perimeter, weighted measures, or quantities from physics as capacity,
eigenvalues for boundary value problems, torsional rigidity.
The term isoperimetric is generally used in the cases where it is possible to
identify the configuration which realizes the maximum or the minimum, in
correspondence of which the inequality hold as an equality.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (I)
CIMPA-UNESCO-EGYPT
9 / 14
Isoperimetric inequalities
The term isoperimetric inequality is used in the literature in order to
identify inequalities which come from maximum or minimum problems in
which not necessarily perimeter and area are involved. For example, they
can be about geometrical quantities as diameter, general notions of
perimeter, weighted measures, or quantities from physics as capacity,
eigenvalues for boundary value problems, torsional rigidity.
The term isoperimetric is generally used in the cases where it is possible to
identify the configuration which realizes the maximum or the minimum, in
correspondence of which the inequality hold as an equality.
Typically the equality is achieved for an unique configuration which often
is the spherical one.
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St. Venant (1856)
Among all elastic beams with given area of the cross-section the one with
circular cross-section has the largest torsional rigidity.
[G. Pólya, 1948]
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St. Venant (1856)
Among all elastic beams with given area of the cross-section the one with
circular cross-section has the largest torsional rigidity.
[G. Pólya, 1948]
Lord Rayleigh (1877)
Among all the elastic membranes with constant density and given area the
circular one has the smallest principal frequency.
[G. Faber, 1923], [E. Krahn, 1924], [L. Tonelli, 1930]
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St. Venant (1856)
Among all elastic beams with given area of the cross-section the one with
circular cross-section has the largest torsional rigidity.
[G. Pólya, 1948]
Lord Rayleigh (1877)
Among all the elastic membranes with constant density and given area the
circular one has the smallest principal frequency.
[G. Faber, 1923], [E. Krahn, 1924], [L. Tonelli, 1930]
Poincaré (1903)
Among all the bodies with given volume the ball has the smallest
electrostatic capacity.
[G. Szegö, 1930]
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Symmetrization of a function
Let Ω be an open bounded set of Rn and let f : Ω → R be a measurable
function.
Schwarz symmetrization of f (spherically symmetric decreasing
rearrangement), denoted by f ] , is such that its level sets are balls of Rn
centered at the origin having the same measure of the corresponding level
sets of f .
In other words, we have:
{x ∈ Ω] : f ] (x) > t} = {x ∈ Ω : |f (x)| > t}]
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t ≥ 0.
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Symmetrization of a function
f]
f
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Symmetrization of a function
f]
f
The decreasing rearrangement of f is the function f ∗ of one variable such
that
f ] (x) = f ∗ (ωn |x|n ),
x ∈ Ω] .
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Some properties of rearrangements
It is not difficult to see thet, roughly speaking,
f ∗ = µ−1
f
where µf is the distribution function of f
µf (t) = |{x ∈ Ω : |f (x)| > t}|,
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t ≥ 0.
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Some properties of rearrangements
It is not difficult to see thet, roughly speaking,
f ∗ = µ−1
f
where µf is the distribution function of f
µf (t) = |{x ∈ Ω : |f (x)| > t}|,
t ≥ 0.
f , f ] and f ∗ are equimeasurable, that is,
µf (t) = µf ] (t) = µf ∗ (t),
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t ≥ 0.
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Some properties of rearrangements
It is not difficult to see thet, roughly speaking,
f ∗ = µ−1
f
where µf is the distribution function of f
µf (t) = |{x ∈ Ω : |f (x)| > t}|,
t ≥ 0.
f , f ] and f ∗ are equimeasurable, that is,
µf (t) = µf ] (t) = µf ∗ (t),
kf kLp (Ω) = kf ] kLp (Ω] ) = kf ∗ kLp (0,|Ω|)
V. Ferone (Univ. di Napoli “Federico II”)
t ≥ 0.
1≤p≤∞
Symmetrization methods (I)
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Some properties of rearrangements
Z
Z
|f | dx ≤
E
|E |
f ∗ (s) ds,
E ⊂Ω
0
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Some properties of rearrangements
Z
Z
|E |
|f | dx ≤
E
f ∗ (s) ds,
Z
Z
µf (t)
|f | dx =
|f |>t
E ⊂Ω
0
f ∗ (s) ds
0
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Some properties of rearrangements
Z
Z
|E |
|f | dx ≤
E
f ∗ (s) ds,
Z
Z
µf (t)
|f | dx =
|f |>t
E ⊂Ω
0
f ∗ (s) ds
0
Hardy-Littlewood inequality:
Z
Z
|f g | dx ≤
Ω
V. Ferone (Univ. di Napoli “Federico II”)
|Ω|
f ∗ (s)g ∗ (s) ds
0
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Some properties of rearrangements
Z
Z
|E |
|f | dx ≤
E
f ∗ (s) ds,
E ⊂Ω
0
Z
µf (t)
Z
|f | dx =
|f |>t
f ∗ (s) ds
0
Hardy-Littlewood inequality:
Z
Z
|f g | dx ≤
Ω
Z
s
∗
Z
s
f (r ) dr ≤
0
|Ω|
f ∗ (s)g ∗ (s) ds
0
g ∗ (r ) dr ,
s ∈ (0, |Ω|)
0
⇓
kf kLp (Ω) ≤ kg kLp (Ω)
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1≤p≤∞
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