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 (III)
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A classical question
Let E ⊂ R2 be a bounded subset with finite perimeter. If the difference
Per (E )2 − 4π|E |
“is small”,
is it possible to say that E is “close” to a disk?
[F. Bernstein, 1905], [T. Bonnesen, 1921, 1924, 1929], [R. Osserman, 1979], [B.
Fuglede, 1989], [H. Groemer - R. Schneider, 1991], [R.R. Hall, 1992], [R.R. Hall W.K. Hayman, 1993], [N. Fusco - F. Maggi - A. Pratelli, 2007]
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Bonnesen type inequalities
In 1924 Bonnesen has proved the following inequality concerning convex
sets E ∈ R2 :
Per (E )2 − 4π|E | ≥ 4πd 2 ,
where d is the width of the smallest circular annulus which contains ∂E .
The constant 4π which appears on the right-hand side is sharp.
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Bonnesen type inequalities
In 1924 Bonnesen has proved the following inequality concerning convex
sets E ∈ R2 :
Per (E )2 − 4π|E | ≥ 4πd 2 ,
where d is the width of the smallest circular annulus which contains ∂E .
The constant 4π which appears on the right-hand side is sharp.
It is evident that the above inequality implies isoperimetric inequality and
gives the isoperimetric property of the disk.
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Two definitions
Fraenkel asymmetry of a set
α(E ) = minn
x∈R
|E ∆Br (x)|
|E |
where Br (x) denotes the ball centered at the point x of radius r , such that
|Br (x)| = |E |,
(1)
and, as usual, E ∆F stands for the symmetric difference of two sets E and
F.
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Two definitions
Fraenkel asymmetry of a set
α(E ) = minn
x∈R
|E ∆Br (x)|
|E |
where Br (x) denotes the ball centered at the point x of radius r , such that
|Br (x)| = |E |,
(1)
and, as usual, E ∆F stands for the symmetric difference of two sets E and
F.
Isoperimetric deficit of a set
If we choose r in such a way that (1) holds true, we put:
∆P(E ) =
V. Ferone (Univ. di Napoli “Federico II”)
Per (E ) − Per (Br )
Per (Br )
Symmetrization methods (III)
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Question
The isoperimetric inequality can be written as
∆P(E ) ≥ 0.
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Question
The isoperimetric inequality can be written as
∆P(E ) ≥ 0.
Is it possible to estimate α(E ) in terms of ∆P(E )?
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Question
The isoperimetric inequality can be written as
∆P(E ) ≥ 0.
Is it possible to estimate α(E ) in terms of ∆P(E )?
Is it possible to obtain an inequality in the form
∆P(E ) ≥ f (α(E ))
for some function f ≥ 0?
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Quantitative isoperimetric inequality
Theorem There exists a constant c(n) such that for every Borel set
E ⊂ Rn with 0 < |E | < ∞ it holds:
∆P(E ) ≥ c(n)α(E )2 ,
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Quantitative isoperimetric inequality
Theorem There exists a constant c(n) such that for every Borel set
E ⊂ Rn with 0 < |E | < ∞ it holds:
∆P(E ) ≥ c(n)α(E )2 ,
1/n
or, recalling that Per (Br ) = nωn |E |1−1/n ,
1/n
Per (E ) ≥ nωn |E |1−1/n (1 + c(n)α(E )2 ).
[B. Fuglede, 1989], [R.R. Hall, 1992], [R.R. Hall - W.K. Hayman, 1993], [N.
Fusco - F. Maggi - A. Pratelli, 2007]
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A quantitative isoperimetric inequality in the plane
Theorem If E ⊂ R2 is convex it holds:
∆P(E ) ≥
π
α(E )2 − cα(E )3 ,
8(4 − π)
where c is an absolute constant, while the constant
π
is sharp.
8(4 − π)
[R.R. Hall - W.K. Hayman, 1993]
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A quantitative isoperimetric inequality in the plane
Theorem If E ⊂ R2 is convex it holds:
∆P(E ) ≥
π
α(E )2 − cα(E )3 ,
8(4 − π)
where c is an absolute constant, while the constant
π
is sharp.
8(4 − π)
[R.R. Hall - W.K. Hayman, 1993]
Remark The proof is based on an inequality concerning the coefficients of
the Fourier series of the support function of the set E . The constant c is
not known and it could also be zero.
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Applications of the quantitative isoperimetric inequality
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Applications of the quantitative isoperimetric inequality
Stability results for Pólya-Szegö inequalities
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Applications of the quantitative isoperimetric inequality
Stability results for Pólya-Szegö inequalities
Stability results for Sobolev or Hardy inequalities
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Applications of the quantitative isoperimetric inequality
Stability results for Pólya-Szegö inequalities
Stability results for Sobolev or Hardy inequalities
Stability results for comparison inequalities for solutions of PDE’s
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Applications of the quantitative isoperimetric inequality
Stability results for Pólya-Szegö inequalities
Stability results for Sobolev or Hardy inequalities
Stability results for comparison inequalities for solutions of PDE’s
Stability results for radial symmetry of solutions to overdetermined
boundary value problems
Aftalion, Bianchi, Brandolini, Busca, Cianchi, Egnell, Esposito, A. Ferone, Fusco,
Henrot, Maggi, Nitsch, Pratelli, Reichel, Rosset, Salani, C. Trombetti, . . .
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An isoperimetric inequality for the deficit
Theorem Let E ⊂ R2 be convex. There exists an unique set Eb in a
certain class C such that
b)
α(E ) = α(E
b ).
e ∆P(E ) ≥ ∆P(E
[A. Alvino - V.F. - C. Nitsch, preprint]
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An isoperimetric inequality for the deficit
Theorem Let E ⊂ R2 be convex. There exists an unique set Eb in a
certain class C such that
b)
α(E ) = α(E
b ).
e ∆P(E ) ≥ ∆P(E
[A. Alvino - V.F. - C. Nitsch, preprint]
As a consequence of this result it follows that there exists a function f
such that for every convex set E ⊂ R2 it holds
∆P(E ) ≥ f (α(E )).
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An isoperimetric inequality for the deficit
Theorem Let E ⊂ R2 be convex. There exists an unique set Eb in a
certain class C such that
b)
α(E ) = α(E
b ).
e ∆P(E ) ≥ ∆P(E
[A. Alvino - V.F. - C. Nitsch, preprint]
As a consequence of this result it follows that there exists a function f
such that for every convex set E ⊂ R2 it holds
∆P(E ) ≥ f (α(E )).
Furthemore, for any admissible value of α, there exists an unique convex
set for which equality holds in the above inequality.
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The family C
We denote by C the family of convex sets satisfying the following
properties:
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The family C
We denote by C the family of convex sets satisfying the following
properties:
they are symmetric with respect to two orthogonal axes
(2-symmetric);
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The family C
We denote by C the family of convex sets satisfying the following
properties:
they are symmetric with respect to two orthogonal axes
(2-symmetric);
their boundary is of class C 1 and it is made of 4 arcs of circumference
(two of them can degenerate in parallel segments);
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The family C
We denote by C the family of convex sets satisfying the following
properties:
they are symmetric with respect to two orthogonal axes
(2-symmetric);
their boundary is of class C 1 and it is made of 4 arcs of circumference
(two of them can degenerate in parallel segments);
if the arcs of the boundary are proper arcs, then two of them are
internal to the optimal circle Br and the other two are external.
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The family C
We denote by C the family of convex sets satisfying the following
properties:
they are symmetric with respect to two orthogonal axes
(2-symmetric);
their boundary is of class C 1 and it is made of 4 arcs of circumference
(two of them can degenerate in parallel segments);
if the arcs of the boundary are proper arcs, then two of them are
internal to the optimal circle Br and the other two are external.
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A set in the family C
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The family C
The family C
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α(θ), ∆P(θ), ρ(θ) =
∆P(θ)
α(θ)2
0 ≤ θ ≤ π/4
ρ
π
8(4−π)
0.405585
θ
∆P
α
arctan(π/4)
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α(θ), ∆P(θ), ρ(θ) =
∆P(θ)
α(θ)2
0 ≤ θ ≤ π/4
ρ
0.457474
0.405585
θ
∆P
α
arctan(π/4)
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α(θ), ∆P(θ), ρ(θ) =
∆P(θ)
α(θ)2
0 ≤ θ ≤ π/4
ρ
π
8(4−π)
0.405585
θ
∆P
α
arctan(π/4)
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α(θ), ∆P(θ), ρ(θ) =
∆P(θ)
α(θ)2
0 ≤ θ ≤ π/4
ρ
π
8(4−π)
0.405585
θ'0.575026
∆P
α
arctan(π/4)
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α(θ), ∆P(θ), ρ(θ) =
∆P(θ)
α(θ)2
0 ≤ θ ≤ π/4
ρ
π
8(4−π)
0.405585
θ
∆P
α
arctan(π/4)
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General strategy
Let E ⊂ R2 be a generic convex set. We will transform it in such a way
that at each step of the transformation the following properties are
fulfilled:
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General strategy
Let E ⊂ R2 be a generic convex set. We will transform it in such a way
that at each step of the transformation the following properties are
fulfilled:
the asymmetry index does not change;
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General strategy
Let E ⊂ R2 be a generic convex set. We will transform it in such a way
that at each step of the transformation the following properties are
fulfilled:
the asymmetry index does not change;
the isoperimetric deficit does not increase;
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General strategy
Let E ⊂ R2 be a generic convex set. We will transform it in such a way
that at each step of the transformation the following properties are
fulfilled:
the asymmetry index does not change;
the isoperimetric deficit does not increase;
the symmetry properties “improve”.
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Steiner symmetrization cannot be used as the first step
Let us consider a set and the circle having the same measure.
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Steiner symmetrization cannot be used as the first step
Let us consider the circle in the optimal position.
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Steiner symmetrization cannot be used as the first step
Let us symmetrize the set with respect to the vertical axis.
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Schwarz symmetrization for functions of one variable
||u||p = ||u ] ||p ,
||u 0 ||∞ ≥ ||(u ] )0 ||∞
if #{u = t} ≥ 2
u
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u]
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Circular symmetrization
E?
E
π
0
π
0
r ] (θ)
r (θ)
−π
0
π
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θ
−π
0
Symmetrization methods (III)
π
θ
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Circular symmetrization
E?
E
π
r ]− (θ)
r (θ)
−π
0
0
π
0
π
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θ
−π
r ]+ (θ)
0
Symmetrization methods (III)
π
θ
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The reduction to a 2-symmetric set
b such that:
Let E ⊂ R2 be convex. We build a 2-symmetric set E
b ),
α(E ) = α(E
V. Ferone (Univ. di Napoli “Federico II”)
b ).
∆P(E ) ≥ ∆P(E
Symmetrization methods (III)
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(2)
18 / 37
The reduction to a 2-symmetric set
b such that:
Let E ⊂ R2 be convex. We build a 2-symmetric set E
b ),
α(E ) = α(E
b ).
∆P(E ) ≥ ∆P(E
(2)
For a given convex set E we consider a circle B (|B| = |E |) which realizes
the minimum in Fraenkel asymmetry. From now on such a circle will be
fixed and its center will be chosen as the origin of reference axes.
E
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The reduction to a 2-symmetric set
b such that:
Let E ⊂ R2 be convex. We build a 2-symmetric set E
b ),
α(E ) = α(E
b ).
∆P(E ) ≥ ∆P(E
(2)
For a given convex set E we consider a circle B (|B| = |E |) which realizes
the minimum in Fraenkel asymmetry. From now on such a circle will be
fixed and its center will be chosen as the origin of reference axes.
B
E
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The reduction to a 2-symmetric set
b such that:
Let E ⊂ R2 be convex. We build a 2-symmetric set E
b ),
α(E ) = α(E
b ).
∆P(E ) ≥ ∆P(E
(2)
For a given convex set E we consider a circle B (|B| = |E |) which realizes
the minimum in Fraenkel asymmetry. From now on such a circle will be
fixed and its center will be chosen as the origin of reference axes.
The simplest case is shown in the
figure: the boundary of E is cut by
the boundary of B exactly in four
pieces, two pieces are external to B,
two pieces are internal.
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B
E
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The reduction to a 2-symmetric set
b such that:
Let E ⊂ R2 be convex. We build a 2-symmetric set E
b ),
α(E ) = α(E
b ).
∆P(E ) ≥ ∆P(E
(2)
B
Using the fact that B realizes the
minimum in Fraenkel asymmetry, one
immediately has that the red and the
blue arcs in the figure have to be
equal.
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E
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The reduction to a 2-symmetric set
b such that:
Let E ⊂ R2 be convex. We build a 2-symmetric set E
b ),
α(E ) = α(E
b ).
∆P(E ) ≥ ∆P(E
(2)
B
Using the fact that B realizes the
minimum in Fraenkel asymmetry, one
immediately has that the red and the
blue arcs in the figure have to be
equal.
E
At this point we can apply Steiner Symmetrization with respect to the
b which satisfies (2).
coordinate axes obtaining a set E
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The reduction to a 2-symmetric set
Remark We observe that the conclusion about the equality of the
circumference arcs comes from the property about the projections of such
arcs described below.
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The reduction to a 2-symmetric set
Remark We observe that the conclusion about the equality of the
circumference arcs comes from the property about the projections of such
arcs described below.
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The reduction to a 2-symmetric set (continued)
Let us analyze the general case. Typical situations are given in the figures.
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The reduction to a 2-symmetric set (continued)
Let us analyze the general case. Typical situations are given in the figures.
E is an equilateral
triangle
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The reduction to a 2-symmetric set (continued)
Let us analyze the general case. Typical situations are given in the figures.
E is an equilateral
∂E has a coincidence
triangle
part
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The reduction to a 2-symmetric set (continued)
Let us analyze the general case. Typical situations are given in the figures.
E is an equilateral
∂E has a coincidence
∂E has one coincidence
triangle
part
part, one external part
and one internal part
How is it possible to reduce these general situations to the simple one we
have already studied?
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A lemma about projections
E
B
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A lemma about projections
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Symmetrization methods (III)
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A lemma about projections
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Symmetrization methods (III)
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A lemma about projections
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Symmetrization methods (III)
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A lemma about projections
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Symmetrization methods (III)
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A lemma about projections
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A lemma about projections
2
1
4
3
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p1 + p2 ≤ p3 + p4
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The reduction to a 2-symmetric set (continued)
General case
Let us introduce the following notation:
Γ+ = ∂B ∩ E ;
Γ− = ∂B \ Ē ;
Γ0 = ∂B ∩ ∂E ;
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The reduction to a 2-symmetric set (continued)
General case
Let us introduce the following notation:
Γ+ = ∂B ∩ E ;
Γ− = ∂B \ Ē ;
Γ0 = ∂B ∩ ∂E ;
i
γ+
(i ∈ I ≡ {1, 2, . . . } ⊆ N) connected components of Γ+ ;
k
γ−
(k ∈ K ≡ {1, 2, . . . } ⊆ N) connected components of Γ− ;
L length of the arcs.
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The reduction to a 2-symmetric set (continued)
General case
Let us introduce the following notation:
Γ+ = ∂B ∩ E ;
Γ− = ∂B \ Ē ;
Γ0 = ∂B ∩ ∂E ;
i
γ+
(i ∈ I ≡ {1, 2, . . . } ⊆ N) connected components of Γ+ ;
k
γ−
(k ∈ K ≡ {1, 2, . . . } ⊆ N) connected components of Γ− ;
L length of the arcs.
Lemma If L(Γ0 ) = 0, for every i ∈ I and k ∈ K it holds:
i
L(γ+
)≤
X
j
L(γ+
),
j6=i
k
L(γ−
)≤
X
j
L(γ−
).
j6=k
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The reduction to a 2-symmetric set (continued)
General case
Let us introduce the following notation:
Γ+ = ∂B ∩ E ;
Γ− = ∂B \ Ē ;
Γ0 = ∂B ∩ ∂E ;
i
γ+
(i ∈ I ≡ {1, 2, . . . } ⊆ N) connected components of Γ+ ;
k
γ−
(k ∈ K ≡ {1, 2, . . . } ⊆ N) connected components of Γ− ;
L length of the arcs.
Lemma There exists 0 ≤ η̄ ≤ 1 such that, for every i ∈ I e k ∈ K, it holds:
i
L(γ+
)≤
X
j
L(γ+
) + η̄L(Γ0 )
j6=i
k
L(γ−
)≤
X
j
L(γ−
) + (1 − η̄)L(Γ0 ).
j6=k
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (continued)
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Symmetrization methods (III)
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The family B
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (end)
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Symmetrization methods (III)
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The reduction to a 2-symmetric set (end)
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Symmetrization methods (III)
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Regularity C 1
If the boundary is not of class C 1 , there are two possible cases.
convex sharp corner
concave sharp corner
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Symmetrization methods (III)
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Regularity C 1
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Symmetrization methods (III)
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Regularity C 1
we smooth the sharp corner
with an arc of circumference
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Symmetrization methods (III)
CIMPA-UNESCO-EGYPT
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Regularity C 1
we smooth the sharp corner
with an arc of circumference
It is not difficult to restore the initial values of area and asymmetry index
without increasing the perimeter.
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Symmetrization methods (III)
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Regularity C 1
ε
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Symmetrization methods (III)
ε
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Regularity C 1
we smooth the sharp corner
with an arc of circumference
ε
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (III)
ε
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Regularity C 1
we smooth the sharp corner
with an arc of circumference
ε
ε
As in the previous case, it is possible to choose ε, d and s in such a way
that the initial values of area and asymmetry index are restored without
increasing the perimeter.
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Symmetrization methods (III)
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