Seminari per l'esame finale / Seminars for the final exam

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Wednesday, July 5, 2017, at 9 AM in Sala Riunioni

Silja Haffter

Marstrand's theorem

The goal of this seminar is to give a complete proof of the following result, due to Marstrand: If the d-dimensional density of a Radon measure on R^n exists, is positive and finite almost everywhere on a set of positive measure, then d is an integer (smaller or equal to n). Whilst the d-dimensional density of a generic H^d-measurable and finite set E doesn't necessarily exist on E, Marstrand’s theorem, combined with Preiss’ theorem, which states that in the latter case the measure is d-rectifiable, relates its existence to the rectifiability of the set E.

We will follow closely the proof given by Mattila. The key idea is to use a suitable blow-up of the measure to construct a non-zero measure satisfying a stronger condition, whose existence in turn forces d to be an integer smaller or equal to n. In order to describe this procedure, the notion of tangent measures will be introduced. If time permits, we will conclude the seminar by outlining some steps towards Preiss’ theorem.

Marstrand's theorem

The goal of this seminar is to give a complete proof of the following result, due to Marstrand: If the d-dimensional density of a Radon measure on R^n exists, is positive and finite almost everywhere on a set of positive measure, then d is an integer (smaller or equal to n). Whilst the d-dimensional density of a generic H^d-measurable and finite set E doesn't necessarily exist on E, Marstrand’s theorem, combined with Preiss’ theorem, which states that in the latter case the measure is d-rectifiable, relates its existence to the rectifiability of the set E.

We will follow closely the proof given by Mattila. The key idea is to use a suitable blow-up of the measure to construct a non-zero measure satisfying a stronger condition, whose existence in turn forces d to be an integer smaller or equal to n. In order to describe this procedure, the notion of tangent measures will be introduced. If time permits, we will conclude the seminar by outlining some steps towards Preiss’ theorem.

Wednesday, July 5, 2017, at 10 AM in Sala Riunioni

Ondřej Bouchala

Subsets with finite and positive Hausdorff measure of a compact metric

The object of this seminar is the following result, due to Besicovitch: Every closed set in the Euclidean space R^n with infinite d-dimensional Hausdorff measure contains a subset with finite and positive measure. The proof relies on the notion of "comparable network measure".

Subsets with finite and positive Hausdorff measure of a compact metric

The object of this seminar is the following result, due to Besicovitch: Every closed set in the Euclidean space R^n with infinite d-dimensional Hausdorff measure contains a subset with finite and positive measure. The proof relies on the notion of "comparable network measure".

Thursday, July 6, 2017, at 9 AM in Sala Riunioni

Federico Glaudo

The isoperimetric inequality through Steiner symmetrization

The isoperimetric inequality states that among all sets with given volume (Lebesgue measure) in R^n, the sphere has minimal perimeter. This theorem has a long history and many proofs. In this seminar we present the proof given by De Giorgi in the 1958 paper "Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita".

The main tool, which plays a key role also in the proof of the isodiametric inequality, is Steiner symmetrization, that is, the transformation which makes a set symmetric with respect to a given hyperplane without changing its volume. We show that the symmetrized set has lower perimeter than the original set, and using this fact we prove the isoperimetric inequality on the strong form.

The isoperimetric inequality through Steiner symmetrization

The isoperimetric inequality states that among all sets with given volume (Lebesgue measure) in R^n, the sphere has minimal perimeter. This theorem has a long history and many proofs. In this seminar we present the proof given by De Giorgi in the 1958 paper "Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita".

The main tool, which plays a key role also in the proof of the isodiametric inequality, is Steiner symmetrization, that is, the transformation which makes a set symmetric with respect to a given hyperplane without changing its volume. We show that the symmetrized set has lower perimeter than the original set, and using this fact we prove the isoperimetric inequality on the strong form.

Thursday, July 6, 2017, at 10 AM in Sala Riunioni

Giada Franz

Decomposition of a finite perimeter sets in terms of "indecomposable components"

We present a notion of connectedness for finite perimeter sets, first introduced by Federer in the more general setting of currents. In particular we say that a finite perimeter set E is decomposable if we can write E as union of in two non-negligible, disjoint subsets E_0, E_1 such that the sum of the perimeters of E_0, E_1 is equal to the perimeter of E. Conversely, we say that E is indecomposable if it is not decomposable. The main result we present states that every finite perimeter set can be written as a disjoint countable union of indecomposable subsets. Finally we will see how these indecomposable components are linked with the classical connected components of an open set.

Decomposition of a finite perimeter sets in terms of "indecomposable components"

We present a notion of connectedness for finite perimeter sets, first introduced by Federer in the more general setting of currents. In particular we say that a finite perimeter set E is decomposable if we can write E as union of in two non-negligible, disjoint subsets E_0, E_1 such that the sum of the perimeters of E_0, E_1 is equal to the perimeter of E. Conversely, we say that E is indecomposable if it is not decomposable. The main result we present states that every finite perimeter set can be written as a disjoint countable union of indecomposable subsets. Finally we will see how these indecomposable components are linked with the classical connected components of an open set.

Wednesday, July 26, 2017, at 10 AM in Sala Seminari

Claudio Afeltra

Proof of the existence of Besicovitch sets via Baire's Theorem

Besicovitch sets (in the R^d) are Lebesgue-negligible compact set which contain segments of a given length in every direction. Such sets play a significant role in harmonic analysis, and it is known that their Hausdorff dimension must agree with the dimension of the space for d=2; it is an important conjecture that the same holds also for d larger than 2.

Indeed, even the existence of Besicovitch sets is a non-trivial fact. In this talk we present a proof of existence given by T.W. Körner in the paper "Besicovitch via Baire" (2003). This proof relies on the Baire's theorem, and shows that Besicovitch sets are generic in a suitable class of compact sets in the plane, endowed with the Hausdorff distance.

Proof of the existence of Besicovitch sets via Baire's Theorem

Besicovitch sets (in the R^d) are Lebesgue-negligible compact set which contain segments of a given length in every direction. Such sets play a significant role in harmonic analysis, and it is known that their Hausdorff dimension must agree with the dimension of the space for d=2; it is an important conjecture that the same holds also for d larger than 2.

Indeed, even the existence of Besicovitch sets is a non-trivial fact. In this talk we present a proof of existence given by T.W. Körner in the paper "Besicovitch via Baire" (2003). This proof relies on the Baire's theorem, and shows that Besicovitch sets are generic in a suitable class of compact sets in the plane, endowed with the Hausdorff distance.

Thursday, July 28, 2017, at 10 AM in Sala Riunioni

Lorenzo Portinale

Wirtinger’s inequality and minimality of complex submanifold of C^n

In this seminar we prove that every complex k-submanifold of C^n (that is, any real 2k-submanifold of R^{2n} such that in any point the tangent plane is a complex subspace of C^n via the canonical identification of R^{2n} and C^n) is area-minimizing.

This result is a consequence of the so-called Wirtinger's inequality, which shows that the Kähler form is a calibration (in the sense of the differential forms) for every complex k-submanifold.

Besides Wirtinger's inequality, the proof of the minimality relies on a direct application of the Stokes theorem, which means that some additional difficulty must be overcome when the manifold contains singular points.

We will prove the result in the smooth case first, and then consider some particular cases where singularities are allowed.

Wirtinger’s inequality and minimality of complex submanifold of C^n

In this seminar we prove that every complex k-submanifold of C^n (that is, any real 2k-submanifold of R^{2n} such that in any point the tangent plane is a complex subspace of C^n via the canonical identification of R^{2n} and C^n) is area-minimizing.

This result is a consequence of the so-called Wirtinger's inequality, which shows that the Kähler form is a calibration (in the sense of the differential forms) for every complex k-submanifold.

Besides Wirtinger's inequality, the proof of the minimality relies on a direct application of the Stokes theorem, which means that some additional difficulty must be overcome when the manifold contains singular points.

We will prove the result in the smooth case first, and then consider some particular cases where singularities are allowed.

Wednesday, August 2, 2017, at 9 AM in Sala Riunioni

Daniele Tiberio

Capacity and Hausdorff dimension

The capacity is a measure of the size of a set which plays an important role in many areas of Analysis (and even beyond). After introducing the notions of p-capacity of a set and of capacitary dimension, we state the main result, Frostman lemma, which establishes some relation between p-capacity and d-dimensional Hausdorff measure, and shows that Hausdorff dimension and capacitary dimension agree (this result will not be proved in full generality).

We conclude the seminar with two applications. The first one concerns the Hausdorff dimension of the product of two sets, the second one concerns the behaviour of the Hausdorff dimension of the projections of a given set on subspaces of R^n.

Capacity and Hausdorff dimension

The capacity is a measure of the size of a set which plays an important role in many areas of Analysis (and even beyond). After introducing the notions of p-capacity of a set and of capacitary dimension, we state the main result, Frostman lemma, which establishes some relation between p-capacity and d-dimensional Hausdorff measure, and shows that Hausdorff dimension and capacitary dimension agree (this result will not be proved in full generality).

We conclude the seminar with two applications. The first one concerns the Hausdorff dimension of the product of two sets, the second one concerns the behaviour of the Hausdorff dimension of the projections of a given set on subspaces of R^n.

Wednesday, August 2, 2017, at 10 AM in Sala Riunioni

Valerio Pagliari

Existence of minimal clusters with prescribed volumes

A classical problem of Calculus of Variations is the isoperimetric one, which means finding the set with minimal perimeter among those with a prescribed volume. It is well known that this problem admits solutions (namely the balls).

In this talk we discuss how this existence result can be extended to the minimization of the perimeter of a collection made of more than one set. More precisely, we consider N-clusters, i.e., disjoint families of finite perimeter sets E_1, ... , E_N in R^d, and consider the following problem: fixed N and strictly positive numbers v_1, ... , v_N, find the N-cluster that minimizes the perimeter under the constraint that each E_i has volume v_i. Here, the perimeter is the sum of the (d-1)-dimensional Hausdorff measures of the interfaces between the components E_i (that is, the intersections of the essential boundaries). This is a quite natural choice that allows both for physical interpretations and good analytical properties.

The main difficulty in the proof is the lack of compactness for minimizing sequences, which in principle may "escape to infinity".

Existence of minimal clusters with prescribed volumes

A classical problem of Calculus of Variations is the isoperimetric one, which means finding the set with minimal perimeter among those with a prescribed volume. It is well known that this problem admits solutions (namely the balls).

In this talk we discuss how this existence result can be extended to the minimization of the perimeter of a collection made of more than one set. More precisely, we consider N-clusters, i.e., disjoint families of finite perimeter sets E_1, ... , E_N in R^d, and consider the following problem: fixed N and strictly positive numbers v_1, ... , v_N, find the N-cluster that minimizes the perimeter under the constraint that each E_i has volume v_i. Here, the perimeter is the sum of the (d-1)-dimensional Hausdorff measures of the interfaces between the components E_i (that is, the intersections of the essential boundaries). This is a quite natural choice that allows both for physical interpretations and good analytical properties.

The main difficulty in the proof is the lack of compactness for minimizing sequences, which in principle may "escape to infinity".

Friday, September 29, 2017, at 2.30 PM in Sala Riunioni

Francesco Paolo Maiale

Besicovitch-Federer projection theorem

The Besicovitch-Federer projection theorem states that a Borel set E in R^n with finite m-dimensional Hausdorff measure H^m is purely m-unrectifiable (with m < n) if and only if the orthogonal projection of E on V is H^m-negligible for almost every subspace V in the Grassmannian Gr(n,m).

In this seminar, we show the "only if" part following the proof given in Mattila's book, "Geometry of sets and measures in Euclidean spaces."

Besicovitch-Federer projection theorem

The Besicovitch-Federer projection theorem states that a Borel set E in R^n with finite m-dimensional Hausdorff measure H^m is purely m-unrectifiable (with m < n) if and only if the orthogonal projection of E on V is H^m-negligible for almost every subspace V in the Grassmannian Gr(n,m).

In this seminar, we show the "only if" part following the proof given in Mattila's book, "Geometry of sets and measures in Euclidean spaces."

Friday, September 29, 2017, at 3.30 PM in Sala Riunioni

Gioacchino Antonelli

Existence of Besicovitch sets in the plane, and their Hausdorff dimension

In the first part of the seminar I will show the existence of a Besicovitch set in the plane (i.e., a Lebesgue-negligible set which contains a line in every direction).

In particular I will show that the construction of this set can be reduced to the existence of a H^1-purely unrectifiable set with finite length in the unit square of R^2 whose projection on the x-axis is [0,1].

Then I will construct such a set.

In the second part of the seminar I'll show that every set F in the plane which contains a line in every direction has Hausdorff dimension 2.

The proof is divided in two main steps: in the first step, using some ideas of potential theory (capacity of a set, capacitary dimension) I will deduce, from more general results, that every Borel set in R^2 of Hausdorff dimension greater or equal than 1 projects on almost every direction in a set of Hausdorff dimension greater or equal than 1.

In the second step I will deduce, using some sort of duality and the first step, that (up to subsets) almost every vertical section of F has Hausdorff dimension greater or equal than one.

After that, I will conclude that the Hausdorff dimension of F is 2 by using some estimates which can be shown by means of comparable net measures.

Existence of Besicovitch sets in the plane, and their Hausdorff dimension

In the first part of the seminar I will show the existence of a Besicovitch set in the plane (i.e., a Lebesgue-negligible set which contains a line in every direction).

In particular I will show that the construction of this set can be reduced to the existence of a H^1-purely unrectifiable set with finite length in the unit square of R^2 whose projection on the x-axis is [0,1].

Then I will construct such a set.

In the second part of the seminar I'll show that every set F in the plane which contains a line in every direction has Hausdorff dimension 2.

The proof is divided in two main steps: in the first step, using some ideas of potential theory (capacity of a set, capacitary dimension) I will deduce, from more general results, that every Borel set in R^2 of Hausdorff dimension greater or equal than 1 projects on almost every direction in a set of Hausdorff dimension greater or equal than 1.

In the second step I will deduce, using some sort of duality and the first step, that (up to subsets) almost every vertical section of F has Hausdorff dimension greater or equal than one.

After that, I will conclude that the Hausdorff dimension of F is 2 by using some estimates which can be shown by means of comparable net measures.

Wednesday, December 20, 2017, at 10 AM in Sala Riunioni

Nicola Picenni

Structure of finite perimeter sets in the plane

Every finite perimeter set in the plane can be obtained by suitably adding a subtracting countably many closed sets whose boundary is a simple closed curve with finite length. This result was first proved by Ambrosio, Caselles, Masnou and Morel. The proof presented in this seminar is obtained by showing that the class of all sets that can be written in this form is actually closed in the class of finite perimeter sets (with respect to convergence in the L^1 distance with a uniform bound on perimeters).

Structure of finite perimeter sets in the plane

Every finite perimeter set in the plane can be obtained by suitably adding a subtracting countably many closed sets whose boundary is a simple closed curve with finite length. This result was first proved by Ambrosio, Caselles, Masnou and Morel. The proof presented in this seminar is obtained by showing that the class of all sets that can be written in this form is actually closed in the class of finite perimeter sets (with respect to convergence in the L^1 distance with a uniform bound on perimeters).

Friday, March 9, 2018, at 2.30 PM

Federico Franceschini

Federer's characterization of finite perimeter sets

The essential boundary of a (Borel) set E in R^n is the set of all p[point in R^n where E has neither density equal to 1 nor density equal to 0.

It was shown during the course that when E has finite perimeter then the essential boundary of E has finite (n-1)-dimensional Hausdorff measure.

In this seminar I will prove the converse statement, namely that if the essential boundary of a set E is H^{n-1}-finite then E has finite perimeter. This result is due to H. Federer; the proof presented here is taken from the book by Evans and Gariepy, and is based on the characterization of functions with bounded variation in terms of their essential variation along lines.

In this presentation I will try to explain the geometric intuition behind this proof, and discuss the technical details as well.

Federer's characterization of finite perimeter sets

The essential boundary of a (Borel) set E in R^n is the set of all p[point in R^n where E has neither density equal to 1 nor density equal to 0.

It was shown during the course that when E has finite perimeter then the essential boundary of E has finite (n-1)-dimensional Hausdorff measure.

In this seminar I will prove the converse statement, namely that if the essential boundary of a set E is H^{n-1}-finite then E has finite perimeter. This result is due to H. Federer; the proof presented here is taken from the book by Evans and Gariepy, and is based on the characterization of functions with bounded variation in terms of their essential variation along lines.

In this presentation I will try to explain the geometric intuition behind this proof, and discuss the technical details as well.

Friday, April 27, 2018, at 3 PM

Mattia Magnabosco

Peter W. Jones's Travelling Salesman Theorem

In the first part of this seminar I construct a compact set in the plane with arbitrarily small Hausdorff dimension that cannot be covered by countably many images of Lipschitz curves (the existence of such sets shows that the H^1 null subset that appears in the definition of 1-rectifiable sets cannot be omitted without affecting the definition).

In the second part of the talk I present a result due to Peter W. Jones, known as the Travelling Salesman Theorem. This result characterizes the planar sets that can be covered by a curve with finite length in terms of a purely geometric criterion related to the distribution of the set in dyadic squares.

I will give a complete proof of the fact that if a set satisfies this criterion then it can be covered by a curve with finite length and I will outline a proof of the converse statement. I follow the proof given by John B. Garnett and Donald E. Marshall.

Peter W. Jones's Travelling Salesman Theorem

In the first part of this seminar I construct a compact set in the plane with arbitrarily small Hausdorff dimension that cannot be covered by countably many images of Lipschitz curves (the existence of such sets shows that the H^1 null subset that appears in the definition of 1-rectifiable sets cannot be omitted without affecting the definition).

In the second part of the talk I present a result due to Peter W. Jones, known as the Travelling Salesman Theorem. This result characterizes the planar sets that can be covered by a curve with finite length in terms of a purely geometric criterion related to the distribution of the set in dyadic squares.

I will give a complete proof of the fact that if a set satisfies this criterion then it can be covered by a curve with finite length and I will outline a proof of the converse statement. I follow the proof given by John B. Garnett and Donald E. Marshall.

Friday, June 22, 2018, at 4 PM in Sala Riunioni

Clara Antonucci

What can we say about a measure if we know its values on balls?

Consider two measures defined on a metric space X that agree on every ball.

If X is the Euclidean space we know that the measures agree, but this is not true in a general metric space without further hypotheses on the measures: we show this by describing an explicit example due to Roy Davies.

In order to prove that the measures agree the fundamental tool is a Vitali-type covering lemma, which holds under the assumption that the underlying measure is "asymptotically doubling" (a locally finite Borel measure m on a metric space X is called asymptotically doubling if for m-almost every point x in X the ratio m(B(x,2r)) over m(B(x,r)) has finite limsup as r tends to zero).

We show with a counterexample that not all measures satisfy this assumption, even when X=[0,1], but that the weaker request of having finite liminf for m-almost every point is always fulfilled when X=R^n.

What can we say about a measure if we know its values on balls?

Consider two measures defined on a metric space X that agree on every ball.

If X is the Euclidean space we know that the measures agree, but this is not true in a general metric space without further hypotheses on the measures: we show this by describing an explicit example due to Roy Davies.

In order to prove that the measures agree the fundamental tool is a Vitali-type covering lemma, which holds under the assumption that the underlying measure is "asymptotically doubling" (a locally finite Borel measure m on a metric space X is called asymptotically doubling if for m-almost every point x in X the ratio m(B(x,2r)) over m(B(x,r)) has finite limsup as r tends to zero).

We show with a counterexample that not all measures satisfy this assumption, even when X=[0,1], but that the weaker request of having finite liminf for m-almost every point is always fulfilled when X=R^n.

Friday, June 18, 2019, at 6 PM

Vincenzo Scattaglia

The Gaussian isoperimetric inequality.

Consider the isoperimetric inequality for sets in R^n where both the volume and the perimetr are computed taking into account a gaussian density (actually, the standard Gaussian density g with baricenter 0 and variance 1) that is, vol(E) is the integral over E of g, and per(E) is the integral over the boundary of E of g (at least of the boundary of E is sufficiently regular).

In this setting the isoperiemetric sets (namely the sets that minimizes the gaussian perimeter among all sets with prescribed gaussian volume) are half-spaces.

The Gaussian isoperimetric inequality.

Consider the isoperimetric inequality for sets in R^n where both the volume and the perimetr are computed taking into account a gaussian density (actually, the standard Gaussian density g with baricenter 0 and variance 1) that is, vol(E) is the integral over E of g, and per(E) is the integral over the boundary of E of g (at least of the boundary of E is sufficiently regular).

In this setting the isoperiemetric sets (namely the sets that minimizes the gaussian perimeter among all sets with prescribed gaussian volume) are half-spaces.

Not assigned

Monotonicity formula for
varifolds
with bounded curvature

If a d-dimensional varifold V has bounded mean curvature then the measure of a ball with center x and radius r divided by r^d modulo a suitable correction is increasing in r. Using this fact one can prove that the sets of points of positive density of the varifold is closed; this result extends a similar result for minimal finite perimeter sets proved in one of the last lectures of the course.

If a d-dimensional varifold V has bounded mean curvature then the measure of a ball with center x and radius r divided by r^d modulo a suitable correction is increasing in r. Using this fact one can prove that the sets of points of positive density of the varifold is closed; this result extends a similar result for minimal finite perimeter sets proved in one of the last lectures of the course.