FUNCTIONS THEORY 2024-2025 - SCQ0094119

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• Seleziona sezione SCQ0094119 - FUNCTIONS THEORY 2024-2025 - PROF. ANNALISA CESARONI

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  SCQ0094119 - FUNCTIONS THEORY 2024-2025 - PROF. ANNALISA CESARONI

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  Schedule: Monday, Tuesday and Wednesday, 14.30-16.10, room 1BC50 Plan of the course Part 0: Preliminaries  (calculus and measure theory) [F, chapter 1], [E, appendices] Part 1: Elements of Geometric Measure Theory. Radon measures and Riesz theorem. Hausdorff measures.  [EG, chapters 1,2] [F, chapter 7] Part 2: The theory of distributions [F, chapter 9] Part 3: Sobolev spaces [E chapter 5] Part 4: Functions of  bounded variation [EG, chapter 5] Main references [EG] Evans L.C.,and   Gariepy,  F.C., Measure theory and fine properties of functions, Boca Raton, CRC Press, 1992. [E] Evans, L.C., Partial Differential Equations, American Mathematical Soc., 2010.  [F] Folland, G.B., Real Analysis. New York: Wiley Interscience, 1999. Other references:  Ambrosio, L.Corso introduttivo alla teoria geometrica della misura e alle superfici minime, Pisa, Scuola Normale Superiore, 2009. Ambrosio, L., Fusco, N., and Pallara, D., Functions of bounded variations and free discontinuity problems, Oxford Mathematical Monographs, 2000. Brezis, H.,Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer 2011 Dates of the exams: Monday December 22, from 11, room 2AB40 Friday January 9,  from 14.30, room 1BC50 Monday January 20, from 14 (in the afternoon), room 1BC50  Friday February 14, from 10 (in the morning), room 1BC50. It is possible to take the exam also in other dates (also in December!), by prior agreement with me.

  • Seleziona attività General information on the course and dates of exams

    

    General information on the course and dates of exams Pagina
  • Seleziona attività List of questions from past exams.

    

    List of questions from past exams. File
  • Seleziona attività Detailed program

    

    Detailed program File
  • Seleziona attività Pagina dell'offerta Formativa

    

    Pagina dell'offerta Formativa URL
  • Seleziona attività Annunci

    

    Annunci Forum
• Seleziona sezione Course Record

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  Course Record

  • Seleziona attività September 30: presentation of the course. Space of Holder and Lipschitz functions. Spaces of C^infty functions.

    

    September 30: presentation of the course. Space of Holder and Lipschitz functions. Spaces of C^infty functions. File
  • Seleziona attività October 1: space of test functions (C^\infty_c). Regularization by convolution. DuBois Reymond lemma and 1d variant.

    

    October 1: space of test functions (C^\infty_c). Regularization by convolution. DuBois Reymond lemma and 1d variant. File
  • Seleziona attività Oct 2: partitions of units. Boundaries and integration by parts formula

    

    Oct 2: partitions of units. Boundaries and integration by parts formula File
  • Seleziona attività October 7: signed distance function and regularity of boundaries. Definition of Radon measures

    

    October 7: signed distance function and regularity of boundaries. Definition of Radon measures File
  • Seleziona attività October 8: Caratheodory theorem, total variation measure, Lebesgue Radon Nykodim theorem

    

    October 8: Caratheodory theorem, total variation measure, Lebesgue Radon Nykodim theorem File
  • Seleziona attività October 14: the theorem of differentiation of measures. Definition of density. Covering theorem (Vitali, Besikovitch). Computation of the volume of the n dimensional ball

    

    October 14: the theorem of differentiation of measures. Definition of density. Covering theorem (Vitali, Besikovitch). Computation of the volume of the n dimensional ball File
  • Seleziona attività October 15: Hausdorff measures.

    

    October 15: Hausdorff measures. File
  • Seleziona attività On the Hausdorff dimension of the Cantor set. Possible reference: Falconer, Geometry of Fractal Sets, Cambridge Univ. press, 1985

    

    On the Hausdorff dimension of the Cantor set. Possible reference: Falconer, Geometry of Fractal Sets, Cambridge Univ. press, 1985 File
  • Seleziona attività October 16: Steiner symmetrization. Isodiametric inequality. |E|\leq H^n_d(E) for every d

    

    October 16: Steiner symmetrization. Isodiametric inequality. |E|\leq H^n_d(E) for every d File
  • Seleziona attività On the measurability of subgraphs of measurable functions

    

    On the measurability of subgraphs of measurable functions File
  • Seleziona attività October 21: H^n is the Lebesgue measure (end of the proof). Area formula (no proof) H^k corresponds on C^1 -dim. submanifold to the surface measure. Rectifiable sets.

    

    October 21: H^n is the Lebesgue measure (end of the proof). Area formula (no proof) H^k corresponds on C^1 -dim. submanifold to the surface measure. Rectifiable sets. File
  • Seleziona attività On sets with H^1(C)=0 and dim_H(C)=1. Purely 1 unrecifiable sets. Reference Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Univ. Press, 2012

    

    On sets with H^1(C)=0 and dim_H(C)=1. Purely 1 unrecifiable sets. Reference Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Univ. Press, 2012 File
  • Seleziona attività October 22: Riesz (Markov) theorem on Radon measures (no proof).

    

    October 22: Riesz (Markov) theorem on Radon measures (no proof). File
  • Seleziona attività Oct 23: distributions and order of a distribution

    

    Oct 23: distributions and order of a distribution File
  • Seleziona attività Oct 28: order of a distribution, support of a distribution, distributions of compact support have finite order. Derivatives of distribution. Weak derivatives of functions.

    

    Oct 28: order of a distribution, support of a distribution, distributions of compact support have finite order. Derivatives of distribution. Weak derivatives of functions. File
  • Seleziona attività Oct 29: derivative in the sense of distribution and weak derivatives. Examples of functions not admitting wea derivatives. Fundamental solution of a linear diff. operator.

    

    Oct 29: derivative in the sense of distribution and weak derivatives. Examples of functions not admitting wea derivatives. Fundamental solution of a linear diff. operator. File
  • Seleziona attività Oct 30: fundamental solutions of the laplacian. If f in R is monotone non decreasing, its derivative in the sense of distribution is a Radon measure

    

    Oct 30: fundamental solutions of the laplacian. If f in R is monotone non decreasing, its derivative in the sense of distribution is a Radon measure File
  • Seleziona attività Nov 4: convergence in the sense of distribution. Convolution between a distribution and a test function, main properties. Density.

    

    Nov 4: convergence in the sense of distribution. Convolution between a distribution and a test function, main properties. Density. File
  • Seleziona attività Nov 5: conclusion of the proof. Spherical mean property and characterization of harmonic functions. Weyl lemma (harmonic distributions are smooth).

    

    Nov 5: conclusion of the proof. Spherical mean property and characterization of harmonic functions. Weyl lemma (harmonic distributions are smooth). File
  • Seleziona attività November 6: Sobolev spaces. Characterization of Sobolev spaces in dimension 1 in terms of absolutely continuous functions.

    

    November 6: Sobolev spaces. Characterization of Sobolev spaces in dimension 1 in terms of absolutely continuous functions. File
  • Seleziona attività Nov 11: duals of Sobolev spaces, theorem of Meyers Serrin. Corollary:W^{k,p}_0(\R^n)=W^{k,p}(\R^n)

    

    Nov 11: duals of Sobolev spaces, theorem of Meyers Serrin. Corollary:W^{k,p}_0(\R^n)=W^{k,p}(\R^n) File
  • Seleziona attività Nov 12: extension theorem, density of C^\infty(\bar U) in W^{1,p}(U). Proof of the lemma (extension by reflection).

    

    Nov 12: extension theorem, density of C^\infty(\bar U) in W^{1,p}(U). Proof of the lemma (extension by reflection). File
  • Seleziona attività Nov 13: extension theorem and trace theorem (no proof). Characterization of W_0^{1,p} no proof. Continuous and compact embeddings. Case n=1.

    

    Nov 13: extension theorem and trace theorem (no proof). Characterization of W_0^{1,p} no proof. Continuous and compact embeddings. Case n=1. File
  • Seleziona attività Nov 18: Gagliardo Nirenberg Sobolev embedding. Local case for W_0^{1,p}(U), U bounded.

    

    Nov 18: Gagliardo Nirenberg Sobolev embedding. Local case for W_0^{1,p}(U), U bounded. File
  • Seleziona attività Nov 19: local version of GNS inequality and embeddings. Morrey inequality.

    

    Nov 19: local version of GNS inequality and embeddings. Morrey inequality. File
  • Seleziona attività Nov 20: local version of Morrey. Generalized Rademacher theorem. Functions in W^{1,\infty}_loc are locally Lipschitz.

    

    Nov 20: local version of Morrey. Generalized Rademacher theorem. Functions in W^{1,\infty}_loc are locally Lipschitz. File
  • Seleziona attività Nov 25: conclusion of the proof on characterization of W^1,infty functions. Embeddings for W^{2,p}. Compact embeddings for p>n. Rellich Kondrachov theorem, counterexample for p^*.

    

    Nov 25: conclusion of the proof on characterization of W^1,infty functions. Embeddings for W^{2,p}. Compact embeddings for p>n. Rellich Kondrachov theorem, counterexample for p^*. File
  • Seleziona attività Nov 27: Rellich Kondrachov theorem. Main application: W^{1,p} is compactly embedded in L^p.

    

    Nov 27: Rellich Kondrachov theorem. Main application: W^{1,p} is compactly embedded in L^p. File
  • Seleziona attività December 2: Harmonic extension. Poincare' inequality (statement)

    

    December 2: Harmonic extension. Poincare' inequality (statement) File
  • Seleziona attività December 3: Poincare' inequality. Application: W^{1,n} are BMO functions. An example from calculus of the variations.

    

    December 3: Poincare' inequality. Application: W^{1,n} are BMO functions. An example from calculus of the variations. File
  • Seleziona attività Dec 4: conclusion of the example of calculus of variations. Introduction of the space of BV functons.

    

    Dec 4: conclusion of the example of calculus of variations. Introduction of the space of BV functons. File
  • Seleziona attività Dec 9: BV norm. LSC of the total variation wrt to L^1 conv. Strong, strict and weak star convergence in BV. Density in strict sense of smooth functions. GNS and Helly theorems.

    

    Dec 9: BV norm. LSC of the total variation wrt to L^1 conv. Strong, strict and weak star convergence in BV. Density in strict sense of smooth functions. GNS and Helly theorems. File
  • Seleziona attività Dec 10: BV functions in dimension 1

    

    Dec 10: BV functions in dimension 1 File
  • Seleziona attività Dec 11: sets of finite perimeter

    

    Dec 11: sets of finite perimeter File
  • Seleziona attività December 16: compactness for perimeters. The Plateau problem

    

    December 16: compactness for perimeters. The Plateau problem File
• Seleziona sezione Problems sheets

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  Problems sheets

  • Seleziona attività Problem sheet 1: preliminaries on calculus

    

    Problem sheet 1: preliminaries on calculus File
  • Seleziona attività Sketch of solutions, sheet 1.

    

    Sketch of solutions, sheet 1. File
  • Seleziona attività Problem sheet 2: measure theory

    

    Problem sheet 2: measure theory File
  • Seleziona attività Sketch of solutions, sheet 2

    

    Sketch of solutions, sheet 2 File
  • Seleziona attività Problem sheet 3: distribution theory

    

    Problem sheet 3: distribution theory File
  • Seleziona attività Sketch of solutions, sheet 3

    

    Sketch of solutions, sheet 3 File
  • Seleziona attività Problem sheet 4: Sobolev spaces

    

    Problem sheet 4: Sobolev spaces File
  • Seleziona attività Sketch of solutions sheet 4

    

    Sketch of solutions sheet 4 File
  • Seleziona attività Problem sheet 5: inequalities and embedding in Sobolev spaces

    

    Problem sheet 5: inequalities and embedding in Sobolev spaces File
  • Seleziona attività Sketch of solutions, sheet 5

    

    Sketch of solutions, sheet 5 File
  • Seleziona attività Problem sheet 6: BV functions, and sets of finite perimeter

    

    Problem sheet 6: BV functions, and sets of finite perimeter File
  • Seleziona attività Sketch of solutions, sheet 6

    

    Sketch of solutions, sheet 6 File
• Seleziona sezione Accessibilità e Questionario intermedio sull’organizzazione e l’efficacia della didattica

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  Accessibilità e Questionario intermedio sull’organizzazione e l’efficacia della didattica

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• Seleziona sezione Registrations

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  Registrations

  • Seleziona attività registration oct 14

    

    registration oct 14 Kaltura Video Resource
  • Seleziona attività Registration october 21

    

    Registration october 21 Kaltura Video Resource
  • Seleziona attività Registration october 28

    

    Registration october 28 Kaltura Video Resource
  • Seleziona attività Registration november 4

    

    Registration november 4 Kaltura Video Resource