FUNCTIONS THEORY 2024-2025 - SCQ0094119

Topic outline

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

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  Schedule: Monday, Tuesday and Wednesday, 14.30-16.10, room 1BC50 Office hours Wednesday 12.30-14  Torre Archimede, 5 floor,  office 5DA9. Feel free to contact the teachers at annalisa.cesaroni@unipd.it. 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

  
  

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    General information on the course and dates of exams Page
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    List of questions from past exams. File
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    Detailed program- to be updated ... File
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    Pagina dell'offerta Formativa URL
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    Annunci Forum
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    Class of Tuesday november 26 is cancelled Page

• Course Record
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    September 30: presentation of the course. Space of Holder and Lipschitz functions. Spaces of C^infty functions. File
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    October 1: space of test functions (C^\infty_c). Regularization by convolution. DuBois Reymond lemma and 1d variant. File
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    Oct 2: partitions of units. Boundaries and integration by parts formula File
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    October 7: signed distance function and regularity of boundaries. Definition of Radon measures File
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    October 8: Caratheodory theorem, total variation measure, Lebesgue Radon Nykodim theorem File
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    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
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    October 15: Hausdorff measures. File
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    On the Hausdorff dimension of the Cantor set. Possible reference: Falconer, Geometry of Fractal Sets, Cambridge Univ. press, 1985 File
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    October 16: Steiner symmetrization. Isodiametric inequality. |E|\leq H^n_d(E) for every d File
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    On the measurability of subgraphs of measurable functions File
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    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
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    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
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    October 22: Riesz (Markov) theorem on Radon measures (no proof). File
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    Oct 23: distributions and order of a distribution File
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    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
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    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
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    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
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    Nov 4: convergence in the sense of distribution. Convolution between a distribution and a test function, main properties. Density. File
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    Nov 5: conclusion of the proof. Spherical mean property and characterization of harmonic functions. Weyl lemma (harmonic distributions are smooth). File
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    November 6: Sobolev spaces. Characterization of Sobolev spaces in dimension 1 in terms of absolutely continuous functions. File
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    Nov 11: duals of Sobolev spaces, theorem of Meyers Serrin. Corollary:W^{k,p}_0(\R^n)=W^{k,p}(\R^n) File
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    Nov 12: extension theorem, density of C^\infty(\bar U) in W^{1,p}(U). Proof of the lemma (extension by reflection). File
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    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
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    Nov 18: Gagliardo Nirenberg Sobolev embedding. Local case for W_0^{1,p}(U), U bounded. File
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    Nov 19: local version of GNS inequality and embeddings. Morrey inequality. File
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    Nov 20: local version of Morrey. Generalized Rademacher theorem. Functions in W^{1,\infty}_loc are locally Lipschitz. File

• Problems sheets
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    Problem sheet 1: preliminaries on calculus File
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    Sketch of solutions, sheet 1. File
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    Problem sheet 2: measure theory File
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    Sketch of solutions, sheet 2 File
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    Problem sheet 3: distribution theory File
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    Sketch of solutions, sheet 3 File
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    Problem sheet 4: Sobolev spaces File
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    Sketch of solutions sheet 4 File

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

  Not available

• Registrations
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    registration oct 14 Kaltura Video Resource
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    Registration october 21 Kaltura Video Resource
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    Registration october 28 Kaltura Video Resource
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    Registration november 4 Kaltura Video Resource