FUNCTIONS THEORY 2025-2026 - SCQ0094119, Sezione: Course record

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Schema della sezione

- Seleziona attività 29/09: Space of continuous functions, and sup norm. Ascoli Arzela' compactness theorem. The space of test functions
  
  29/09: Space of continuous functions, and sup norm. Ascoli Arzela' compactness theorem. The space of test functions File
- Seleziona attività 6/10: test functions are dense in L^p. DuBois Reymond lemma (added a page)
  
  6/10: test functions are dense in L^p. DuBois Reymond lemma (added a page) File
- Seleziona attività 7/10: partitions of unity. Open sets of class C^k, signed distance, exterior and interior sphere condition
  
  7/10: partitions of unity. Open sets of class C^k, signed distance, exterior and interior sphere condition File
- Seleziona attività 8/10: integration on sphere formula. Borel measures and Radon measures.
  
  8/10: integration on sphere formula. Borel measures and Radon measures. File
- Seleziona attività 13/10: Positive linear functionals on C_c(U). Riesz theorem on Radon measure. Weak compactness in the space of Radon measure. Recall of L^p weak, weak^* compactness theorems.
  
  13/10: Positive linear functionals on C_c(U). Riesz theorem on Radon measure. Weak compactness in the space of Radon measure. Recall of L^p weak, weak^* compactness theorems. File
- Seleziona attività 14/10: embedding of L^1 in the space of signed Radon measures and weak compactness thm; Lebesgue theorem, Meaure theoretic boundary and topological boundary.
  
  14/10: embedding of L^1 in the space of signed Radon measures and weak compactness thm; Lebesgue theorem, Meaure theoretic boundary and topological boundary. File
- Seleziona attività 15/10: absolutely continuous measures and singular measures, density of an a.c. measuure, Lebesgue-Radon-Nikodym decomposition. Hausdorff measures.
  
  15/10: absolutely continuous measures and singular measures, density of an a.c. measuure, Lebesgue-Radon-Nikodym decomposition. Hausdorff measures. File
- Seleziona attività 20/10:Hausdorff dimension of a set. Example of the Cantor set and devil's staircase (without proof of the Hausdorff dimension)
  
  20/10:Hausdorff dimension of a set. Example of the Cantor set and devil's staircase (without proof of the Hausdorff dimension) File
- Seleziona attività 21/10: Isodiametric inequality in R^n by Steiner symmetrization. Proof that H^n is Lebesgue on R^n
  
  21/10: Isodiametric inequality in R^n by Steiner symmetrization. Proof that H^n is Lebesgue on R^n File
- Seleziona attività 22/10: H^k rectifiable sets.Definition of distribution. Order of a distribution.
  
  22/10: H^k rectifiable sets.Definition of distribution. Order of a distribution. File
- Seleziona attività 27/10: order of a distribution, example of distributions, principal value. Derivative of a distribution
  
  27/10: order of a distribution, example of distributions, principal value. Derivative of a distribution File
- Seleziona attività 28/10: weak derivatives and derivatives in the sense of distributions.
  
  28/10: weak derivatives and derivatives in the sense of distributions. File
- Seleziona attività 29/10: Cantorian measures. Convolution of a distribution and a test function
  
  29/10: Cantorian measures. Convolution of a distribution and a test function File
- Seleziona attività Cantor function (Ambrosio, Fusco, Pallara book)
  
  Cantor function (Ambrosio, Fusco, Pallara book) File
- Seleziona attività 3/11: density of test functions in the space of distributions. Fundamental solution of a differential operator.
  
  3/11: density of test functions in the space of distributions. Fundamental solution of a differential operator. File
- Seleziona attività 4/11: Fundamental solution of the laplacian. Introduction to Sobolev spaces.
  
  4/11: Fundamental solution of the laplacian. Introduction to Sobolev spaces. File
- Seleziona attività 4/11 problem session
  
  4/11 problem session File
- Seleziona attività 5/11: Sobolev spaces in dimension 1.
  
  5/11: Sobolev spaces in dimension 1. File
- Seleziona attività 17/11: density of smooth functions. Gagliardo, Nirenberg, Sobolev inequality.
  
  17/11: density of smooth functions. Gagliardo, Nirenberg, Sobolev inequality. File
- Seleziona attività 18 and 19 november: GNS inequality and Poincare' inequality. First eigenvalue of the laplacian with Dirichlet bdry conditions (Ref. Evans PDE ch, 6, thm 2)
  
  18 and 19 november: GNS inequality and Poincare' inequality. First eigenvalue of the laplacian with Dirichlet bdry conditions (Ref. Evans PDE ch, 6, thm 2) File
- Seleziona attività 24/11: Morrey inequality and consequences
  
  24/11: Morrey inequality and consequences File
- Seleziona attività 25/11: Lipschitz continuous functions and W^{1, infty}, Rademacher theorem
  
  25/11: Lipschitz continuous functions and W^{1, infty}, Rademacher theorem File
- Seleziona attività 26/11: extension and traces for Sobolev functions
  
  26/11: extension and traces for Sobolev functions File
- Seleziona attività 1/12: Continuous embeddings for W^{1,p}(U), U bdd of class C^1. Compact embeddings.
  
  1/12: Continuous embeddings for W^{1,p}(U), U bdd of class C^1. Compact embeddings. File
- Seleziona attività 2/12: proof of Rellich-Kondrachov theorem. Corollary (W^{1,p} is compactly embedded in L^p)).
  
  2/12: proof of Rellich-Kondrachov theorem. Corollary (W^{1,p} is compactly embedded in L^p)). File
- Seleziona attività 3/12: a classical example from Calculus of Variations. Poincare' inequality
  
  3/12: a classical example from Calculus of Variations. Poincare' inequality File
- Seleziona attività 9/12: W^1,n is continuously embedded in BMO. The harmonic extension of a W^1,2 function.
  
  9/12: W^1,n is continuously embedded in BMO. The harmonic extension of a W^1,2 function. File
- Seleziona attività 10/12: the space of BV functions
  
  10/12: the space of BV functions File
- Seleziona attività 15/12:strong convergence, strict convergence and weak star convergence in BV. BV functions in dimension 1.
  
  15/12:strong convergence, strict convergence and weak star convergence in BV. BV functions in dimension 1. File
- Seleziona attività 16/12: BV functions in dim>1: density in strict sense of W^{1,1}, GNS inequality, compactness theorem
  
  16/12: BV functions in dim>1: density in strict sense of W^{1,1}, GNS inequality, compactness theorem File
- Seleziona attività 16/12: problem session
  
  16/12: problem session File
- Seleziona attività 17/12: Sets of finite perimeter. Main properties.
  
  17/12: Sets of finite perimeter. Main properties. File