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2024-SC2651-002PD-2024-SCQ0094119-N0-SC2651
Course Record
Course Record
Section outline
Select activity 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.
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Select activity 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.
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Select activity Oct 2: partitions of units. Boundaries and integration by parts formula
Oct 2: partitions of units. Boundaries and integration by parts formula
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Select activity 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
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Select activity October 8: Caratheodory theorem, total variation measure, Lebesgue Radon Nykodim theorem
October 8: Caratheodory theorem, total variation measure, Lebesgue Radon Nykodim theorem
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Select activity 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
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Select activity October 15: Hausdorff measures.
October 15: Hausdorff measures.
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Select activity 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
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Select activity 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
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Select activity On the measurability of subgraphs of measurable functions
On the measurability of subgraphs of measurable functions
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Select activity 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.
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Select activity 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
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Select activity October 22: Riesz (Markov) theorem on Radon measures (no proof).
October 22: Riesz (Markov) theorem on Radon measures (no proof).
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Select activity Oct 23: distributions and order of a distribution
Oct 23: distributions and order of a distribution
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Select activity 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.
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Select activity 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.
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Select activity 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
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Select activity 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.
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Select activity 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).
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Select activity 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.
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Select activity 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)
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Select activity 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).
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Select activity 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.
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Select activity 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.
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Select activity Nov 19: local version of GNS inequality and embeddings. Morrey inequality.
Nov 19: local version of GNS inequality and embeddings. Morrey inequality.
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Select activity 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.
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Select activity 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^*.
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Select activity 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.
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Select activity December 2: Harmonic extension. Poincare' inequality (statement)
December 2: Harmonic extension. Poincare' inequality (statement)
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Select activity 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.
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Select activity 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.
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Select activity 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.
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Select activity Dec 10: BV functions in dimension 1
Dec 10: BV functions in dimension 1
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Select activity Dec 11: sets of finite perimeter
Dec 11: sets of finite perimeter
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Select activity December 16: compactness for perimeters. The Plateau problem
December 16: compactness for perimeters. The Plateau problem
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