Vai al contenuto principale
Se prosegui nella navigazione del sito, ne accetti le politiche:
Condizioni di utilizzo e trattamento dei dati
Prosegui
x
Italiano (it)
English (en)
Français (fr)
Italiano (it)
Ospite
Login
Macroarea STEM
Home
Calendario
Unipd
Portale Didattica
Orari
Uniweb
Webmail
My Media
Home
Calendario
Unipd
Portale Didattica
Orari
Uniweb
Webmail
My Media
Altro
Espandi tutto
Minimizza tutto
Apri indice del corso
2024-SC2651-002PD-2024-SCQ0094119-N0-SC2651
Course Record
Course Record
Schema della sezione
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