‌

Elliptic Partial Differential Equations: Existence and Regularity of Solutions

Registro delle lezioni

Anno accademico 2022/2023
Docente Luigi Ambrosio

Lecture

  • 04 Nov 2022 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Basic introduction to the theory of Sobolev spaces

  • 11 Nov 2022 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Basic examples of PDEs, boundary conditions, systems

  • 18 Nov 2022 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Rellich compactness theorem and Poincare' inequality. Proof of Garding inequality and of Tonelli lower semicontinuity theorem

  • 25 Nov 2022 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Morrey's theorem and quasi convexity

  • 02 Dic 2022 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Differentiation of measures, Hausdorff measures, introduction to Caccioppoli inequality

  • 09 Dic 2022 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Proof of Caccioppoli inequality, Sobolev embeddings, Widman's technique

  • 16 Dic 2022 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Nirenberg's method, means and medians, decay estimates for elliptic systems with constant coefficients

  • 13 Gen 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Regularity up to the boundary. Interior estimates for nonlinear problems. Introduction to Campanato and Morrey spaces.

  • 20 Gen 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Proof of Schauder theorem, propagation of regularity in Morrey and Campanato spaces

  • 02 Feb 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    The space BMO and its characterizations. Weak L^p spaces, Maximal operators and Calderon-Zygmund decomposition

  • 10 Feb 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Interpolation theorems, Stampacchia interpolation. Helmholtz decomposition

  • 17 Feb 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Elliptic systems by duality. The case p=1. Coarea formula, Sobolev embedding and isoperimetric inequalities

  • 24 Feb 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    De Giorgi's solution of Hilbert's XIX problem. Step 1: L^2 to L^infty estimate

  • 03 Mar 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Holder regularity of solutions, completion of proof. De Giorgi's counterexample to full regularity. Introduction to Evans' theorem

  • 10 Mar 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Completion of the proof of Evans' theorem. Nonlinear Caccioppoli inequality. Estimates on the dimension of the singular set.

  • 17 Mar 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Introduction to the theory of viscosity solutions. Perron's method

  • 24 Mar 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Comparison principle for sub and super solutions of first order. Characterization of the distance function in terms of solutions to the Eikonal equation. Preliminaries on convex functions

  • 03 Apr 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Alexandrov theorem on differentiability of convex functions, Jensen's maximum principle, inf and sup convolutions

  • 14 Apr 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Proof of the comparison principle and applications to existence and uniqueness results for boundary value problems in the viscosity sense

  • 21 Apr 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Proof of the Alexandrov-Bakelman-Pucci estimate

  • 28 Apr 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Proof of the Krylov-Safonov regularity theorem

Vai alla Ricerca avanzata