Course Unit


Advanced analysis

  • Unit Coordinator: Corrado Lattanzio
  • Programme: Erasmus Mundus, Double Degrees
  • ECTS Credits: 6
  • Semester: 1
  • Year: 2
  • Campus: University of L'Aquila
  • Language: English
  • Aims:

    Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.

  • Content:
    • Distributions. Locally integrable functions. The space of test function D(Ω). Distributions associated to locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.;
    • Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).;
    • Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and embedding theorem for p n. Sobolev in- equalities in the general case. Compact embeddings: Rellich–Kondrachov theorem, Poincaré inequalities. Embedding theorem for p = n. Characterization of the dual space H−1.;
    • Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.;
      First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem.
  • Pre-requisites:

    Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

  • Reading list:

    H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universi- text, Springer.;
    C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer.;
    L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.;
    G. Gilardi, Analisi 3. McGraw–Hill.;
    V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc.


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