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.


Print  

Related Articles