Hilbert spaces: topological and geometrical properties; dual space; bounded and compact operators; Fredholm alternative; spectrum. Fourier series: pointwise convergence. Fourier transform: L2-theory. Distributional derivative; distributions with compact support; fundamental theorem of calculus for distributions. Harmonic functions: mean value property; maximum principle; Harnack’s inequality; analiticity. Laplace operator: existence and uniqueness for the Dirichlet problem; eigenvalues.
Evans, Partial Differential Equations, Graduate Studies in Mathematics, AMS, Providence, RI, USA.
Garabedian, Partial Differential Equations, AMS Chelsea Publishing, Providence, RI, USA.
Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, AMS, Providence, RI, USA.
Learning Objectives
Knowledge acquired:
The fundamental ideas of functional analysis in Hilbert spaces, of the theory of distributions, of harmonic analysis, the corresponding techniques of proof and some of their relevant applications.
Competence acquired:
The basic methods of proof in functional and harmonic analysis and the connections of this topics to other disciplines of mathematics.
Skills acquired (at the end of the course):
The student is expected to correctly state, prove and apply the main results of functional and harmonic analysis and the theory of distribution, and to solve exercises and problems related to those disciplines.
Prerequisites
Courses required: Mathematical Analysis 1, 2 and 3; Geometry 1 and 2.
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 153
Hours for lectures: 72
FUNCTIONAL ANALYSIS IN HILBERT SPACES
• Definition of dot product, Cauchy-Schwarz inequality, the parallelogram identity.
• Definition of Hilbert space and examples.
• Projection on a convex set.
• Orthonormal systems. Series and Fourier coefficients. Bessel's inequality.
• Completeness of orthonormal systems. Parseval identity.
• Linear and bounded functionals. Dual space. Riesz representation theorem. Weak convergence.
Banach-Alaoglu's theorem.
• The theorems of Stampacchia and Lax-Milgram.
• Open mapping theorem and its consequences.
• Banach-Steinhaus's theorem.
• Linear and continuous operators.
• Compact operators: examples and first properties.
• Fredholm alternative's theorem.
• Spectrum of a compact operator.
• Spectral decomposition of a symmetric and compact operator.
Fourier series
• Trigonometric polynomials.
• Series and Fourier coefficients.
• Bessel's inequality.
• Riemann-Lebesgue's lemma.
• Dirichlet’s kernel.
• Dini's and Jordan's criteria.
• Uniform convergence of Fourier series.
• Completeness of trigonometric system in the space of square integrable functions.
• Parseval identity.
• Other types of convergence. Fejer's and Poisson's kernels.
• Uniform approximation by polynomials: Weierstrass theorem.
Fourier Transform
• Fourier transform of a summable function.
• Behavior of Fourier transform with respect to dilation, translation, rotation and convolutions.
• The space of Schwarz. Derivation and transform.
• Transform of a Gaussian.
• Plancherel theorem and Parseval identity. Definition of Fourier transform in the space of square integrable functions.
• Inversion formula.
• Summabillity kernels: Dirichlet, Fejer-Weiestrass Gauss, Abel-Poisson.
• Transform of Poisson's kernel..
• Poisson’s addition formula.
• Solution of some boundary problems for partial differential equations by separation of variables and by using Fourier's transform. Fundamental solution of Laplace equation and the heat equation.
DISTRIBUTIONS
• Space of test functions and its topology.
• Definition of distribution and of his order.
• Space of distributions and its topology.
• Distributional derivative. Definition of Sobolev spaces.
• Operations on distributions.
• Regularity of convolution.
• Characterization of the space of distributions with compact support. Metric space of infinitely differentiable functions.
• Fundamental theorem of calculus for distributions. Distributions with null gradient are constants.
• Temperate distributions and Fourier transform.
HARMONIC FUNCTIONS
• Mean value property.
• Harmonic functions are infinitely differentiable.
• Maximum principle.
• Hopf principle.
• Harnack's inequality.
• Liouville's theorem.
• Harnack's convergence theorems.
BOUNDARY Value PROBLEMS FOR LAPLACE AND POISSON's EQUATIONS
• Poisson’s equation in Euclidean space and its resolution.
• Fundamental solution for Laplace's equation.
• Uniqueness theorems for Dirichlet, Neumann and Robin problems.
• Boundary value problems in unbounded domains. Kelvin transformation.
• Stokes identity and definition Green’s function.
• Representation formula for the solution of Dirichlet's problem.
• Symmetry of Green’s function.
• Construction of Green's function and Poisson's kernel for half space and sphere.
• Analyticity of harmonic functions.
• Subharmonic and superarmonic functions.
• Perron's method.
• Barrier functions. Regular and exceptional points.
• Existence of Green’s function.
• Dirichlet's principle and Lax-Milgram's theorem.
• Conversion Dirichlet or Neumann problem into an integral equation and application of Fredholm's alternative theorem.
• Eigenvalues and eigenfunctions of Laplace's operator.