1) L. C. Evans. Partial Differential Equations. AMS 1998.
2) F. John. Partial Differential Equations. Springer (4th edition).
3) Notes provided by the teacher.
Learning Objectives
a) Get acquaintance with some basic problem related to partial differential equations and, especially, with the Cauchy problem.
b) Get acquaintance with the inverse problems
Prerequisites
Functional Analysis. Normed spaces and continuous linear maps. Hahn-Banach theorem. Banach spaces. Hilbert spaces. Differential calculus in R ^ n. Theory of Lebesgue measure. L ^ p spaces. Holder spaces. Convolution. Distributions. Fourier transform. Sobolev spaces. Harmonic functions. Boundary value problems theory for elliptic equations : L ^ 2 theory.
Teaching Methods
Lectures
Type of Assessment
Oral examination at the end of course
Course program
1)BASIC TOOLS.
a) Functional Analysis. Normed spaces and continuous linear maps. Hahn-Banach theorem. Banach spaces. Hilbert spaces. Lax-Milgram theorem. Compact operators. Spectral theorem.
b) Differential calculus in normed spaces. Frechét derivative. Inverse and implicit function theorem. Regular open set in R ^ n.
c) Spaces L ^ p. Holder spaces. Convolution. Distributions. Fourier transform. Sobolev spaces.
d) Harmonic functions. Heat equation. Equation of vibrating strings.
e) Boundary value problems theory for elliptic equations:L ^ 2 theory. Mention to boundary value problems for parabolic equations.
2) THE CAUCHY PROBLEM FORPARTIAL DIFFERENTIAL EQUATIONS.
a) First order equations.
b) Equations in the analytical field. Cauchy Kovalevski Theorem. Holmgren Theorem.
c) Hadamard’s definition of well-posed problem
d) Uniqueness and continuous dependence of solutions to equations with nonanalytic coefficients. Stability estimates for elliptic equations of second order. Three sphere inequality of the three.
3. EXAMPLES OF INVERSE PROBLEMS. Tomography. Backward problem for the heat equation. Inverse problems in potential theory.
4. THE INVERSE PROBLEM OF CONDUCTIVITY '
a) Formulation of the Problem. Definition of the Dirichlet to Neumann map and its main properties.
b) Proof of uniqueness for inverse problems: Kohn-Vogelius proof, the method of singular solutions, the method of oscillating solutions.
c) Stability estimates.
5 REGULARIZATION METHODS FOR ILL-POSED PROBLEMS