LINEAR PARTIAL DIFFERENTIAL EQUATIONS
Academic year and teacher
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- Versione italiana
- Academic year
- 2017/2018
- Teacher
- ANDREA CORLI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- The purpose of the course is to provide an analytic introduction to linear partial differential equations, in particular to transport, Laplace, heat and wave equations.
The main knowledge gained will be the following.
• To Have a first introduction to partial differential equations and some of the methods used for their study.
• Understanding the fundamental difference existing between elliptic, parabolic and hyperbolic equations by studying the three types above.
• To acquire a basic knowledge on the main results related to the equations above.
The main skills developed will be:
• Mastering some classical techniques related to the differential and integral calculus in several variables, with particular reference to the integration on surfaces, to the formulas Gauss-Green and the averages of functions.
• To interpret physically the analytical results obtained.
• To solve explicitly some basic types of partial differential equations. Prerequisites
- It is absolutely needed a good knowledge of differential and integral calculus for functions of several variables and the rudiments of linear algebra. The knowledge of the physical interpretation of the equations, usually treated in a course as "Partial Differential Equations of Mathematical Physics" or "Continuum Mechanics" can be very useful but not essential.
Course programme
- The course includes 48 hours of teaching, including exercises. The main topics of the course are as follows.
• Introduction. General information on partial differential equations. Some analytical results: Gamma function, measure of balls and n-dimensional spheres, various results on integration. (6 hours)
• The transport equation. The initial value problem. The non-homogenous problem. Interpretation of the solution. (4 hours)
• The Laplace equation. Physical deduction of the equation. The fundamental solution. The Poisson equation. Average formulas. The principle of maximum; uniqueness of solutions. Regularity of solutions of the Laplace equation. Estimates on the derivatives of the solutions (P). The Liouville theorem. Representation of the solutions of the Poisson equation. Analytical solutions of the Laplace equation. The Harnack inequality. Green's function: general deduction and property. The Green's function for a half-space; Poisson's formula. The Green's function for a ball (*); Poisson's formula. Methods of energy. Uniqueness of solutions to the Poisson equation with the boundary values. The Dirichlet principle. (16 hours)
• The heat equation. Physical deduction of the equation. The fundamental solution. The initial value problem. The non-homogeneous problem: the Duhamel principle. Average formula (*). The principle of maximum; uniqueness of solutions. The maximum principle for the initial value problem; uniqueness of solutions. Nonphysical solutions (*). Regularity of solutions of the heat equation. Local estimates for the solutions (*); analytic and Gevrey regularity (*). Methods of energy: uniqueness of solutions. (12 hours)
• The wave equation. Physical deduction of the equation. The case n = 1: d'Alembert formula. The d'Alembert formula in a quadrant. The case n greater or equal to 2: the method of descent. The spherical averages and the Euler-Poisson-Darboux equation. The case n = 3: reduction of the Euler-Poisson-Darboux equation to the wave equation. The case n = 2 deduced from the case n = 3. The case n odd, n greater than or equal to 3: representation formula of the solution of the initial value problem (*); theorem of existence and uniqueness (*). The case n even, n greater than or equal to 2: representation formula of the solution of the initial value problem (*); Theorem of existence and uniqueness (*). The non-homogeneous problem. The special cases n = 1 and n = 3. Methods of energy. Uniqueness of solutions. The domain of dependence.(10 hours)
All statements above are completed by their proofs, except those where it appears (*); (P) indicates that the demonstration was carried out only in part. The course was completed by various exercises, many of which were taken from the book of Evans quoted below. Didactic methods
- The course is organized by classroom instruction and exercises. The exercises, at various levels, proposed week to week, are individually corrected by the teacher and discussed the following week with the students. Students are strongly advised to engage in this activity, in order to have direct control of their learning level and not only a purely theoretical knowledge of the course.
Learning assessment procedures
- As noted above, a first Assessment will be through the weekly resolution of simple exercises.
The final exam consists of an oral exam on the course contents. The student will be required
(a) to choose at least two important results for equation and discuss in detail the proof with the teacher (up to 20 points).
(b) a general knowledge of the theoretical content of the course (5);
(c) the ability to link the different parts of the program (5);
(d) the resolution of simple exercises, analogous to those already proposed in classroom (5). Reference texts
- L.C. Evans: Partial differential equations, second edition. American Mathematical Society (2010).
Notes are also provided by the teacher.
