Subject: PARTIAL DIFFERENTIAL EQUATIONS (A.A. 2020/2021)
Unit Partial differential equations
Related or Additional Studies (lesson)
The course provides the student with the basic methods of Potential Theory in order to solve boundary value problems for uniformly elliptic second order Partial Differential Equations or second order equations with non-negative characteristic form. The course aims at introducing the students into the research activity. Some open problems are described in order to start a research activity to be developed in the thesis and possibili in a PhD program.
Calculus for functions of several real variables. Lebesgue integral. A basic knowledge of Functional Analysis.
- Mean value formulas for harmonic functions. Strong maximum principle. Weak solutions, regularity properties of weak solutions Harnack inequality, Liouville theorems.
- Fundamental solution to the Laplace equation. Poisson kernel. Green function.
- Perron method for boundary value problems on bounded open sets. Wiener theorem. Barrier functions, Zaremba criterion for the regularity of a boundary point.
- Heisenberg group. Sub-laplacian operator. Hormander hypoellipticty condition.
- Mean value formula for the heat equation. Parabolic maximum principle.
- Degenerate Kolmogorov equations. Fundamental solution.
Lectures, synchronous remote delivery will be guaranteed. Based on the evolution of the health emergency COVID19, the option of the theaching activity in presence will be evaluated.
The course will end with an oral exam. The exams will be carried out either in presence or in remote, depending on the evolution of the COVID19 situation.
1. Knowledge and understanding:
At the end of the course the student will have a deep knowledge of the Regularity Theory for Partial Differentiale Equatons.
2. Applying knowledge and understanding
At the end of the course the student will be able to solve boundary value problems for Elliptic Equations and Initial-boundary value problems for Parabolic Equations.
3. Making judgements:
Attitude towards a methodological approach that leads to verify claims and methods through rigorous arguments.
Ability to self-assessment of their skills and abilities.
4. Communication skills:
Ability to deal with a dialectical discourse in a timely and consistent way, arguing with precision.
Lasting acquisition of mathematical knowledge to be used at any time of their cultural journey.
5. Learning skills:
Attitude towards a methodological approach that leads to an improvement in the method of study in order to deepen learning capacity.
- L.C. Evans - Partial differential equations - AMS, Providence, RI
- D. Gilbarg, N.S. Trudinger - Elliptic partial differential equations of second order - Springer