96758 - PDEs

Learning outcomes

At the end of the course, students will be able to study linear PDEs of first and second order, mainly by classical methods. This knowledge is fundamental to all the theoretical and modelling. applications.

Course contents

Syllabus — Partial Differential Equations

 A. Laplace Equation and Harmonic

• Divergence theorem and Green’s identities; mollification tools

• Subharmonic functions: mean–value inequality and monotone averages

• Mean–value expansion up to o ( ε 2 ) o(ε2)

• Maximum principles (weak and strong versions)

• Comparison principle; Dirichlet uniqueness; unbounded-domain counterexamples

• Harnack inequality and the Harnack principle (monotone limits)

• Weak Harnack inequality and the local maximum principle

• Interior gradient bounds and the differential Harnack inequality

• Liouville theorems and compactness of harmonic families

• Poisson equation: nonhomogeneous gradient estimates via maximum principle

• Barrier constructions; Hopf lemma; quantitative Hopf–Oleinik lemma

• Removable singularities for harmonic/subharmonic functions

• Fundamental solution; Green functions; Poisson kernel; representation formulas

• Local subharmonicity and equivalence of notions (mean / distribution / potential / viscosity)

• Perron’s method and boundary regularity via potential theory

B. Heat Equation

• Heat kernel and representation formula (Fourier-light derivation)

• Initial-value problems and Duhamel’s principle

• Parabolic maximum principles and comparison; uniqueness

• Growth conditions and maximum principle on unbounded domains

• Interior gradient estimates via maximum principle

• Parabolic Harnack inequality for positive solutions

C. Wave Equation

• Finite propagation speed and domain of dependence

• The 1D wave equation and d’Alembert’s formula

• Spherical means and representation formulas in higher dimensions

• Energy identity, uniqueness, and stability estimates

Readings/Bibliography

Han, Qing. A Basic Course in Partial Differential Equations.Graduate Studies in Mathematics, Vol. 120. Providence, RI: American Mathematical Society (AMS), 2011. ISBN 978-0-8218-5255-2.

Evans, Lawrence C. Partial Differential Equations. Second edition. Graduate Studies in Mathematics, Vol. 19. Providence, RI: American Mathematical Society, 2010. xxii + 749 pp. ISBN-13: 978-0-8218-4974-3.

Teaching methods

The course consists of lessons describing the fundamental concepts of the program. Lessons are completed with examples illuminating the theoretical content.

Assessment methods

Assessment Methods (PDE Course)

The examination consists of a written examination followed by an oral examination.

1. Written Examination

The written examination lasts 2 hours and 30 minutes and is graded on a scale of 0–18 points.

The written exam focuses on standard problems covering the main theoretical and applied aspects of the course.

In order to be admitted to the oral examination, students must obtain at least 8/18 in the written exam.

Students who do not reach this threshold must retake the exam in a subsequent session.

2. Oral Examination

The oral examination lasts up to 30 minutes and is graded on a scale of 0–12 points.

The oral exam consists of three specific questions:

• Question 1: A conceptual question concerning definitions, fundamental properties, or qualitative principles discussed during the course.

• Question 2: Reproduction or discussion of the solution to one of the problems from the written examination (selected by the instructor), with emphasis on the key ideas and logical structure of the argument.

• Question 3: Either a simple exercise on a course topic or a discussion of the main idea behind a proof, method, or example presented during the lectures.

The oral examination is intended to assess conceptual understanding, clarity of exposition, logical rigor, and mathematical maturity.

3. Final Grade

The final grade (expressed on a 0–30 scale) is calculated as the sum of:

• the written examination score (maximum 18 points), and

• the oral examination score (maximum 12 points).

Students pass the examination if they obtain a final score of at least 18/30.

Honors (30 cum laude) may be awarded at the discretion of the instructor to students who achieve a final score of 30/30and demonstrate outstanding mastery of the subject.

4. Problem Sets and Preparation

Representative problem sets will be posted every 2–3 weeks on the Virtuale platform. These sets, together with the examples discussed in class, serve as the primary reference guide for the types of problems expected in both the written and oral examinations.

Students are encouraged to attend office hours to discuss these problems and to clarify any questions related to the course material.

Teaching tools

Set of Exercises on Virtuale (UNIBO) every 2 - 3 weeks

Office hours

See the website of Diego Ribeiro Moreira