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

Course Contents

 A. Laplace Equation and Harmonic Functions

• Divergence theorem and Green’s identities; mollification tools.
• Subharmonic functions: mean-value inequality and monotonicity of averages.
• Maximum principles: weak and strong versions.
• Comparison principle; uniqueness for the Dirichlet problem; counterexamples in unbounded domains.
• Harnack inequality and the Harnack principle for monotone limits.
• Weak Harnack inequality and the local maximum principle.
• Interior gradient estimates and the differential Harnack inequality.
• Liouville theorems and compactness of families of harmonic functions.
• Barrier constructions; Hopf’s lemma; quantitative Hopf–Oleinik lemma.
• Removable singularities for harmonic and subharmonic functions.
• Fundamental solution; Green functions; Poisson kernel; representation formulas.
• Basic characterizations of subharmonic functions.
• Perron’s method and boundary regularity via potential theory.
B. Heat Equation
• The Cauchy problem: heat kernel and representation formula.
• Qualitative properties of the heat flow: positivity and order preservation, smoothing estimates, conservation of mass, energy dissipation, decay, and infinite speed of propagation.
• The heat equation on a bounded interval: initial-boundary value problems, separation of variables, and Fourier series.
• Parabolic maximum and comparison principles, with applications to uniqueness.
• Energy estimates and uniqueness by the energy method.
C. Wave Equation
• The one-dimensional wave equation and d’Alembert’s formula.
• Finite speed of propagation and domain of dependence.
• 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 is based on lectures combining theoretical developments, detailed proofs, and worked examples. Particular emphasis will be placed on the analytical methods used to derive qualitative properties of solutions. Exercise sheets may be assigned occasionally, and selected problems may be discussed during the lectures or office hours.

Assessment methods

Assessment methods

The written examination is compulsory, lasts two hours and thirty minutes, and is graded on a 21-point scale. It focuses on standard problems involving the main concepts, methods, and qualitative principles developed during the course.

Students who do not take the written examination are not eligible to take the oral examination. Only students who take the written examination and obtain a score of at least 10/21 may proceed to the oral examination. Students who obtain a score below 10/21 must retake the written examination in a subsequent examination session.

Students who obtain a score between 10/21 and 17/21 must take the oral examination in order to complete the examination and reach the minimum passing grade of 18/30.

Students who obtain a score between 18/21 and 21/21 may either accept the written examination score as their final grade or take the optional oral examination in order to improve it.

The oral examination lasts up to 30 minutes and is graded on a 9-point scale. It consists of a discussion structured around three main questions, each worth 3 points, concerning the concepts, methods, and results developed during the course. Follow-up questions may be used to clarify the student’s answers and assess their understanding more accurately.

The final grade is obtained by adding the scores awarded in the written and oral examinations, up to a maximum of 30/30.

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

Selected problem sets and model problems may be assigned occasionally through the Virtuale platform. These materials, together with the examples discussed in class, will provide an excellent source of preparation for both the written and oral examinations and will illustrate the types of problems and arguments students are expected to master.

Teaching tools

Lecture materials, supplementary notes, and occasional exercise sheets will be made available through the Virtuale platform.

Office hours

See the website of Diego Ribeiro Moreira