Partial Differential Equations — 2^nd Semester — 2025/2026

Lecturer:	José Natário

	Email: jnatar@math.tecnico.ulisboa.pt

	Office: Mathematics Building, 4th floor, room 4.29

Classes:	Tuesdays from 13:30 to 15:30 in room V1.10, Thursdays from 9:30 to 11:30 in room V0.07, and Fridays from 14:30 to 15:30 in room V1.32.
Office Hours:	Drop by my office or send me an email

Annoucements

The final grades are now available (see side bar). If you want to take a look at your graded exams come to my office at your convenience.

First order equations: The transport equation: the homogeneous and nonhomogeneous cases. Linear and quasilinear equations: the characteristic methods for the Cauchy problem. One-dimensional conservation laws. Application to traffic dynamics (shock waves). Reference to the notion of weak solution in this context.

Second order linear equations: Classification of linear second order PDEs in dimension two: parabolic, elliptic and hyperbolic equations. Brief notions about characteristics and solutions to the Cauchy problem: statement and applications of Cauchy-Kovalevskaya's theorem. The Laplace and Poisson equations: fundamental solution, solution in the whole space, maximum principles, mean value property, uniqueness of solution to boundary value problems. Harmonic functions: estimates for the derivatives and analyticity. Green's function and Poisson's formula in the ball. Harnack inequality. The Dirichlet principle. The heat equation: fundamental solution and heat kernel, maximum principle and uniqueness of solution to initial-boundary value problems. Mass conservation. Duhamel's formula. The wave equation: D'Alembert's solution in dimension one. Duhamel's formula. Domains of dependence and influence. Solution to the Cauchy problem in dimensions two and three. Energy methods for the Poisson, heat and wave equations.

Distributions and Sobolev spaces: Brief introduction to the theory of distributions. Sobolev spaces and their basic properties. Sobolev inequalities. The space W₀^m,p(U), where U is an open set in R^N. Poincaré's inequality. The notion of trace. The extension theorem. Characterization of the dual space H^-1(U).

Theory of weak solutions for elliptic problems: Elliptic problems: second order elliptic operators, weak formulation. Application of Riesz' and Lax-Milgram's theorems to solve elliptic problems with different types of boundary conditions. Regularity results: inner and boundary L²-regularity. Statements for the general case.

Biliography

Main

H. Tavares, Partial Differential Equations

Secondary

Evans, Partial Differential Equations
Salsa, Partial Differential Equations in Action
Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations

Week 1 (due on February 27)
Week 2 (due on March 6)
Week 3 (due on March 13)
Week 4 (due on March 20)
Week 5 (due on March 27)
Week 6 (due on April 2)
Week 7 (due on April 24)
Week 8 (due on April 30)
Week 9 (due on May 8)
Week 10 (due on May 15)
Week 11 (due on May 22)
Week 12 (due on May 29)
Week 13 (due on June 5)

Exams

Exams from previous years: