07/10/2005, 15:00 — 16:00 — Conference Room, Instituto de Sistemas e Robótica, North Tower, 7th floor, IST
Diogo Gomes, CAMGSD/IST
The strange world of partial differential equations
This will be the first of a series of introductory talks on the modern theory of Partial Differential Equations both for engineers and (non-PDE) mathematicians. The plan of the first three talks is as follows:
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Formulas, Examples and Counterexamples
We will discuss general methods for constructing solutions of PDEs (First-order, Laplace, Heat, and wave equations), present several examples concerning existence, uniqueness or regularity of solutions, which, we hope, motivate the need for considering weak solutions, study existence, uniqueness and regularity issues for those weak solutions. -
Calculus of Variations and PDE
In this lecture we will survey several connections between calculus of variations (including control theory as a subset of CV) and PDEs. We will discuss a typical problems in calculus of variations: the Dirichlet integral and Laplace's equation, control theory and Hamilton-Jacobi equations, gradient-flows and dissipative equations, and the role of symmetries and conservation laws in other classes of equations such as wave or KdV. -
Probability and PDE
In the last lecture of this series we will discuss how ideas from probability arise in the study of elliptic and parabolic equations, stochastic control and Hamilton-Jacobi equations, as well as in certain models in fluid mechanics. Hopefully, at this point, other(s) speaker(s) will take over, and in this second part other we will consider (also depending on the speaker) topics such as: -
Functional analysis methods
Abstract results for solutions of linear equations such as Lax-Milgram theorem (elliptic equations) or semigroup theory (evolution equations). -
Harmonic Analysis and PDE
Introduction to Fourier transform methods and the modern theory of dispersive non-linear equations.
