Introduction to the theory of pdes for applied
first order pdes and single
pdes of higher order, examples from continuum mechanics
Symbol of a pde and of systems; characteristics; existence, uniqueness and
continuous dependence on the data; well- and ill-posedness.
(Brief exposition of necessary functional analysis, e.g. operator theory,
distributions, Sobolev spaces,
see below *).
Weak and strong solutions.
Maximum principles for elliptic and parabolic pde's, existence of solutions.
Linear elliptic pde's, coercivity/energy estimates; Lax-Milgram lemma,
Garding's inequality, existence and uniqueness of weak solutions.
Evolutionary pde's - abstract parabolic initial value problems, energy
methods, uniqueness and existence.
Nonlinear elliptic pde's, monotone operators, existence of a weak solution.
Systems of hyperbolic equations; Symmetrisable systems; well-posedness.
Introduction to semi-group methods.
Undergraduate courses on real analysis and partial differential equations
(methods courses) will be assumed without explicity mention. * Functional
analysis is more problematic (as applied mathematics students may not have
taken such options) but time constraints prevent assuming no prior
knowledge. Probably the best way forward is to present some necessary
functional analysis briefly during the lectures and to provide ädditional"
notes online and together with careful page references to books covering the
material in the hope that students who have little or no functional analysis
will wish to learn more in ßelf-study" as a means to coming to a deeper
understanding of the "theory" of pde's
(eg other courses which this course fits in with)
Other Magic courses will/may provide useful pre-requisites:
MAGIC003: Introduction to Linear Analysis
MAGIC018: Linear Differential Operators in Mathematical Physics
No syllabus information is available yet.