MAGIC: Theory of Partial Differential Equations (MAGIC058)

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Introduction to the theory of pdes for applied mathematics.

Undergraduate courses on real analysis and partial differential equations(methods courses) will be assumed without explicit mention. Functionalanalysis is more problematic (as applied mathematics students may not havetaken such options) but time constraints prevent assuming no priorknowledge. Probably the best way forward is to present some necessaryfunctional analysis briefly during the lectures and to provide ädditional"notes online and together with careful page references to books covering thematerial in the hope that students who have little or no functional analysiswill wish to learn more in ßelf-study" as a means to coming to a deeperunderstanding of the "theory" of PDEs.

• Systems of 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).
• 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.

There are two assessments. They are equally weighted. Assessment 1 was due April 9th. Assessment 2 is due May 30th.