5 cu.

=20Advanced studies

=20The course is about Sobolev mappings between domains in the Euclidean&nb=
sp; *n*-space which satisfy certain nonlinear partial differential e=
quations (PDEs). They appear in Geometric Function Theory (GFT) as quasicon=
formal mappings and as mappings of finite distortion. In materials science,=
on the other hand, such mappings are studied as deformations of elastic bo=
dies. Both theories are governed by the energy functionals defined on a cla=
ss of Sobolev mappings, predominantly homeomorphisms. The central issues ar=
e about existence, uniqueness and regularity of the energy-minimal deformat=
ions, also about the associated phenomena such as occurrence of cracks and =
fractures. Understanding the topological curiosities of cracks seems =
to be worth the effort of developing this subject into a viable theory. The=
oretical prediction of failure of bodies caused by cracks and fractures is =
a good motivation that should appeal to mathematical analysts and researche=
rs in the engineering fields.

** Moral of the Lectures.** If one dark=
rainy night you find yourself in the midst of Sobolev Mappings, Nonlinear =
PDEs and Variational Integrals, then you should remind yourself=
that you may cleverly exploit the ideas and methods of

*=E2=80=9CEvery block of stone has a statue inside it and it is the t=
ask of the sculptor to discover it.=E2=80=9D*

* &n=
bsp; =
; &n=
bsp; =
; &n=
bsp; Michelangelo di Lodovico Buonarroti Simoni*

Heart-felt welcome to everybody

=20**Tadeusz Iwaniec**

FiDiPro, Helsinki University

=20I will assume basic knowledge of measure theory, integration, analytic f=
unctions and some functional analysis.

=20

To make these lectures available to students whose mathematical knowledg= e may be limited, every effort will be made to reduce to a minimum the tech= nical details and mathematical requirements. No prior exposure to these top= ics is assumed.

=20Weeks 3-9 and 11-18, Friday 12-14 in room C123. The first lecture is on = Friday the 20th of January.

=20Easter holiday 5.4.-11.4.

=20There will be no exams or exercise sessions held. Instead, students will= be evaluated based on their participation on the lectures. You are invited= to pursue the exercises given occasionally during the course.

=20Notes will be provided during the course.