The role of conservation laws and integrability by compensation in the analysis of conformally invariant variational problems
speaker: Tristan Rivière (ETH, Zürich)abstract: In this mini-course we propose to explore regularity and compactness properties for critical points to conformally invariant lagrangian playing a special role in differential geometry : Harmonic maps into riemannian and pseudo-riemannian manifolds, minimal surfaces, prescribed mean curvature surfaces, Willmore surfaces, 12-harmonic maps and bi-harmonic maps into manifolds ...etc. We will isolate common features to all these objects and in particular show the existence of "hiden" conservation laws which do not enter in the classical framework of Noether's Theorem. We will give an introduction to the theory of integrability by compensation and we will show how, combined with the existence of conservation laws, this theory permits to overcome a number of analysis difficulties of these particular elliptic partial differential systems. Prerequisit : Elementary Functional and Fourier Analysis. Classical Function Spaces from Calculus of variations (Lp, Sobolev, Hoelder). Elementary theory of Linear elliptic PDE in these spaces. Basic concepts from differential geometry (1st and 2nd fundamental form of a submanifold in Rn, mean curvature, Gauss map...)
timetable:
Thu 10 Sep, 9:30 - 10:30, Aula Dini
Thu 10 Sep, 11:00 - 12:00, Aula Dini
Thu 10 Sep, 15:00 - 16:00, Aula Dini
Mon 14 Sep, 9:30 - 10:30, Aula Dini
Mon 14 Sep, 11:00 - 12:00, Aula Dini
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