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The aim of the course is to study a variational model for the energy of dislocations’ configurations in metal plasticity.
Metals are crystals which exhibit a local order in regions, called grains, where the underlying structure is nearly a rotation of a reference lattice. These grains are separated by boundaries and the incompatibilities of the crystal lattices along these boundaries are often resolved by means of defects, such as dislocations.
Among the various mathematical models introduced to explain such phenomena, a remarkable one can be found in a recent paper by Lauteri-Luckhaus, where the authors give the first rigorous proof of the Read-Shockley logarithmic energy estimates for the line tension between small rotated planar grain boundaries. The Lauteri-Luckhaus energy functional has a geometric flavor, which captures the basic energetic competition behind the emergence of crystalline defects. Moreover, their analysis is a masterpiece of mathematics, combining various mathematics ranging from harmonic analysis to PDEs techniques, to geometric measure theory, to discrete mathematics.
The main objective of the course is to explain the Lauteri-Luckhaus results and, time permitting, further recent developments and perspectives.