PDE Seminar A selection criterion for solutions for a system of eikonal equations

2010-5-17 16:30 - 2010-5-17 17:30
Faculty of Science Building #3 Room 202
Giovanni Pisante (Hokkaido University)
We deal with the system of eikonal equations |du/dx|=1 , |du/dy|=1 in a
planar Lipschitz domain with zero boundary condition. Exploiting the
classical pyramidal construction introduced by Cellina, it is easy to
prove that there exist infinitely many Lipschitz solutions.
Then, the natural problem that has arisen in this framework is to find a
way to select and characterize a particular meaningful class of solutions.
We propose a variational method to select the class of solutions which
minimize the discontinuity set of the gradient. More precisely we select
an optimal weighted measure for the jump set of the second derivatives of
a given solution of the system and we prove the existence of minimizers of
the corresponding variational problem.
This is a joint work with G. Croce from the University of Le Havre.