PDE seminar: On Korn-type inequalities

2013-11-25 16:30 - 2013-11-25 17:30
Faculty of Science Building #3, Room 309
Hiroya Ito (The University of Electro-Communications)
Korn’s inequality, famous for its fundamental role in mathematical elasticity, asserts
that there exists a positive constant C such that

\|\epsilon(u)\|^2+\|u\|^2\geq C\|\nabla u\|^2

(\epsilon(u):=\frac{1}{2}(\nabla u+(\nabla u)^T), \|\cdot\|:=\|\cdot\|_{L^2(\Omega)}

for all H^1 vector fields u on a bounded Lipschitz domain \Omega in R^n. In this talk, after reviewing known facts about Korn’s inequality, we generalize it by replacing \epsilon(u) for a vector field u with P(\partial)u for a vector-valued function u where P(\xi) is a matrix whose entries are homogeneous polynomials of \xi = (\xi_1;\xi_2,\cdots,\xi_n) of degree 1. From Neˇcas’s work we know some necessary and sufficient conditions for such an inequality to hold. Our main result is obtaining another new condition, which is in general easier to check than the conditions by Neˇcas and enables us to rewrite his arguments to get them more clearly. Some other interesting consequences and important examples will be referred to as well.