偏微分方程式セミナー: On Korn-type inequalities

開催日時
2013年   11月 25日 16時 30分 ~ 2013年   11月 25日 17時 30分
場所
北海道大学理学部3号館309室
講演者
伊東裕也氏 (電気通信大学共通教育部数学教室)
 
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.

関連項目

研究集会・セミナー・集中講義の一覧へ