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Korn type inequalities in Orlicz spaces

Banff International Research Station for Mathematical Innovation and Discovery

A standard form of the Korn inequality amounts to an estimate for the $L^p$ norm ($1<p<\infty$) of the full gradient of a vector-valued function in terms of the same norm of just its symmetric part. It is well known that a result of this kind may fail if the $L^p$ norm is replaced by a more general Orlicz norm $L^A$ associated with a Young function $A$. We shall show that a Korn type inequality in Orlicz spaces can be restored if possibly different norms $L^A$ and $L^B$ are allowed on the two sides of the inequality, provided that the Young functions $A$ and $B$ satisfy suitable, necessary and sufficient balance conditions. Related inequalities for trace-free symmetric gradients, for the Bogovskii operator, and for negative Orlicz-Sobolev norms will also be discussed. Part of this talk is based on collaborations with D.Breit and L.Diening.

Mathematics; Partial differential equations; Calculus of variations and optimal control; optimization; Global analysis

video/mp4

Author affiliation: Università di Firenze (Italy)

2017-02-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60307

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/