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BIRS Workshop Lecture Videos

Korn type inequalities in Orlicz spaces Cianchi, Andrea


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.

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