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Moreover, we consider Korn-type inequalities involving the trace-free part of the symmetric gradient εD(u) :=ε(u)−1 ddivu I and corresponding variational integrals

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Korn type inequalities in Orlicz spaces

Many problems in the mathematical theory of Generalized Newtonian fluids and in the mechanics of solids lead to the following question: it is possible to bound a suitable energy depending on∇uby the corresponding one in depen- dence onε(u) := 12 ∇u+∇uT

, i.e., Z

Ω

|∇u|pdx≤c(p,Ω) Z

Ω

|ε(u)|pdx

for all u ∈ W˚1,p(Ω,Rd)? This is true for all 1 < p < ∞. We discuss gen- eralizations of Korn’s inequality above to Orlicz spaces and their applications.

Moreover, we consider Korn-type inequalities involving the trace-free part of the symmetric gradient

εD(u) :=ε(u)−1 ddivu I and corresponding variational integrals.

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