Sharp conditions for Korn inequalities in Orlicz spaces
As is well-known, Korn's inequality is a crucial tool in the mathematical approach to the behavior of Newtonian fluids. In the paper under consideration, the authors show (Theorem 1.1) that Korn's inequality in Orlicz spaces is valid if and only if the corresponding Orlicz function \(\varphi\) satisfies both the \(\Delta_2\) and the \(\nabla_2\) conditions. We remind the reader that a convex, left-continuous function \(\varphi\) such that \(\varphi(0)= 0\), \(\lim_{t\to 0^+}\varphi(t)= 0\), and \(\lim_{t\to\infty} \varphi(t)=\infty\) verifies the \(\Delta_2\) condition if \(\varphi(2t)\leq K\varphi(t)\), for all \(t\geq 0\), with \(K>0\) a constant. The \(\nabla_2\) condition is the \(\Delta_2\) condition of the conjugate function \(\varphi^*\). It is interesting to point out that Korn's inequality fails on \(L^1\), \(L\log L\), Exp and \(L^\infty\). Another application of Theorem 1.1 is to the solvability of the Poisson equation with zero boundary data in an Orlicz space generated by \(\varphi\).
scientific article; zbMATH DE number 6122077
scientific article