On Korn’s Inequality

For β ∈ R, the authors consider the evolution system in the unknown variables u and α { ttu＋ xxxxu＋ xxtα＋（β＋｜｜ xu｜｜L2^2） xxu=f, ttα- xxα- xxtα- xxtu=0} describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.

Regular Global Attractors of Type III Thermoelastic Extensible Beams

The author first reviews the classical Korn inequality and its proof. Following recent works of S. Kesavan, P. Ciarlet, Jr., and the author, it is shown how the Korn inequality can be recovered by an entirely different proof. This new proof hinges on appropriate weak versions of the classical Poincaré and Saint-Venant lemma. In fine, both proofs essentially depend on a crucial lemma of J. L. Lions, recalled at the beginning of this paper.