Korn's inequality and eigenproblems for the Lamé operator
The paper deals with some extensions of Korn's inequality in the context of eigenproblems in elasticity to \(n\) spatial dimensions, with \(n \geq 2\). Two different sets of boundary conditions are considered, leading to the so-called tangential-normal and normal-tangential eigenproblems. In the first case, the tangential trace of the tractions and normal trace of the displacements vanish on the boundary of the domain. It is known that this problem enjoys a countable spectrum in the case of smooth domains. This result is extended to Lipschitz domains. In the second case, the normal trace of the tractions and the tangential trace of the displacements vanish on the boundary of the domain. It is shown that this problem also enjoys a countable spectrum in the case of Lipschitz domains. In addition, a third eigenproblem accounting for mixed boundary conditions is also considered, in which both conditions are assumed in different parts of the boundary of the domain. Finally, a nice set of analytic solutions is presented, supporting the theoretical finding reported in the paper.
scientific article; zbMATH DE number 7612438
scientific article