On the optimal constants in Korns and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions

We are concerned with the optimal constants: in the Korn inequality under tangential boundary conditions on bounded sets $Omega subset mathbb{R}^n$, and in the geometric rigidity estimate on the whole $mathbb{R}^2$. We prove that the latter constant equals $sqrt{2}$, and we discuss the relation of the former constants with the optimal Korns constants under Dirichlet boundary conditions, and in the whole $mathbb{R}^n$, which are well known to equal $sqrt{2}$. We also discuss the attainability of these constants and the structure of deformations/displacement fields in the optimal sets.

A diophantine equation for sums of consecutive like powers

We show that the diophantine equation $n^ell+(n+1)^ell + ...+ (n+k)^ell=(n+k+1)^ell+ ...+ (n+2k)^ell$ has no solutions in positive integers $k,n ge 1$ for all $ell ge 3$.