Improved Rellich inequalities for the polyharmonic operator G. Barbatis 35J3535P1535J4026D10Polyharmonic operatorHardy-Rellich inequalitydistance to the boundarysharp constants We prove two improved versions of the Hardy-Rellich inequality for the polyharmonic operator $(-\Delta)^{m}$ involving the distance to the boundary. The first involves an infinite series improvement using logarithmic functions, while the second contains $L^{2}$ norms and involves as a coefficient the volume of the domain. We find explicit constants for these inequalities, and we prove their optimality in the first case. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2752 10.1512/iumj.2006.55.2752 en Indiana Univ. Math. J. 55 (2006) 1401 - 1422 state-of-the-art mathematics http://iumj.org/access/