Title: Improved Rellich inequalities for the polyharmonic operator

Authors: Gerassimos Barbatis

Issue: Volume 55 (2006), Issue 4, 1401-1422

Abstract: 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.