# Estimate of a general integral involving laplacians

I have two real functions $$u,\eta$$ defined on $$R^n$$ and with compact support so we can do all integrations by parts we want, and we need to estimate $$\int | \nabla \eta \cdot \nabla u|^2$$ by $$\int | \eta \Delta u|^2 + \int | u \Delta \eta|^2 +\int | u \eta^{(k)}|^2$$ (with any factor depending on the dimension for the inequality), and where by $$\int | u \eta^{(k)}|^2$$ I mean any integral with $$u$$ and any derivative of $$\eta$$ ($$\eta$$ is just a smooth cut-off function). Thanks !