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 !


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