interpolation inequality- how we use it to get into this inequality

Question

I was working to get some inequality, and the author use the following inequality and call it "interpolation inequality"

$$\|u\|_{L^2} \leq c\|u\|_{H^{-1}}^{\frac{1}{2}}\|\nabla u\|_{L^2}^{\frac{1}{2}}\leq c' \|u\|_{H^{-1}}^{\frac{1}{2}}\|\Delta u\|_{L^2}^{\frac 1 2}$$

set $\| \cdot \|_{H^{-1}}=\|(-\Delta)^{-\frac{1}{2}}\ \cdot\|_{L^2}$, where $-\Delta$ is the minus laplace operator associated with the Neumann boundary conditions and acting on functions with null average.

I try to find how he get it but i didn't know, i read the usual interpolation inequality and interpolation inequality of Gagliardo-Nirenberg.... but i didn't know how he use interpolation inequality to get into this inequality.

Please any ideas?

Answer 1

the answer of my question is that w make an interpolation between the two spaces $H^1$ and $H^{-1}$ so we get

$$H^0=L^2=[H^{-1},H^1]_{\frac{1}{2}}$$

so, we get the result.