I'm concerned with a generic uniformly elliptic operator $L$ on $\mathbb{R}^n$. If $L$ is uniformly elliptic and I am studying the equation $Lu=f$ then the way I can deduce regularity on $\mathbb{R}^n$ is via the Fourier transform: $\hat{Lu} = \hat{f}$ which leads to $P(\xi)\hat{u} = \hat{f}$. From this finally I use the assumption that $P(\xi) \geq c |\xi|^2$ to deduce along with Parseval that $\|u\|_{H^2} \lesssim \|f\|_{L^2} + \|u\|_{L^2}$. My question is, why does this become so complicated on bounded domains?

**Question:** Why can't we simply write $u = \sum_k \phi_k \hat{u}(k)$ as a Fourier series and deduce from the equation that $\|k\|^2|\hat{u}(k)|^2 = |f(k)|^2$ (the Fourier coefficients) and use this to deduce that $$\sum_k (1+|k|^2)^2 |\hat{u}(k)|^2 \lesssim \|u\|_{L^2} + \|f\|_{L^2}$$ where again I've used Parseval's identity. In other words, doesn't everything from the $\mathbb{R}^n$ case just get converted into statements about the Fourier *series*? (as opposed to tranform).

Hope this is clear! Thanks!