If $U$ is an open region of $\mathbb{R}^n $ with smooth boundary, $u\in W_0^{1,2}(U)$ satisfies $$\int_U \nabla u \nabla v dx =\lambda \int_U uvdx $$ where the test function $v\in W_0^{1,2} (U)$ (defined as the closure of $C_c^{\infty} (U)$ under $W^{1,2}$ norm), prove $u\in W_0^{m,2}(U)$ for all $m\in \mathbb{Z}^+$.
From Evans' PDE book and my teacher I know the weak solution for $u\in W^{1,2}(U)$ condition can gives $u\in W^{m,2}(U)$, by choosing suitable test function $v$ and operating on the boundary. But since $W_0^{m,2}(U)$ is defined by smooth function approximation, I tried to replace $u$ by its convolution with mollifier $\rho_{\varepsilon}$ but the weak solution condition seems fails. Any help?