Let $\Omega$ be a domain limited with smooth boundary in $\mathbb{R}^{n}$ and consider the Sobolev space $H^{1}_{0}(\Omega)$ equiped with the norm $||u||=\int_{\Omega}|\nabla u|^{2}dx$. Let $\{\phi_{k}\}_{k\in\mathbb{N}}\subset H^{1}_{0}(\Omega)$ be the eigenfunctions of the problem \begin{equation} \begin{cases} -\Delta u=\lambda u, & \mbox{in $\Omega$}\\ u=0, & \mbox{on $\partial\Omega$}. \end{cases} \end{equation} Set $W=\langle\phi_{k}\rangle_{k=1}^{j}$ and $V=W^{\perp}$. Note that $H^{1}_{0}(\Omega)=W\oplus V$. I wnat to proof that for each $\epsilon >0$ there is a $v\in V$, with $||v||=\epsilon$, so that the set $$A_{v}=\{x\in\Omega: w(x)+v(x)>1, w\in W, ||w||\leq 1\}$$ has positive lebesgue measure.
I tried to prove by contradiction, but could not.
I thank the help!