2
$\begingroup$

Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e. $$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\partial \Omega}=0.$$ It follows from the spectral theorem for compact operators that the set $\{\phi_i\}$ forms a basis for $L^2(\Omega)$. Thus there exists $\phi_{i_k}$ such that $$1=\sum_{k=1}^{\infty}c_{i_k}\phi_{i_k}(x), \ \ x\in \Omega, \ \ c_{i_k}\neq 0.$$ Now let $X$ be the closure of the subspace of $L^2(\Omega)$ spanned by $\{\phi_{i_k}: k\in \mathbb{N}\}$, and define $L: X \cap H^1(\Omega) \rightarrow H^{1/2}(\partial \Omega)$ by $L(f)=f|_{\partial \Omega}$.

Is $L$ surjective? If not, what is the range of $L$?

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ Typically, $i_k = k$ and $X=L^2$, so $L$ is onto. However, if $\Omega$ is symmetric (say, $\Omega = (-1,1)$ in dimension $1$), then $X$ only contains symmetric (i.e. even) $L^2$ functions, and thus the image of $L$ also has this property. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Dec 13, 2019 at 8:49

0

Reset to default

You must log in to answer this question.