Let $\Omega\subset \mathbb R^k$ be compact with smooth boundary, $T$ denote the trace operator to its boundary (i.e. a continuous extension of the restriction operator), and define the following subspaces of $L^2(\Omega)$: $$\mathcal{D}_D=\{u \in H^2 \textrm{ such that }Tu=0\}\quad \textrm{and}\quad \mathcal{D}_N=\{u \in H^2 \textrm{ such that }T(\partial_nu)=0\},$$ the Dirichlet and Neumann domains, respectively. Does there exist a continuous bijection from $L^2$ to itself which restricts to a bijection from $\mathcal{D}_D$ to $\mathcal{D}_N$?
Some comments: I know that such a map exists if, instead of the Neumann domain, we consider functions which satisfy some APS boundary condition, see for example section 3 of [BL]. However, the Neumann boundary condition requires information in a neighborhood of the boundary, so the methods presented in that paper don’t work. I can’t prove that no other way works though, but this example shows that we need to look at information which is more refined than simply considering the functional analysis perspective.
Define $L$ as the bijection which is continuous.
Let $k = 1$, $\Omega = [0,1]$
For $u$ continuous such that $u(0) = u(1) = 0$, define, $$L(u) = \int_0^x u(y) \ dy $$
Hence we have, $$T(u) = 0,T(\partial L(u)) = 0$$
This is also a bijection on continuous functions upto addition of constant functions.
Now extend this to $L^2[0,1]$ since continuous functions are dense in $L^2[0,1]$. The bijection follows from general version of Lebesgue differentiation theorem.
This can be tried to be extended to $k>1$ for $\Omega = [0,1]^k$ by using higher dimensional analogue of fundamental theorem of calculus.
For example, if we can extend the normal vector at the surface to a smooth vector $N(x)$ in $\Omega$ then define:
$$L(u) = \int_{0}^1 u(\gamma(t)) dt$$
$$Path \ \gamma(t) : \gamma(0) = a, \gamma(1) = x, \gamma'(t) = N(\gamma(t))$$ Pick shortest path,
For surjection, for function $u \in D_N$, choose $$w(x) = <\nabla u(x),N(x)>$$
$$L(w) = \int_{0}^1 <\nabla u((\gamma(t))),N(\gamma(t))> dt = u(\gamma(1)) - u(\gamma(0)) = u(x) - u(a).$$
So we have surjection upto a additive constant function.
But unfortunately $$D_N \supseteq L(D_D) \supseteq H^3 \cap D_N$$ upto additive constant functions. Not sure of equality in the first subset inclusion.