Let $\Omega$ be a convex open subset of $\mathbb{R}^d$ with a smooth boundary. Is there an example of a one to one and onto mapping of the form $$L^{d+1}(\Omega) \to W^{1,d+1}(\Omega)$$
$\begingroup$
$\endgroup$
2
-
1$\begingroup$ Should your map be a linear isomorphism? Or anything else apart of bijection? Else the question is bit strange: of course, these spaces both have cardinality continuum. $\endgroup$– Fedor PetrovCommented Aug 18, 2017 at 5:47
-
$\begingroup$ @FedorPetrov : bijection. I am seeking examples. not existence. $\endgroup$– Rajesh DCommented Aug 18, 2017 at 6:06
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
The map $(-\Delta + 1)^{-\frac{1}{2}}: L^p (\Omega) \to W^{1, p}_0 (\Omega)$ is a linear bijection when $\Omega$ is smooth and $1 < p < +\infty$, where $\Delta$ is the Laplacian with Dirichlet boundary conditions.
When $\Omega=\mathbb{R}^d$, this operator correspond to the convolution with the Bessel kernel of order $1$.
answered Aug 21, 2017 at 10:59
-
$\begingroup$ Is there a name for this function? It looks like a square root of sum of Laplacian and itself. $\endgroup$– Rajesh DCommented Aug 21, 2017 at 11:24
-
$\begingroup$ $\Delta$ on $\Omega$ needs a boundary condition. $\endgroup$ Commented Aug 21, 2017 at 18:53
-
$\begingroup$ I have added the boundary condition. $\endgroup$ Commented Aug 22, 2017 at 11:39