# Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.

If $1 <p< \infty$ we know that $\Delta:W_0^{2,p}(B) \rightarrow L^p(B)$ is an isomorphism (one to one, onto, continuous, inverse continous). Here I am using some non standard notation in the sense that by $W^{2,p}_0(B)$ i mean the functions in $W^{2,p}(B)$ which are zero on the boundary. Now we know the above does not hold for $p=1,\infty$.

I am interested in having the target space to be $L^\infty(B)$. So, as well known counterexamples show, there are examples of $f$ bounded such that the solution $\Delta u =f$ in $B$ with $u=0$ are not in $W^{2,\infty}$. So the inverse isn't continuous.

So I would like to just try and define the appropriate space on the left using some graph norm type idea. So , for instance, pick $N<p< \infty$ and set $$X:=\{ u \in W^{2,p}_0(B): \Delta u \in L^\infty\}$$ and lets put the norm

$\| u\| = \| \Delta u \|_{L^\infty}$.

So it appears $X$ is a Banach space and $\Delta :X \rightarrow L^\infty$ should be an isomorpism. Is this correct?

If this is the case I realize that $X$ is slightly bigger than $W^{2,\infty}_0$ (as defined above).

My functional analysis has decayed to zero and so any comments would be greatly appreciated.

thanks Craig

• As a slightly off-topic comment, if you are interested in the Laplacian in $W^{k,p}$, are you aware of the concept of $p$-Laplacian? It has also been studied for $p=\infty$. Aug 24, 2014 at 10:32
• Thanks for the comment. I am not interested in this case you mentioned directly, but maybe it would help clarify my situation. I will attempt to google it. Thanks. Aug 24, 2014 at 14:08
• In addition to googling it, you can use the links in my comment. Unfortunately links in comments are next to invisible here. There is a link to Lindqvist's notes in the first Wikipedia article, and that might be a good introduction to the topic if Wikipedia seems insufficient. Aug 24, 2014 at 14:14
• I will take a look at links and notes you recommended (and your right, i hadn't scene the link). Thanks Aug 24, 2014 at 14:27
• You write that $\Delta(W^{2,p}_0)=L^p(B)$, which contains $L^{\infty}$, so $X$ is indeed (by definition) isomorphic to this space. I don't see any relation whatsoever of your question to $p$-Laplacians other than the fact that the letter $p$ is used both times. Aug 25, 2014 at 3:27