# $\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?

I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$.

Does the following inequality (or something similar hold) for $u \in W^{2,p}\cap W^{1,p}_0$: $$\lVert u \rVert_{W^{2,p}} \leq C(\lVert \Delta_p u \rVert_{L^2})$$ (with possibly a $\lVert u \rVert_{L^s}$ contribution on the RHS). So essentially I am looking for a elliptic regularity result.

I tried looking for elliptic regularity results regarding the $p-$laplace equation but had no good luck.

Consider for example $p=2$. Then $\lVert u \rVert_{H^2}$ is bounded above by a factor of $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$. In fact they are equivalent norms in $H^2$.

• The $p$-Laplacian is nonlinear (and is $p-1$-homogeneous), so $u\mapsto\|\Delta_pu\|_{L^2}$ is not a norm. Your question seems interesting once you specify what you mean by the norm (or equivalence) and make sure that it is well defined and scales properly. Are you sure you want the $p$-Laplacian instead of the usual $2$-Laplacian? Commented Dec 9, 2014 at 15:46
• @JoonasIlmavirta You're quite right, I hope the question is correct now. I am I suppose asking for a sort of regularity of solutions to $$-\Delta_p u = f, \quad u|_{\partial\Omega} = 0$$ for $f \in L^2$. Commented Dec 9, 2014 at 16:01
• Have you looked at the one-dimensional case? Doing so should answer your question. Commented Dec 9, 2014 at 16:20

Since you work in a bounded domain there is also the matter of boundary regularity. Even for $p=2$ for example you either need a $C^{4,\alpha}$ smooth boundary (classical Schauder estimates), or an exterior ball condition on the boundary (Adolfsson Math. Scand.70:146-160, 1992) or that your domain is a planar curvilinear polygon (corners with obtuse angles do not matter in that case, see Grivard's book: Singularities in Boundary Value Problems, or the books by Mazya and Kozlov). There are also results for more general domains such as domains with cusps but I just wanted to point out the fact that the matter is far from straightforward.
• I realized that boundary regularity is probably not interesting since it is mentioned without any concerns that the inequality holds for $p=2$. However, it was not explicitly mentioned and I only wanted to point it out as possible issue. Commented Dec 15, 2014 at 10:33