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Let $\mathbb{D}^2$ be the closed two-dimensional unit disk. Let $V \in \Omega^1(\mathbb{D}^2)$ be a smooth one-form.

Does there exist a constant $C$, such that

$$ \vert\vert V\vert \vert_{W^{2,p}(\Omega)} \leq C(\vert\vert \Delta V \vert\vert_{L^p(\Omega)} + \vert\vert V\vert\vert_{L^p(\Omega)}),$$

holds for all one-forms $V$, where $C$ does not depend on the boundary values of $V$?

Here $\Delta=\delta d+d\delta$ is the usual Hodge Laplacian on differential forms*.

The point is that I don't assume any specific boundary conditions on $V$. (So, what is written here for instance does not seem to apply).

Comment:

*I am actually interested in a more general case where $\mathbb{D}^2$ is endowed with some smooth Riemannian metric $g$, and the Euclidean Laplacian $\Delta$ is replaced by the Riemannian one w.r.t $g$ (the adjoint $\delta$ depends on $g$).

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1 Answer 1

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No, there is no such $C.$ Define $V_n(x,y)=\mathrm{Re}[f_n(x+iy)]\;dx + \mathrm{Im}[f_n(x+iy)]\;dy$ where $f_n(z)=z^n.$ Then

$$\nabla V=0$$ and

\begin{align*} \|V_n\|_{L^p(\Omega)} &=\left(\int_\Omega |f_n(x+iy)|^p\right)^{1/p}\\ &=\left(\int_\Omega r^{np}\right)^{1/p}\\ &=\left(2\pi \int_0^1 r^{1+np}dr\right)^{1/p}\\ &=\left(\frac{2\pi}{2+np}\right)^{1/p}\\ &\sim c_p n^{-1/p}\text{ for some constant $c_p$} \end{align*}

But

\begin{align*} \|V_n\|_{W^{2,p}(\Omega)} &\geq c'\left(\int_\Omega \left|\frac{\partial^2 f_n}{\partial x^2}(x+iy)\right|^p\right)^{1/p}\\ &= c'_pn(n-1)\left(\int_\Omega r^{np-2}\right)^{1/p}\\ &\sim c'_pc_p n^{2-1/p} \end{align*} with some other constant $c'_p$ depending on how the Sobolev norm is defined. So $$\|V_n\|_{W^{2,p}(\Omega)}\gg \|\nabla V_n\|_{L^p(\Omega)} +\|V_n\|_{L^p(\Omega)}.$$

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