i consider a map $u$ of class $W^{2,2}(\Omega,\mathcal{N})$, where $\mathcal{N}$ is a compact submanifold of $R^n$ without boundary and $\Omega\subset \mathbb{R}^m$ is a bounded domain with smooth boundary. Then at each point $a\in \partial \Omega$ there is a ball $B_r(a)$ and a one-to-one mapping $\phi$ from $B_r(a)$ onto $B_r(0)\subset \mathbb{R}^m$ such that

(1) $\phi(B_r(a)\cap \Omega)\subset \mathbb{R}^m_+$

(2) $\phi(B_r(a)\cap \partial \Omega)\subset \partial\mathbb{R}^m_+$

(3) $\phi\in C^{\infty}(B_r(a))$, $\phi^{-1}\in C^{\infty}(B_r(0))$.

Now i want to know which properties would this mapping $\phi:B_r(a)\rightarrow B_r(0)$ satisfy.

It's obvious that $\vert \phi\vert\leq Cr$. But what inequalities follow for $$ \vert D\phi(x)\vert\leq \ldots $$ and $$ \vert \Delta\phi(x)\vert\leq \ldots? $$

Thank you very much.


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