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Defining functions for unbounded \(C^m\) domains. (English) Zbl 1288.26008
For an open set \(\Omega \subset \mathbb{R}^n\) a real-valued \(C^m\) function \(\rho\) defined on a neighborhood \(U\) of \(\partial \Omega\) such that \(\{x\in U\mid\rho(x)<0\}=\Omega \bigcap U\) and \(\nabla \rho \neq 0\) on \(\partial \Omega\) is called \(C^m\) defining function (\(m\geq 1\)), \(\Omega\) is a \(C^m\) domain if it has a \(C^m\) defining function. If \(\rho\) is a \(C^m\) defining function for \(\Omega \subset \mathbb{R}^n\) defined on a neighborhood \(U\) of \(\partial \Omega\) and such that 1) dist\((\partial \Omega, \partial U)>0\), 2) \(\|\rho\|_{C^m(U)}<\infty\), 3) \(\inf _U|\nabla \rho|>0 \), then \(\rho\) is \(C^m\) uniformly. A \(C^m\) uniformly function for all \(m\in \mathbb{N}\) is a \(C^{\infty}\) uniformly function. The following notions are introduced:
For \(\Omega \subset \mathbb{R}^n\) with \(C^m\) boundary
\[ \widetilde{\delta}(x)=\begin{cases} d(x,\partial \Omega), \;\;x \overline{\in} \Omega\,,\\ -d(x,\partial \Omega), \;\;x \in \overline{\Omega}\,, \end{cases} \] \(\mathrm{Unp}(\partial \Omega)=\{x\in \mathbb{R}^n\mid \text{there is a unique point}\; y\in \partial \Omega, \text{such that}\; \delta(x)=|y-x| \}\).
For \(y\in \partial \Omega\), \(\mathrm{Reach}(\partial \Omega,y)=\sup\{r\geq 0\mid B(y,r)\subset \mathrm{Unp}(\partial \Omega)\}\).
Reach\((\partial \Omega)=\inf\{\text{Reach}(\partial \Omega,y)\mid y\in \partial \Omega\}\).
The main result of the article is presented in the following
{Theorem. } Let \(\Omega \subset \mathbb{R}^n\) be a \(C^m\) domain, \(m>2\). Then the following assertions are equivalent:
\(\Omega\) has a uniformly \(C^m\) defining function.
\(\partial \Omega\) has positive reach, and for any \(0<\varepsilon<\text{Reach}(\partial \Omega) \) the signed distance function satisfies \(\|\widetilde{\delta}\|_{C^m(U_{\varepsilon})}<\infty\) on \(U_{\varepsilon}=\{x\in \mathbb{R}^n\mid\delta(x)<\varepsilon\}\).
There exists a \(C^m\) defining function \(\rho\) for \(\Omega\) and a constant \(C>0\) such that for every point \(p \in \partial \Omega\) with local coordinates \(\{y_1,\ldots,y_n\}\) satisfying \( \frac{\partial \rho}{\partial y_j(p)}=0\) for \(1\leq j\leq n-1\) \[ |\nabla \rho(p)|^{-1}|\frac{\partial^k \rho(p)}{\partial y^I}|<C \] where \(I\) is a multi-index of length \(k\), \(2\leq k\leq m\), and \(0\leq I_n\leq \min\{m-k,k\}\).

26B25 Convexity of real functions of several variables, generalizations
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
32T15 Strongly pseudoconvex domains
Full Text: DOI arXiv
[1] Chen, S. C. and Shaw, M. C.: Partial Differential equations in several complex variables. AMS/IP Studies in Advanced Mathematics 19, American Mathematical Society, Providence, RI, 2001. · Zbl 0963.32001
[2] Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418-491. · Zbl 0089.38402
[3] Gansberger, K.: An idea on proving weighted Sobolev embeddings. In preparation, ArXiv: 1007.3525.
[4] Gansberger, K.: On the weighted \? \partial -Neumann problem on unbounded domains. In preparation, ArXiv: 0912.0841v1.
[5] Gilbarg, D. and Trudinger, N. S.: Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin, 2001. · Zbl 1042.35002
[6] Haslinger, F. and Helffer, B.: Compactness of the solution operator to \partial in weighted L2-spaces. J. Funct. Anal. 243 (2007), no. 2, 679-697. · Zbl 1144.32023
[7] Herbig, A. K. and McNeal, J. D.: Convex defining functions for convex domains. J. Geom. Anal. 22 (2012), no. 2, 433-454. · Zbl 1254.26020
[8] Harrington, P. and Raich, A.: Sobolev spaces and elliptic theory on unbounded domains in Rn. Submitted, ArXiv: 1209.4044.
[9] Krantz, S. and Parks, H.: Distance to Ck hypersurfaces. J. Differential Equations 40 (1981), no. 1, 116-120. · Zbl 0431.57009
[10] Weinstock, B. M.: Some conditions for uniform H-convexity. Illinois J. Math. 19 (1975), 400-404. · Zbl 0303.32012
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