Lipschitz area

In mathematics , a Lipschitz area - or area with a Lipschitz edge - is an area in Euclidean space whose edge is “sufficiently regular” in the sense that it is locally the graph of a Lipschitz continuous function . Lipschitz domains are used in the theory of partial differential equations . The term is named after the German mathematician Rudolf Lipschitz .

The areas described here are also referred to as strong Lipschitz areas to avoid confusion with the weak Lipschitz areas, which represent a more general class of areas.

definition

A region of Euclidean space is called a (strong) Lipschitz region if both positive numbers and exist and there is a locally finite coverage of the boundary such that there is a real-valued function of variables for each , so that the following conditions hold: ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle \delta }$ ${\displaystyle M}$ ${\displaystyle (U_{i})_{i}}$ ${\displaystyle \partial \Omega }$ ${\displaystyle U_{i}}$ ${\displaystyle f_{i}}$ ${\displaystyle n-1}$

1. For a number , every subfamily of with elements has the empty set as a common intersection . ${\displaystyle R}$ ${\displaystyle (U_{i})_{i}}$ ${\displaystyle R+1}$

2. For every pair of points with there exists such that ${\displaystyle x,y\in \Omega _{\delta }:=\{a\in \Omega :\operatorname {dist} (a,\partial \Omega )<\delta \}}$ ${\displaystyle |x-y|<\delta }$ ${\displaystyle i}$

${\displaystyle x,y\in V:=\{a\in U_{i}:\operatorname {dist} (x,\partial U_{i})>\delta \}}$

applies.

3. Every function fulfills a Lipschitz condition ${\displaystyle f_{i}}$

${\displaystyle |f_{i}(\xi _{i,1},\ldots ,\xi _{i,n-1})-f_{i}(\nu _{i,1},\ldots ,\nu _{i,n-1})|<M|\xi _{i,1}-\nu _{i,1},\ldots ,\xi _{i,n-1}-\nu _{i,n-1}|}$

with the Lipschitz constant . ${\displaystyle M}$

4. For a Cartesian coordinate system in , the set is described by ${\displaystyle (\xi _{i,1},\ldots ,\xi _{i,n-1})}$ ${\displaystyle U_{j}}$ ${\displaystyle \Omega \cup U_{i}}$

${\displaystyle \xi _{i,n}<f_{i}(\xi _{i,1},\ldots ,\xi _{i,n-1})}$.

Limited Lipschitz areas

If is a restricted area, then the above definition simplifies to a single condition. The restricted area is a Lipschitz area if and only if the edge is locally a Lipschitz edge. This means that there is a neighborhood for every boundary point , so that the set is the graph of a Lipschitz continuous function. ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle x\in \partial \Omega }$ ${\displaystyle U_{i}}$ ${\displaystyle \partial \Omega \cap U_{i}}$

properties

• Every area with is also a Lipschitz area. ${\displaystyle C^{k}}$ ${\displaystyle k\geq 1}$
• According to Rademacher's theorem, tangential vectors can be found almost everywhere on a Lipschitz boundary .

Examples

• The open circular area is an area and thus also a Lipschitz area. ${\displaystyle C^{\infty }}$
• The area of ​​an open rectangle is a Lipschitz area, but not an area. ${\displaystyle C^{1}}$
• Slotted surfaces, such as the slotted circular surface

${\displaystyle \Omega :=\{x\in \mathbb {R} ^{d}:|x-x_{0}|<R,x\neq x_{0}+\lambda e_{1}\ {\text{for}}\ 0\leq \lambda <R\}}$,

where a basis vector is the canonical basis des , are not Lipschitz domains. ${\displaystyle e_{1}}$ ${\displaystyle \mathbb {R} ^{d}}$

Partial differential equation theory

The concept of the Lipschitz area appears in the theory of Sobolev spaces . For example, some variants of Sobolev's embedding theorem require that the investigated areas are Lipschitz areas. Thus, many domains of definition are also Lipschitz domains, which are examined in the context of certain partial differential equations and problems of variation .

Individual evidence

1. ^ RA Adams: Sobolev spaces . 1st edition. Academic Press, New York, San Francisco, London 1975, ISBN 978-0-12-044150-1 , pp. 66 .
2. ^ ^{A b} R. A. Adams: Sobolev spaces . 1st edition. Academic Press, New York, San Francisco, London 1975, ISBN 978-0-12-044150-1 , pp. 67 .
3. ^ Giovanni Leoni: A First Course in Sobolev Spaces: Second Edition . 2nd Edition. American Mathematical Society, Pittsburgh 2017, ISBN 978-1-4704-2921-8 , pp. 274 .
4. ↑ ^{a b c} Peter Knabner, Lutz Angerman: Numerical Methods for Elliptic and Parabolic Partial Differential Equations . Springer-Verlag, New York 2003, ISBN 978-1-4419-3004-0 , pp. 96 .