Sphere condition

In mathematics , a spherical condition is a property of a subset of a metric space , usually des . The spherical condition is clearly fulfilled if one can create a sphere at each edge point in such a way that the intersection of the edge with this sphere is only this point. Depending on whether this sphere is in the set or outside, one speaks of an inner or outer sphere condition. ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \Omega }$

Sphere conditions are used, for example, in the formulation of conditions for solving the Dirichlet problem with the Poisson equation .

Mathematical formulation

Outer sphere condition in , inner (and outer) sphere condition in ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$

${\displaystyle \Omega }$fulfilled in an inner sphere condition if: ${\displaystyle x_{0}}$

${\displaystyle \exists \,x\in \Omega ,\,\varepsilon >0:B_{\varepsilon }(x)\subset {\overline {\Omega }}\land B_{\varepsilon }(x)\cap \partial \Omega =\{x_{0}\}}$

Conversely met in an outer ball condition if: ${\displaystyle \Omega }$ ${\displaystyle x_{0}}$

${\displaystyle \exists \,x\in \Omega ^{c},\,\varepsilon >0:B_{\varepsilon }(x)\cap {\overline {\Omega }}=\{x_{0}\}}$

The ball denotes a radius . If this assertion holds for every point , then one says that the spherical condition is fulfilled. If the same radius can also be used in every point , it is said that the spherical condition is uniformly fulfilled. ${\displaystyle B_{\varepsilon }(x)}$ ${\displaystyle x}$ ${\displaystyle \varepsilon }$ ${\displaystyle x_{0}}$ ${\displaystyle \Omega }$ ${\displaystyle \varepsilon }$

Examples

The presence of a spherical condition ensures a certain smoothness of the edge. Obviously the points on the edges of a cube do not fulfill any inner spherical condition. The inner surface points of a cube surface obviously fulfill a spherical condition, but not uniformly, since one has to become smaller with the radius when one approaches an edge with the point. In the drawing on the right, a uniform outer spherical condition is sufficient, as can easily be seen from the convexity of the set . In the pointed corners like there is no inner sphere condition. ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle x_{1}}$

literature

• Free boundary problems. Theory and applications . In: Pierluigi Colli, Claudio Verdi, Augusto Visintin (eds.): International Series of Numerical Mathematics . No. 147 . Birkhäuser, Basel et al. 2004, ISBN 3-7643-2193-8 , p. 232 (English).