Lebesgue spine

In mathematics, in the area of potential theory, a Lebesgue spine or Lebesgue thorn is a type of set used for discussing solutions to the Dirichlet problem and related problems of potential theory. The Lebesgue spine was introduced in 1912 by Henri Lebesgue to demonstrate that the Dirichlet problem does not always have a solution, particularly when the boundary has a sufficiently sharp edge protruding into the interior of the region.

Definition

A typical Lebesgue spine in ${\displaystyle \mathbb {R} ^{n}}$, for ${\displaystyle n\geq 3,}$ is defined as follows

${\displaystyle S=\{(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{n}>0,x_{1}^{2}+x_{2}^{2}+\cdots +x_{n-1}^{2}\leq \exp(-1/x_{n}^{2})\}.}$

The important features of this set are that it is

in ${\displaystyle \mathbb {R} ^{n}}$ and the origin is a

.

Observations

The set ${\displaystyle S}$ is not closed in the

limit point

of ${\displaystyle S}$, but the set is closed in the

${\displaystyle \mathbb {R} ^{n}}$

In comparison, it is not possible in ${\displaystyle \mathbb {R} ^{2}}$ to construct such a connected set which is thin at the origin.

References

• J. L. Doob. Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, Berlin Heidelberg New York,
  ISBN 3-540-41206-9
  .
• L. L. Helms (1975). Introduction to potential theory. R. E. Krieger
  ISBN 0-88275-224-3
  .

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e