In
mathematics, in the area of
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
, a Lebesgue spine or Lebesgue thorn is a type of
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
used for discussing solutions to the
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
and related problems of potential theory. The Lebesgue spine was introduced in 1912 by
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
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
, for
is defined as follows
:
The important features of this set are that it is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
and
path-connected in the
euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
in
and the origin is a
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
of the set, and yet the set is thin at the origin, as defined in the article
Fine topology (potential theory)
In mathematics, in the field of potential theory, the fine topology is a natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which \Delta u \ge 0, where \Delta is the ...
.
Observations
The set
is not closed in the
euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
since it does not contain the origin which is a
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
of
, but the set is closed in the
fine topology in
.
In comparison, it is not possible in
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, .
* L. L. Helms (1975). ''Introduction to potential theory''. R. E. Krieger .
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Potential theory