Difference between revisions of "Energy of measures"
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A concept in [[Potential theory|potential theory]] that is an analogue of the physical concept of the potential energy of a system of electric charges. For points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356601.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356603.png" />, let | A concept in [[Potential theory|potential theory]] that is an analogue of the physical concept of the potential energy of a system of electric charges. For points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356601.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035660/e0356603.png" />, let | ||
− | + | \[ | |
+ | H(|x|) = \left\{ \begin{array}{rl} | ||
+ | \ln\frac{1}{|x|} & \text{for $n = 2$} \\ | ||
+ | \frac{1}{|x|^{n-2}} & \text{for $n \geq 3$}, | ||
+ | \end{array} \right. | ||
+ | \] | ||
be the fundamental solution of the Laplace equation and let | be the fundamental solution of the Laplace equation and let |
Revision as of 13:39, 20 September 2012
A concept in potential theory that is an analogue of the physical concept of the potential energy of a system of electric charges. For points of a Euclidean space
,
, let
\[ H(|x|) = \left\{ \begin{array}{rl} \ln\frac{1}{|x|} & \text{for '"`UNIQ-MathJax1-QINU`"'} \\ \frac{1}{|x|^{n-2}} & \text{for '"`UNIQ-MathJax2-QINU`"'}, \end{array} \right. \]
be the fundamental solution of the Laplace equation and let
![]() | (2) |
be the Newton (for ) or logarithmic (for
) potential of a Borel measure
on
.
Restricting from now on to the case , one defines the mutual energy of two non-negative measures
and
by
![]() | (3) |
![]() |
Now , but it can happen that
. The energy of the measure
is the number
,
. For two measures
,
of arbitrary sign one can use the canonical decomposition
,
(or any decomposition of the form
,
) and, provided these four measures have finite energy, define the mutual energy of
and
by
![]() |
which may turn out to be negative, but
![]() |
The totality of all measures with finite energy can be made into a pre-Hilbert vector space with the scalar product
and the energy norm
. Here the Bunyakovskii–Cauchy–Schwarz inequality
holds as well as the energy principle: If
, then
. H. Cartan has shown that the space
is not complete, but the set
of non-negative measures is complete in
.
Let be a compact set in
,
. Among all probability measures
on
(that is, those for which
,
) there is an extremal capacitary measure
with minimal energy
, which is connected with the capacity
of
by the relation
![]() | (4) |
If the potential of a measure
has a square-summable gradient, then
![]() | (5) |
where
![]() |
is the Dirichlet norm and ,
. In fact, (5) remains valid for any measure
, and the Dirichlet norm
can be defined by an appropriate limit transition.
In the case of the plane , a direct application of (3) with the logarithmic potential (2) for the definition of the energy of measures is not possible because of the singular behaviour of the logarithmic kernel (1) at infinity. Let
be a bounded domain in
,
, admitting a Green function
, and let
be a Borel measure on
. When one applies Green potentials
and
of the form
![]() |
instead of Newton potentials and
in (3), one obtains for
a definition of the energy of measures on
that is equivalent to the one given above, but which turns out to be suitable also for
, with preservation of all properties described above (and
).
References
[1] | M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) |
[2] | J. Wermer, "Potential theory" , Lect. notes in math. , 408 , Springer (1974) |
[3] | N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) |
Energy of measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Energy_of_measures&oldid=28050