Energy of measures

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 $ x = ( x _ {1} \dots x _ {n} ) $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, 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 (up to dimensional constants) the fundamental solution of the Laplace equation and let

$$ \tag{2 } U _ \mu (x) = \int\limits H ( | x - y | ) d \mu (y) $$

be the Newton (for $ n \geq 3 $) or logarithmic (for $ n = 2 $) potential of a Borel measure $ \mu $ on $ \mathbf R ^ {n} $.

Restricting from now on to the case $ n \geq 3 $, one defines the mutual energy of two non-negative measures $ \mu $ and $ \nu $ by

$$ \tag{3 } ( \mu , \nu ) = \int\limits H ( | x - y | ) d \mu (x) d \nu (y) = $$

$$ = \ \int\limits U _ \mu (y) d \nu (y) = \int\limits U _ \nu (x) d \mu (x) . $$

Now $ ( \mu , \nu ) \geq 0 $, but it can happen that $ ( \mu , \nu ) = + \infty $. The energy of the measure $ \mu $ is the number $ ( \mu , \mu ) $, $ 0 \leq ( \mu , \mu ) \leq + \infty $. For two measures $ \mu $, $ \nu $ of arbitrary sign one can use the canonical decomposition $ \mu = \mu ^ {+} - \mu ^ {-} $, $ \nu = \nu ^ {+} - \nu ^ {-} $( or any decomposition of the form $ \mu = \mu _ {1} - \mu _ {2} $, $ \mu _ {1} , \mu _ {2} \geq 0 $) and, provided these four measures have finite energy, define the mutual energy of $ \mu $ and $ \nu $ by

$$ ( \mu , \nu ) = ( \mu ^ {+} , \nu ^ {+} ) + ( \mu ^ {-} , \nu ^ {-} ) - ( \mu ^ {+} , \nu ^ {-} ) - ( \mu ^ {-} , \nu ^ {+} ) , $$

which may turn out to be negative, but

$$ ( \mu , \mu ) \geq ( \sqrt {( \mu ^ {+} , \mu ^ {+} ) } - \sqrt {( \mu ^ {-} , \mu ^ {-} ) } ) ^ {2} \geq 0 . $$

The totality $ {\mathcal E} $ of all measures with finite energy can be made into a pre-Hilbert vector space with the scalar product $ ( \mu , \nu ) $ and the energy norm $ \| \mu \| _ {e} = \sqrt {( \mu , \mu ) } $. Here the Bunyakovskii–Cauchy–Schwarz inequality $ | ( \mu , \nu ) | \leq \| \mu \| _ {e} \cdot \| \nu \| _ {e} $ holds as well as the energy principle: If $ \| \mu \| _ {e} = 0 $, then $ \mu = 0 $. H. Cartan has shown that the space $ {\mathcal E} $ is not complete, but the set $ {\mathcal E} ^ {+} \subset {\mathcal E} $ of non-negative measures is complete in $ {\mathcal E} $.

Let $ K $ be a compact set in $ \mathbf R ^ {n} $, $ n \geq 3 $. Among all probability measures $ \lambda $ on $ K $( that is, those for which $ \lambda \geq 0 $, $ \lambda (K) = 1 $) there is an extremal capacitary measure $ \lambda _ {0} $ with minimal energy $ ( \lambda _ {0} , \lambda _ {0} ) $, which is connected with the capacity $ C (K) $ of $ K $ by the relation

$$ \tag{4 } ( \lambda _ {0} , \lambda _ {0} ) = \int\limits U _ {\lambda _ {0} } (x) d \lambda _ {0} (x) = \ \frac{1}{C (K) } . $$

If the potential $ U _ \mu $ of a measure $ \mu \in {\mathcal E} $ has a square-summable gradient, then

$$ \tag{5 } c (n) \| \mu \| _ {e} = \| U _ \mu \| , $$

where

$$ \| U _ \mu \| = \left ( \int\limits _ {\mathbf R ^ {n} } {\rm grad} ^ {2} U _ \mu (x) d x \right ) ^ {1/2} $$

is the Dirichlet norm and $ c (n) = ( n - 2 ) 2 \pi ^ {n/2} / \Gamma ( n / 2 ) $, $ n \geq 3 $. In fact, (5) remains valid for any measure $ \mu \in {\mathcal E} $, and the Dirichlet norm $ \| U _ \mu \| $ can be defined by an appropriate limit transition.

In the case of the plane $ \mathbf R ^ {2} $, 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 $ \Omega $ be a bounded domain in $ \mathbf R ^ {n} $, $ n \geq 2 $, admitting a Green function $ g ( x , y ) $, and let $ \mu $ be a Borel measure on $ \Omega $. When one applies Green potentials $ G _ \mu $ and $ G _ \nu $ of the form

$$ G _ \mu (x) = \int\limits g ( x , y ) d \mu (y) $$

instead of Newton potentials $ U _ \mu $ and $ U _ \nu $ in (3), one obtains for $ n \geq 3 $ a definition of the energy of measures on $ \Omega $ that is equivalent to the one given above, but which turns out to be suitable also for $ n = 2 $, with preservation of all properties described above (and $ c (2) = 2 \pi $).

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)