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Difference between revisions of "Riesz potential"

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{{TEX|done}}
 
{{TEX|done}}
  
'' $  \alpha $-
+
'' $  \alpha $-potential''
potential''
 
  
 
A potential of the form
 
A potential of the form
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outside the support  $  S( \mu ) $
 
outside the support  $  S( \mu ) $
 
of  $  \mu $,  
 
of  $  \mu $,  
the potential  $  V( x) = V _ {n-} 2 ( x) $
+
the potential  $  V( x) = V _ {n- 2} ( x) $
 
is a [[Harmonic function|harmonic function]]. When  $  \alpha > n- 2 $,  
 
is a [[Harmonic function|harmonic function]]. When  $  \alpha > n- 2 $,  
 
the Riesz potential  $  V _  \alpha  ( x) $
 
the Riesz potential  $  V _  \alpha  ( x) $
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a more precise maximum principle is valid: If  $  V _  \alpha  ( x) \mid  _ {S( \mu ) }  \leq  M $,  
 
a more precise maximum principle is valid: If  $  V _  \alpha  ( x) \mid  _ {S( \mu ) }  \leq  M $,  
 
then  $  V _  \alpha  ( x) \leq  M $
 
then  $  V _  \alpha  ( x) \leq  M $
everywhere on  $  \mathbf R  ^ {n} $(
+
everywhere on  $  \mathbf R  ^ {n} $ (this statement remains valid also when  $  n= 2 $
this statement remains valid also when  $  n= 2 $
 
 
and  $  \alpha \rightarrow 0 $,  
 
and  $  \alpha \rightarrow 0 $,  
 
that is, for the logarithmic potential).
 
that is, for the logarithmic potential).
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concentrated on  $  K $
 
concentrated on  $  K $
 
and such that  $  \mu ( K) = 1 $;  
 
and such that  $  \mu ( K) = 1 $;  
then the  $  \alpha $-
+
then the  $  \alpha $-capacity is equal to
capacity is equal to
 
  
 
$$  
 
$$  
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If  $  V _  \alpha  ( K) < + \infty $,  
 
If  $  V _  \alpha  ( K) < + \infty $,  
then the infimum is attained on the capacitary measure  $  \lambda $(
+
then the infimum is attained on the capacitary measure  $  \lambda $ (also called equilibrium measure), which is concentrated on  $  K $,  
also called equilibrium measure), which is concentrated on  $  K $,  
 
 
$  \lambda ( K) = 1 $,  
 
$  \lambda ( K) = 1 $,  
 
generating the corresponding capacitary  $  \alpha $-
 
generating the corresponding capacitary  $  \alpha $-
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For  $  n $
 
For  $  n $
 
even and  $  \alpha = n- 2 m \leq  0 $,  
 
even and  $  \alpha = n- 2 m \leq  0 $,  
$  | x - y |  ^ {2m-} n \mathop{\rm log}  | x- y | $
+
$  | x - y |  ^ {2m- n}  \mathop{\rm log}  | x- y | $
 
is a [[Fundamental solution|fundamental solution]] of the polyharmonic equation  $  \Delta  ^ {m} u = 0 $,  
 
is a [[Fundamental solution|fundamental solution]] of the polyharmonic equation  $  \Delta  ^ {m} u = 0 $,  
otherwise  $  | x- y |  ^ {2m-} n $
+
otherwise  $  | x- y |  ^ {2m- n} $
 
is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order  $  > 2 $,  
 
is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order  $  > 2 $,  
 
see [[#References|[a2]]]. A treatment of Riesz potentials in the framework of balayage spaces is given in [[#References|[a1]]].
 
see [[#References|[a2]]]. A treatment of Riesz potentials in the framework of balayage spaces is given in [[#References|[a1]]].

Revision as of 06:00, 12 July 2022


$ \alpha $-potential

A potential of the form

$$ V _ \alpha ( x) = V( x; \alpha , \mu ) = \ \int\limits \frac{d \mu ( y) }{| x- y | ^ \alpha } ,\ \alpha > 0, $$

where $ \mu $ is a positive Borel measure of compact support on the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, and $ | x- y | $ is the distance between the points $ x, y \in \mathbf R ^ {n} $. When $ n \geq 3 $ and $ \alpha = n- 2 $, the Riesz potential coincides with the classical Newton potential; when $ n= 2 $ and $ \alpha \rightarrow 0 $, the limit case of the Riesz potential is in some sense the logarithmic potential. When $ n \geq 3 $ and $ 0 < \alpha \leq n- 2 $, the Riesz potential is a superharmonic function on the entire space $ \mathbf R ^ {n} $; moreover, in the classical case $ \alpha = n- 2 $, outside the support $ S( \mu ) $ of $ \mu $, the potential $ V( x) = V _ {n- 2} ( x) $ is a harmonic function. When $ \alpha > n- 2 $, the Riesz potential $ V _ \alpha ( x) $ is a subharmonic function outside $ S( \mu ) $. For all $ \alpha > 0 $ the Riesz potential $ V _ \alpha ( x) $ is a lower semi-continuous function on $ \mathbf R ^ {n} $, continuous outside $ S( \mu ) $.

Among the general properties of Riesz potentials the following are the most important. The continuity principle: If $ x _ {0} \in S( \mu ) $ and if the restriction $ V _ \alpha ( x) \mid _ {S( \mu ) } $ is continuous at the point $ x _ {0} $, then $ V _ \alpha ( x) $ is continuous at $ x _ {0} $ as a function on $ \mathbf R ^ {n} $. The restricted maximum principle: If $ V _ \alpha ( x) \mid _ {S( \mu ) } \leq M $, then $ V _ \alpha ( x) \leq 2 ^ \alpha M $ everywhere on $ \mathbf R ^ {n} $. When $ n- 2 \leq \alpha < n $, a more precise maximum principle is valid: If $ V _ \alpha ( x) \mid _ {S( \mu ) } \leq M $, then $ V _ \alpha ( x) \leq M $ everywhere on $ \mathbf R ^ {n} $ (this statement remains valid also when $ n= 2 $ and $ \alpha \rightarrow 0 $, that is, for the logarithmic potential).

The capacity theory for Riesz potentials can be constructed, for example, on the basis of the concept of the $ \alpha $- energy of a measures $ \mu $:

$$ E _ \alpha ( \mu ) = \int\limits \int\limits \frac{d \mu ( x) d \mu ( y) }{| x- y | ^ \alpha } ,\ \alpha > 0. $$

One may assume that for a compact set $ K $,

$$ V _ \alpha ( K) = \inf \{ E _ \alpha ( \mu ) \} , $$

where the infimum is taken over all measures $ \mu $ concentrated on $ K $ and such that $ \mu ( K) = 1 $; then the $ \alpha $-capacity is equal to

$$ C _ \alpha ( K) = [ V _ \alpha ( K)] ^ {- 1/ \alpha } . $$

If $ V _ \alpha ( K) < + \infty $, then the infimum is attained on the capacitary measure $ \lambda $ (also called equilibrium measure), which is concentrated on $ K $, $ \lambda ( K) = 1 $, generating the corresponding capacitary $ \alpha $- potential $ V( x; \alpha , \lambda ) $( cf. also Capacity potential). The further construction of $ \alpha $- capacities of arbitrary sets is carried out in the same way as for the classical capacities.

The Riesz potential is called after M. Riesz (see [2]), who obtained a number of important properties of Riesz potentials; for the first time such potentials were studied by O. Frostman (see [1]).

References

[1] O. Frostman, "Potentiel d'equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions" Medd. Lunds Univ. Mat. Sem. , 3 (1935) pp. 1–118
[2] M. Riesz, "Intégrales de Riemann–Liouville et potentiels" Acata Sci. Math. Szeged , 9 (1938) pp. 1–42
[3] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[4] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)

Comments

For $ n $ even and $ \alpha = n- 2 m \leq 0 $, $ | x - y | ^ {2m- n} \mathop{\rm log} | x- y | $ is a fundamental solution of the polyharmonic equation $ \Delta ^ {m} u = 0 $, otherwise $ | x- y | ^ {2m- n} $ is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order $ > 2 $, see [a2]. A treatment of Riesz potentials in the framework of balayage spaces is given in [a1].

The Riesz kernels $ | x- y | ^ {- \alpha } $ are the standard examples of convolution kernels. Thus, Riesz potentials may be regarded as special singular integrals. For more details on this interesting point of view see [a3].

References

[a1] J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2] B.W. Schulze, G. Wildenhain, "Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung" , Birkhäuser (1977)
[a3] E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970)
[a4] L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967)
How to Cite This Entry:
Riesz potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_potential&oldid=48566
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article