Exercise 39 (Power series in  Rn ) Exercise 39 (Power series in  Rn )
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Exercise 41 (Subharmonic functions) Exercise 41 (Subharmonic functions)
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© 2002-2007 Prof. Dr. Hans Wilhelm Alt, University of Bonn, Germany

Exercise 40 (HARNACK inequality)

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HARNACK inequality    [aufgabe:40]
Sei  Ω⊂Rn  offen und  u∈C2(Ω)  eine nicht-negative, harmonische Funktion in  Ω . Zeige, dass für jede (weg-)zusammenhängende Menge  D  mit  clos(D)⊂Ω  kompakt eine positive Konstante  C=C(n,D,Ω)  existiert, so dass
 
sup
x∈D
 u(x)≦C 
 
inf
x∈D
 u(x).
Hinweis: Wende Aufgabe aufgabe:36 auf eine geeignete Kugelkette an, die die Punkte in  D , in denen das Supremum bzw. Infimum angenommen wird, verbindet.
Version 1.5
H.W. Alt - 02.01.2007

Exercise 39 (Power series in  Rn ) Exercise 39 (Power series in  Rn )
Exercises Exercises
Exercise 41 (Subharmonic functions) Exercise 41 (Subharmonic functions)
Exercise 41 (Subharmonic functions) Index
© 2002-2007 Prof. Dr. Hans Wilhelm Alt, University of Bonn, Germany