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DISCRETE MINIMAL ENERGY PROBLEMS
LectureII
E.B.Saff
CenterforConstructiveApproximation
VanderbiltUniversity
UniversityofCrete,HeraklionMay,2017
FundamentalThm: lim (A,N)/N2 =W (A).Everyweak∗
N→∞ K K
E
limitmeasureofthesequence ν(ω∗) ∞ forN-pointK-energy
{ N }N=2
minimizingconfigurationsω∗ isanequilibriummeasureforthe
N
continuousproblemonA.
Recall Notation
ω = x ,...,x A, Acompact,infinite
N 1 N
{ }⊂
K :A A R + , symmetricandl.s.c.
× → ∪{ ∞}
E (ω )= K(x,x)
K N i j
i(cid:54)=j
(cid:88)
(A,N)=min E (ω ): ω A, #ω =N
K K N N N
E { ⊂ }
W (A):= min I [A]= min K(x,y)dµ(x)dµ(y)
K K
µ∈M(A) µ∈M(A)
(cid:90)(cid:90)
A×A
µ equilibriummeasure,I [µ ]=W (A)
A K A K
ν(ω )= 1 δ , normalizedcountingmeasure
N N x∈ωN x
(cid:80)
Recall Notation
ω = x ,...,x A, Acompact,infinite
N 1 N
{ }⊂
K :A A R + , symmetricandl.s.c.
× → ∪{ ∞}
E (ω )= K(x,x)
K N i j
i(cid:54)=j
(cid:88)
(A,N)=min E (ω ): ω A, #ω =N
K K N N N
E { ⊂ }
W (A):= min I [A]= min K(x,y)dµ(x)dµ(y)
K K
µ∈M(A) µ∈M(A)
(cid:90)(cid:90)
A×A
µ equilibriummeasure,I [µ ]=W (A)
A K A K
ν(ω )= 1 δ , normalizedcountingmeasure
N N x∈ωN x
Fundamen(cid:80)talThm: lim (A,N)/N2 =W (A).Everyweak∗
N→∞ K K
E
limitmeasureofthesequence ν(ω∗) ∞ forN-pointK-energy
{ N }N=2
minimizingconfigurationsω∗ isanequilibriummeasureforthe
N
continuousproblemonA.
Riesz s-Energy in Euclidean Space
HereafterA Rp and = x y , Euclideandistance.
⊂ ||·|| | − |
TheRieszs-kernelisdefinedby
1 1
K (x,y):= , s >0; K (x,y):=log , x,y A.
s x y s log x y ∈
| − | | − |
Wewrite
E (ω):=E (ω), (A,N)= (A,N), s >0 or s =log.
s Ks Es EKs
Forp =3,s =1,getCoulombkernel.
ForA Rp ands =p 2,wegetNewtonkernel.
⊂ −
Some Basic Properties of Riesz Energy
• A B (A,N) (B,N)
s s
⊂ ⇒ E ≥E
• (A+x,N)= (A,N)
s s
E E
• (αA,N)= α−s (A,N)
s s
E | | E
• (A,N)iscontinuousins fors >0
s
E
•
(A,N) N(N 1)
lim Es − − = (A,N)
s→0+ s Elog
andforfixedN,everyclusterpointofminimals-energyN-point
configurationsω(s) ass 0+ isanN-pointminimallogenergy
N →
configuration.
Proposition(Proveit)
ForeachfixedN 2,
≥
1
lim (A,N)1/s = .
s→∞Es δN(A)
Moreover,everyclusterconfigurationass ofs-energy
→∞
minimizingN-pointconfigurationsonAisanN-pointbest-packing
configurationonA.
Best-Packing and Riesz Energy (s )
→ ∞
Recall: Separationdistanceofω = x ,...,x A
N 1 N
{ }⊂
δ(ω ):= min x x .
N i j
1≤i(cid:54)=j≤N| − |
N-pointbest-packingdistanceonA,
δ (A):=sup δ(ω ):ω A, #ω =N ,
N N N N
{ ⊂ }
ω∗ isbest-packingconfigurationifδ(ω∗)=δ (A).
N N N
Best-Packing and Riesz Energy (s )
→ ∞
Recall: Separationdistanceofω = x ,...,x A
N 1 N
{ }⊂
δ(ω ):= min x x .
N i j
1≤i(cid:54)=j≤N| − |
N-pointbest-packingdistanceonA,
δ (A):=sup δ(ω ):ω A, #ω =N ,
N N N N
{ ⊂ }
ω∗ isbest-packingconfigurationifδ(ω∗)=δ (A).
N N N
Proposition(Proveit)
ForeachfixedN 2,
≥
1
lim (A,N)1/s = .
s→∞Es δN(A)
Moreover,everyclusterconfigurationass ofs-energy
→∞
minimizingN-pointconfigurationsonAisanN-pointbest-packing
configurationonA.
Amusing Exercise
LetA=S1,theunitcircleintheplaneR2.
Provethatforeachs,0<s < ors =logtheN-throotsofunity(or
∞
anyrotationofthem)areminimals-energypointsforS1. Moreover,
theyaretheonlysetsofminimalenergypoints.
Moregenerally: ProvethisistrueifK(x,y)=f(x y ),wheref is
| − |
strictlyconvexanddecreasingon(0,2].
N =3?
N =4?
S2 = x R3 : x = 1
{ ∈ | | }
WhataboutminimalRieszenergypointsonS2 for
N =2?
N =4?
S2 = x R3 : x = 1
{ ∈ | | }
WhataboutminimalRieszenergypointsonS2 for
N =2?
N =3?