Abstract
We present the derivation of equations for the interval of equilibrium for the constrained energy problem in
the presence of an external field and show that for some time dependence of the
external field Q(y,t) = Q(y,0)yt/2
the endpoints of interval of equilibrium are the solution of Continuum limit of the Toda lattice
Also here we investigate the continuity of the set of equilibrium in
general case.
We shall investigate the interval of equilibrium for the constrained energy problem in the presence of an external field.
In this section we briefly recall the classical theorems dealing with weighted energy problems and constrained energy problems
of potential theory.
Let M be the collection of finite Borel measures,
M
_{S}
^{x}
Ì M
be measures with support in S and total mass x. Let M_{S}^{x,
s} be
the subset of M_{S}^{x}
M_{S}^{x,s} = µ :0£ µ £ s, µ(R) = x , S_{µ} Ì S.
where S_{s}=S;
µ £ s means that s  µ is a measure.
Let Q:S R È {+¥} be a continuous function on the compact S Ì R and
cap({x Î R: Q(x) < ¥}) > 0.
The logarithmic potential of the measure µ from M is defined by
and weighted energy integral is defined by
I_{Q}(µ) = 
ó õ 
ó õ


dµ(y) dµ(z) + 2 
ó õ

Q dµ

and Q is called an external field.
If the weighted energy functionals considered on unbounded
S R, then the external field Q has to satisfy the
additional condition
That provides existence of energy integral
Let us consider an extremal energy problem, i.e. the problem of finding a measure µ
_{Q}
^{x} such that
I_{Q}
(m
_{Q}
^{x})=W_{Q}^{x} = 
inf
m
Î
M
^{x}

I_{Q}
(m).


If in addition s has finite logarithmic energy (energy when Q = 0) over all compact sets and
s({x Î R: Q(x) < ¥}) > x then s is called a constraint.
We can define a constrained extremal energy problem, i.e. the problem of finding a measure µ_{Q}^{x,s} such that
I_{Q}(µ_{Q}^{x,s})=W_{Q}^{x,s} = inf_{µ Î MSx,s}I_{Q}(µ).
Introduction in the theory of constrained extremal problems is in the
paper by Dragnev and Saff [DS] and in the case without constraint there is the monograph by
Saff and Totik [ST]. We introduce here some known facts.
If W_{Q}^{x} £ ¥ then there exists unique measure µ_{Q}^{x} Î M^{x} such that I_{Q}(µ_{Q}^{x}) = W_{Q}^{x}.
Also if W_{Q}^{x,s}£ ¥ then there exists unique measure µ_{Q}^{x,s}Î M^{x,s}
such that I_{Q}(µ_{Q}^{x,s}) = W_{Q}^{x,s}.
The extremal measures are called the equilibrium measures
because they have the following properties (when U^{s} and Q are continuous):
U^{µ
Q
x}
+ Q = 

ì í î 
³ F_{Q}^{x} 
on 
S 
=F
_{Q}
^{x} 
on 
S_{µQx}



(1) 
for some constant F_{Q}^{x}, and correspondingly
U^{µQx,s}+ Q = 

ì í î 
£ F_{Q}^{x,s} 
on 
S_{µQx,s} 
³ F_{Q}^{x,s} 
on 
S_{s µQx,s}



(2) 
for some constant F_{Q}^{x,s}. Here S_{µ} denotes the support of measure µ.
Properties (1) and (2) are sufficient conditions for being equilibrium measures, namely if there is a
constant such that the first or the second relation above holds for some measure with total mass
x then the measure is solution of the extremal problem and it is called respectivelyequilibrium measure associated with Q with total mass
x or
sconstrained equilibrium measure with total mass x associated with Q.
In the paper we use continuous external fields and potentials of constraints,
but they may be lower semicontinuous (see [BR],[ST] and [DS]) and there exists similar equilibrium conditions.
There is a functional introduced in [BR]
F_{Q}^{x}(K):= xlog 
cap(K)+ 
ó õ

Q dw_{K}, 
where w_{K} denotes the equilibrium measure associated with the
regular compact set K.
For the first time the functional for x=1 was considered in [MS]
The support of the equilibrium measure in R minimizes the functional,
more precisely
F_{Q}^{x}
(K) = 

ì í î 
=F_{Q}^{x} 
if 
S_{µQx} K S^{µQx} 
> F_{Q}^{x} 

otherwise



were S^{µQx}={y:(U^{µQx}+ Q)(y) =F_{Q}^{x} }. One can derive this by integration of above equilibrium conditions with respect to
measure w_{K}.
Also V.S.Buyarov and E.A.Rahmanov have proved that S_{µQx} increase with respect to x and have represented the sets
S_{µQx} and S^{µQx} for all x Î R_{+} in the following way
They have proven for unconstrained case that almost everywhere with respect to x we have
cap(S^{µQx}\ S_{µQx})=0. There is an analogical results for the support of the measure in the constrained case
in [K]. In the paper we investigate continuity with respect to x of the set S_{µ}
Ç S_{s µ}.
The paper deals with properties of endpoints of interval S_{µ} Ç S_{s µ}=[a(x),b(x)], were
µ = µ_{Q}^{s, x} is an equilibrium measure with total mass µ (R) = x £ s(R)
for an external field Q and constraint s (see section 4).
Also we reproduce a derivation of equations for the support S_{µQx }= [a(x),b(x)]
of equilibrium measure with total mass x corresponding to external field Q .
In some sense support defines the equilibrium measure and to find the support is a very
essential task in solving the extremal problem in the potential theory.
The set S_{µ} Ç S_{s µ} is very interesting too, it is the set where the
constraint is not hit and also it is the set of equilibrium. Here we restrict ourselves by
the case were this set is an interval.
To justify our interest to the equations for endpoints S_{µ} Ç S_{s µ}=[a(x),b(x)] we mention that
there are applications of the subject to theory of partial differential equations. Namely endpoint functions
a(x,t) and b(x,t) of these extremal problems for some special time dependence t of the external field Q(y,t)
are the solutions of known system of PDE called continuum limit of Toda lattice (see [AV], [DM]). In the section 5 we derive
these equations.
2 
The continuity of the set of equilibrium. 

In this section we consider of continuity of the set of equilibrium which is S_{µQs,
x} Ç S_{s µQs, x}. It is a compact set in the metric space R and we
represent the definition of the notion of continuity for the general compact sets which one can find
in the Bible for the set theory by K.Kuratowski "Topology".
And after the definition we prove a lemma describing the breaking points of
intersection of continuous families of sets and apply the lemma to the set of equilibrium.
Let S(x) be the family of compact sets with parameter x.
The set liminf_{x→ x0+0}S(x)
is called limit interior of the S(x) in x_{0}
from above. Point p belongs to
liminf_{x→ x0+0}S(x) if any neighborhood of p have nonempty intersection with each S(x)
where x_{0}<x<x_{0}+e for some e.
By the same way we define the limit superior. Point p belongs to
limsup_{x→ x0+0}S(x) if any neighborhood of p
have nonempty intersection with infinite number of S(x) where x_{0}<x<x_{0}+e for some e.
Evident that


S_{1}(x)Ç S_{2}(x) Ì 

S_{1}(x)Ç 

S_{2}(x).

The family of sets S(x) converges in x_{0} from above or has a limit in x_{0} from above if
The family of compacts is continuous in x_{0} from above if The converging family of compacts S(x) has jump J^{±} in
x_{0}+0 if
There is the parallel with number theory: if J
_{S}^{±}
(x_{0}
)=Ø then S(x) converges
and lim_{x→ x0± 0}S(x)=S(x_{0}),
but the converse is false.
Evidently, that for increasing family of compact set these definitions have more
simple representation
and for decreasing family of compact set respectively
The situation when x tends to x_{0} from below is defined in analogous way.
Theorem 1 Let Q and s be an external field and constraint respectively on S Ì R.
Let µ = µ_{Q}^{x,s} be the sconstrained equilibrium measure with total mass x associated with Q.
(i)If S_{µ} Ç S_{sµ}=[a(x),b(x)], a(x)<b(x) then the functions a(x) and b(x) are
continuous almost everywhere.
(ii) If S º [A,B]Ì R then the set of equilibrium S_{µ} Ç S_{sµ}
is continuous almost everywhere.
Lemma 1
(i) Let K(x) and L(x) be increasing and decreasing respectively families of compact sets with parameter x in the compact
S Ì R, namely if x_{1}<x_{2} then K(x_{1}) Ì K(x_{2}) and L(x_{1}) É L(x_{2}).
In a neighborhood of x_{0}
K(x) Ç L(x) = M(x).
K(x) and L(x) are continuous in x_{0}, i.e.,
Then the jump M(x) in x_{0}, which we defines
is contained in ¶ M(x_{0}) and, in particular, the jump does not contain an interior point.
(ii) Let K(x) and L(x) be continuous in x_{0} families of compact sets with parameter x in [A,B]Ì R.
In a neighborhood of x_{0}
K(x) Ç L(x) = M(x).
If for any x holds K(x)È L(x) º [A,B] Ì R then M(x) is continuous in x_{0}.
Remark. The point (ii) does not hold for any compact in R. Let
S = {0, 1, 1/2, 1/3, ...} and
L(x) = {1, 1/2, 1/3, ..., [1/x]^{1}},
K(x) = S \ L(x) = {1/[1/x]+1, 1/[1/x]+2, ..., 0}
for x Î (0,1].
L(0) = {0} = K(0).
Then non trivial M(x) is only in 0.
Proof.
(i) Taking in to account that K(x) and L(x) are increasing and decreasing set families
respectively we can easy see that for x > x_{0}
M(x) M(x_{0}) =
(K(x) Ç L(x))(K(x_{0}) Ç L(x_{0})) Ì
Ì (K(x) \ K(x_{0}))È(L(x_{0}) \ L(x)).
Let us define the set C:
and prove that C Ì ¶ K(x_{0}) È ¶ L(x_{0}).
Consider any p from C.
If p Ï K(x_{0}) then there exists K(x'), x'>x_{0} such that p Ï K(x').
Let fix any n_{1} > 1/x'x_{0}.
provide that p Î ¶ K(x_{0})
or
for any n>n_{1}.
If p Î int(L_{0}) then there is L(x''), x''>x_{0} so that p Î L(x''), it is simple sequence of continuity of L(x) in x_{0}.
Let fix any n_{2} >max( 1/x''x_{0}, n_{1}). So, in the case p Î int(L_{0}) for any n>n_{2} we have
and hence p Î ¶ K(x_{0}).
We have got p Î ¶ K(x_{0}) È ¶ L(x_{0}) for any n>n_{2} and
J^{+}Ì ¶ K(x_{0}) È ¶ L(x_{0}).
The above consideration yields that
hence to complete the proof we have to investigate the set


¶ [ 

M(x) M(x_{0})] Ì


¶ [ 

M(x)]È ¶ M(x_{0}).

Let
this means that each neighborhood of y contains point from M(x) = L(x)Ç K(x), hence
p
Î limsup
L(x) Ç
limsupK(x) = M(x_{0}).
To prove the assertion "from below" one should apply above reasoning with the exchange of K(x) and L(x).
(ii)Let p be any point from M(x_{0}), then p is the
limit of K(x) and L(x) in x_{0}. Then there are points
k(x) and l(x) from K(x) and L(x) for all x, xx_{0}<e in each neighborhood of p.
By familiar Cantor method of bisecting (see proof of Lemma 4) there exists a point of M(x) in [k, l].
(or in [l, k]) for each x. Then p Î
liminf_{x→ x0+0}M(x) and
= 

(L(x) Ç K(x)) Ì 

L(x) Ç 

K(x) = M(x_{0})

Now we represent the proof of Theorem 1.
Proof. From [K]
it followers that for continuous external field the map x µ_{Q}^{s, x} is increasing
for positive x £ s(R). That is , if 0<x_{1}< x_{2} £ s(R), then the difference
µ_{Q}^{s, x2}µ_{Q}^{s, x1} is a positive measure. Hence the supports S_{ µQs, x}
form an increasing family with respect to x. Families of increasing compacts on R has at most countable number
of breaking points. The decreasing family S_{s  µQs, x} has the same breaking points
with respect to x. One can see it from
(s µ_{Q}^{s, x1})(s µ_{Q}^{s, x2})= µ_{Q}^{s, x2}µ_{Q}^{s, x1}. So we are allowed to
apply Lemma 1 (i) for S_{ µQs, x} =: K(x) and S_{s  µQs, x} =: L(x) for almost all x_{0}.
Hence the functions a(x) and b(x), that are defined from S_{µ} Ç S_{sµ}=[a(x),b(x)], are continuous on
R\ N were N is at most countable set, since non degenerated interval can have only jump that contain interior points.
(ii) is evident from (i) and Lemma 1(ii).
3 
The equations for the endpoints of equilibrium measure support in case of constrain absence. 

We start with a modification of the theorem from ([ST],p.201).
We deal with an equilibrium measure with total mass x and some special kind of external field
that will be made use in the next section.
Theorem 2
Suppose that Q: S R, S is a compact in R, satisfies the following conditions:
a) Q is continuous function on S ,
b) Q^{'} is differentiated a. e. and Q^{'} Î L^{¥}(S).
Let n be some signed measure, so that
a) U^{n} is continuous,
b) ò_{ R}1/yg dn(y) < +¥ for an arbitrary g Î S .
Let µ = µ_{Q}^{s, x} be the equilibrium measure with total mass x = x_{0} on S corresponding to the
external field Q = Q  U^{n} and the support of µ is a finite closed non degenerated interval
S_{µQx} = [a(x_{0}),b(x_{0})]=[a,b] Í S.
Then we have


ó õ


Q^{'}(y) 

dy
 
ó õ



dn(y)


ì í î 
= x_{0} n(R) 
if 
b Î int(S) 
£ x_{0} n(R) 
if 
b Î ¶(S)





ó õ


Q^{'}(y) 

dy
+ 
ó õ



dn(y) 

ì í î 
= x_{0} n(R) 
if 
a Î int(S) 
³ x_{0} n(R) 
if 
a Î ¶(S)



where [a,b ]^{c} = R \ [a,b] is complement of [a,b] in R.
Proof.
Functional F_{Q}^{x}(K):= xlogcap(K)+òQ dw_{K} satisfies
F_{Q }^{x}(S_{µQx}) = F_{Q}^{x}([a,b])
= 

F_{Q}^{x}([a,b ]),

where w_{K} denotes the Robin distribution associated with the set K
(the equilibrium measure for Q=0 on K).
By known facts about capacity and equilibrium measure of the interval we have
F_{Q}^{x0}([a,b ]):= x_{0}log 
æ
ç
ç
è 

ö
÷
÷
ø 
+



ó
õ 



dy = 
=x_{0}log 
æ
ç
ç
è 


ö
÷
÷
ø 
+



ó
õ 



dy
 
ó õ 
U^{n} dw_{[a,b ]} 
In the last integral the both integrals exist and we are allowed to apply Fubini's theorem:
F_{Q}^{x0}([a,b ])=
x_{0} log 
æ
ç
ç
è 


ö
÷
÷
ø 
+



ó
õ 



dy

ó õ 
U^{w[a,b ]}(y) dn(y)= 
= x_{0}log 
æ
ç
ç
è 


ö
÷
÷
ø 
+



ó
õ 



dy + 
+ 
ó
õ 
é
ê
ê
ë 
log 
æ
ç
ç
è 

ö
÷
÷
ø 
+g_{[a,b ]}(y) 
ù
ú
ú
û 
dn(y)=

= log 
æ
ç
ç
è 

ö
÷
÷
ø 
( n (R)x_{0}) +



ó
õ 



dy +

ó
õ

g_{[a,b ]}(y) dn(y).

On making the change of variable
we can rewrite F_{Q}^{x}([a,b ]) as
log 
æ
ç
ç
è 

ö
÷
÷
ø 
( n (R)x_{0}) +



ó
õ 

Q( 

+ 

cos 
q ) dq+

ó
õ

g_{[a,b ]}(y) dn(y)

Now we consider the case when a(x_{0}) or b(x_{0}) lies in int(S).
We take derivatives ¶ F/¶b and ¶ F/¶a which equal to
= 

+



ó
õ 

Q^{'}( 

+ 

cosq ) (1+cosq ) dq+ 
= 

+ 


ó
õ 

Q^{'}(y) 

dy


ó
õ 


dn(y)

and correspondingly
= 

+



ó
õ 

Q^{'}( 

+ 

cosq ) (1cosq ) dq+ 
= 

+ 


ó
õ 

Q^{'}(y) 

dy+


ó
õ 


dn(y)

were [a,b ]^{c} is R\ [a,b ].
Possibility to differentiate under the integral sign is guaranteed by lemma 2 and lemma 3 which we present
just after the proof.
On taking a=a(x_{0}) and b=b(x_{0}), it follows that
= 

+


ó
õ 

Q^{'}(y) 

dy
 

ó
õ 


dn(y) = 0.

A similar argument yields the same formula for the derivative with respect to a
= 

+ 


ó
õ 

Q^{'}(y) 

dy+


ó
õ 


dn(y) = 0

If b(x_{0}) is not interior point of S, we can apply Saff Totik arguments ([ST],p.202) for our Ffunctional
and will be able to write that
¶ F/¶_{+}b ([a ,b]) tends to same limit when b tends to
b0 and minimizes the Ffunctional in point b. So we have inequalities in the theorem because the derivative of
Ffunctional is not greater than zero


ó
õ 

Q^{'}(y) 

dy
 
ó
õ 


dn(y) £ x_{0} n(R). 
By the same way we have ¶ F/¶_{}a ([a ,b]) ³ 0


ó
õ 

Q^{'}(y) 

dy
+ 
ó
õ 


dn(y)³ n(R)x_{0}. 
To complete the section we prove the above mentioned technical lemmas which deal with changing of integration and
differentiation.
Lemma 2
Suppose that Q is a continuous function on the compact
S Ì R, with Q^{'} Î L^{¥}(S).
Furthermore, let I_{1},I_{2} be some compacts sets, and
y:I_{1}× I_{2} ® S, with y Î C^{1}(I_{1}× I_{2}).
Then for all bÎ I_{1} with (bd,b+d)Ì I_{1}
for some d>0



ó
õ 

Q(y(b,q ))
dq = 
ó
õ 

Q^{'} (y(b, q
)) 

(b, q )
dq,

that is, we may exchange integration and differentiation with respect to
the parameter b.
Proof.
Since Q^{'} Î L^{1}(S), we have for
[b_{1},b_{2}]Ì I_{1}
Q(y(b_{2},q)Q(y(b_{1}, q )
= 
ó
õ 

Q^{'}
(y(b, q )) 

(b, q
) db. 
Consequently,
Q(y(b_{2},q )Q(y(b_{1}, q )
£
Q^{'} _{L¥(S)}·


_{L¥( I1× I2)}·
b_{2}b_{1},

and the claim follows by applying Lebesgue's Dominated
Convergence Theorem to f_{n}^{±}(q)= ±[Q(y(b ± 1/n,
q ))Q(y(b, q ))]/(1/n).
The next Lemma deals with the differentiation of a Green potential of a non degenerate interval
[a,b] with respect to one of the endpoints a or
b.
Lemma 3
Let n be some measure with compact support S(n), and
a<b. Provided that
the Green potential òg_{[a,b]}(y) dn(y) can
be differentiated with respect to b, and
Similarly, provided that
the Green potential òg_{[a,b]}(y) dn(y) can
be differentiated with respect to a, and
Proof.
We will only show the first assertion of the Lemma, the second part
is similar. Notice first that, for y Ï[a,b],
In the case bÏS(n), we notice that this derivative is smooth
for yÎ S(n)\ [a,b], and clearly we may
exchange the order of differentiation and integration, see for
instance the argument used in the preceding Lemma.
By splitting
if necessary the Green potential into two parts, we see that it
only remains to consider the case S(n)Ì
[a,3ba/2].
For the second derivative we obtain for y Ï[a,b]
[ 

]^{2} g_{[a,b]}(y) =
 





= 





,

which
is negative when y Î S(n)\ [a,b].
Consider the two sequences of functions
h_{n}^{±}(y)= ± 
g_{[a,b ± 1/n]}(y) g_{[a,b
]}(y) 

1/n 

,

converging pointwise to the
derivative of g_{[a,b]}(y) with respect to b
for almost all y.
Then we know that the Green potential has lefthand or right hand derivatives
with respect to b, with



ó
õ 
g_{[a,b]}(y) dn(y) =



ó
õ

h_{n}^{±}(y) dn(y) ,

provided that the righthand limits exist.For the righthand and lefthand derivatives,
consider the functions
f_{n}^{+}(y) :=

ì
ï
ï
ï
í
ï
ï
ï
î 

g_{[a,b +1/n]}(y) g_{[a,b
]}(y) 

1/n 

,




g_{[a,y]}(y) g_{[a,b]}(y) 

yb 

,



0, 
yÎ [a, b ].


f_{n}^{}(y) :=

ì
ï
í
ï
î 
 
g_{[a,b 1/n]}(y) g_{[a,b
]}(y) 

1/n 

,


yÎ R\ [a, b ], 
0, 
yÎ [a, b ].


Then by the above sign considerations we may conclude that
f_{n}^{+}(y) and f_{n}^{}(y) is increasing and decreasing respectively in n for y Î
S(n)\ [a,b].
Moreover, f_{n}^{±}(y) tends to the
derivative of g_{[a,b]}(y) with respect to b
for almost all y, and hence



ó
õ 
f_{n}^{±}(y) dn(y) =

ó
õ 

g_{[a,b]}(y)
dn(y)

by Lebesgue's Monotone Convergence Theorem. Thus the assertion
of the Lemma follows by showing that limò[
f_{n}^{±}(y)h_{n}^{±}(y)] dn(y) = 0. Indeed, by construction,
f_{n}^{+}(y)£ h_{n}^{+}(y) and f_{n}^{}(y)³ h_{n}^{}(y), and, since
g_{[a,b]}(y)=O(yb)_{y
b+}, we get for some suitable constant C


0 £ 


ó
õ[
h_{n}^{+}(y)f_{n}^{+}(y)] dn(y)




= 


ó
õ 

[h_{n}^{+}(y) f_{n}^{+}(y)] dn(y)




= 


ó
õ 

[ 

 n ] g_{[a,b]}(y) dn(y)




£ C 


ó
õ 

[ 

 n ] yb dn(y)





and


0 £ 


ó
õ[
f_{n}^{}(y) h_{n}^{}(y)] dn(y)




= 


ó
õ 

[f_{n}^{}(y) h_{n}^{}(y)] dn(y)




= 


ó
õ 

n g_{[a,b1/n]}(y) dn(y)




£ C 


ó
õ 

n y(b  1/n) dn(y)





the last equalities following from the assumption on the finiteness of
the above integral. Thus the assertion of the Lemma follows.
4 
The equations for the endpoints of the interval of equilibrium with presence of constraint. 

The main aim of the paper is to describe the interval of equilibrium, i. e. the set where constrain is not hit.
Theorem 3
Suppose that Q: [A,B] R, ( ¥ < A < B < +¥) satisfies the following conditions:
a) Q is continuous function on the interval [A,B],
b) Q^{'} is differentiated a. e. and Q^{'} Î L^{¥}([A,B]).
Let s be a constraint on [A,B] ([A,B] = S_{s } ), so that
a) U^{s} is continuous,
b) ò_{ R}1/yg ds(y) < +¥ for an arbitrary g Î [A, B] .
Let µ = µ_{Q}^{s, x} be the equilibrium measure for external field Q and constraint s with total
mass µ (R) = x = x_{0}£ s(R).
If S_{µ} Ç S_{s µ}=[a(x_{0}),b(x_{0})] = [a,b], A £ a(x_{0}) < b (x_{0}) £ B then

ó
õ 



ds (l)+



ó
õ 

Q^{'}(l) 

dl=


ì í î 
= x_{0} 
if 
b < B 
£ x_{0} 
if 
b=B



 
ó
õ 



ds (l)+



ó
õ 

Q^{'}(l) 

dl= 

ì í î 
= x_{0} 
if 
a > A 
³ x_{0} 
if 
a = A



where [A,B] \ S_{s  µ } = S_{µ} \ S_{s  µ } is part of
S_{µ} where constraint is hit.
Proof. We consider two cases:
1)The case a=a(x_{0}) or b=b(x_{0}) Î int(S_{s µ}),
2)The case a=a(x_{0}) or b=b(x_{0}) Î int(S_{µ}).
1) We can think about µ as µ = n + t , where
t := µ _{Ss µc} = s _{Ss µc}.
Let us define Q(l) := Q(l ) + U^{t}(l) and will write
the equilibrium conditions for µ and after n.
Q(l )+U^{µ}(l) =
Q(l )+U^{µS µQs, x0\ [a, b]}(l)+U^{µ[a, b]}(l)
=Q
(l) + U
^{n}
(l) =


ì í î 
£ F 
on 
S_{µ} 
³ F 
on 
S_{s µ} = S_{n} 
= F 
on 
S_{µ} Ç S_{s µ} = S_{n} = [a, b]



The last two expressions are the sufficient conditions for n being equilibrium measure on å_{n}
with support [a, b]. Also we have
n (R) =µ (R) t (R)=µ (R) µ (S_{s µ}^{c})
=µ (R) s _{Ss µc} (R) = x_{0}
s(S_{s µ}^{c}) ³ 0.
Clear that if U^{s} is continuous then U^{s K} is continuous for any compact
K Ì S_{s}. It follows from
U^{s K}+U^{s Ss \ K}=U^{s} and from the well known fact that
logarithmic potential is a harmonic function outside support of the measure. So Q is continuous.
By means of Theorem 1 for S_{n}, n and Q we have


ó
õ 

Q^{'}(l) 

dl +

ó
õ 


dt(y)
= n (R) +t (R)
b Î int(S_{s µ})




ó
õ 

Q^{'}(l) 

dl
 
ó
õ 


dt(y)
= t (R)n (R)
a Î int(S_{s µ})

With substituting explicit expressions for t and n one should get for b Î int(S_{s µ})

ó
õ 



ds(l )+



ó
õ 

Q^{'}(l) 

dl =
s(S_{s µ}^{c}) + x s(S_{s µ }^{c}) = x_{0}

and for a Î int(S_{s µ})
 
ó
õ 



ds(l )+



ó
õ 

Q^{'}(l) 

dl= x_{0}.

2)We present s µ =: µ^{*} as µ^{*} = n^{*} + t^{*} , where
t^{*} := µ^{*} _{Ss µ*c} = s _{Sµc}.
Let us define
Q^{*}(l) := Q(l )  U^{s}(l)+ U^{t*}(l)=
Q(l )  U^{sSµ }(l)
and write the equilibrium conditions for µ and then for n^{*} .
Q(l )U^{µ}(l) = Q^{*}(l) + U^{n*}(l) =


ì í î 
³ F 
on 
S_{µ} = S_{s  µ*} = S_{n*} 
£ F 
on 
S_{s µ} 
= F 
on 
S_{µ} Ç S_{s µ} = S_{n*} = [a, b]



The first and the last expressions are the sufficient conditions for n^{*} be to the equilibrium measure on S_{n*}
with support [a,b]. Also we can write
n^{*} (R) = s (R)µ (R) 
s _{Sµc} (R) = s(S_{µ} ) x_{0}.
Take in mind that s is the constrain and
n^{*} (R)= s(S_{µ} ) µ(S_{µ} ) ³ 0.
By means of Theorem 2 for
Q(l) = Q(l) 
U^{sSµ}(l) and n^{*} we have in the case b Î int(S_{µ}).
 

ó
õ 

Q^{'}(y) 

dy

ó
õ 


ds_{Sµ}(y) =
n^{*} (R)  s (S_{µ}) = x_{0}

and for a Î int(S_{µ})
 


ó
õ 

Q^{'}(y) 

dy+

ó
õ 


ds_{Sµ}(y) =
s (S_{µ})  n^{*} (R) = x_{0}

We remember S_{µ}\ [a,b ] = S_{s µ}^{c}.
Thus we have get equations for
a, b Î int(S_{µ}) È int(S_{s µ }) É int(S_{µ} È S_{s µ }) = int([A,B]) by Lemma 4 which
we present after this proof.
Inequalities in the theorem can be derived by the same way.
Now we prove above mentioned Lemma 4.
Lemma 4
Let A, B be compact subsets in R, A Ç B = [a, b] Ì R.
If x Î int(AÈ B) then x Î int(A) or x Î int(B). That is int(AÈ B) Ì int(A)È int(B).
Proof.
Suppose that x Î int(AÈ B) but x Ï int(A) and x Ï int(B).
Then there are three possibilities:
1)x Î ¶ A and x Ï B. Since x Î int(AÈ B) and x Ï B then
$ U(x) Ì (AÈ B) which is an open set containing x,
so that U(x) Ç B = Ø
(by definition of set interior and the fact that R is a Hausdorff topology space).
Since x Î ¶ A then there exists consequence {x_{n}} in U (x) with limit x and
not belonging to A. Since {x_{n}} Ï A and {x_{n}} Ï B then U (x)Ì (AÈ B)
and we have a contradiction.
2)x Ï A and x Î ¶ B. The proof is the same as for the case 1).
3)x Î ¶ A and x Î ¶ B. It means that x Î [a, b].
if x Î (a, b) then the assertion of the lemma is evident.
We will consider the case when x = a, and the case x = b is similar.
Evidently [x, x + ba/2) Ì A and [x, x + ba/2) Ì B and
$ U (x) Ì (AÈ B) which is an open set containing x,
then there is e > 0 such that (ae,a) Ì (AÈ B).
And (ae,a) Ç AÇ B = Ø because of lemma condition.
if (ae,a) Ì A or (ae,a) Ì B then x is an interior
point of A or B, and we have a contradiction.
So there are x Î A and y Î B in the interval (ae,a) , let x<y.
Without the loss of generality we can consider that x+y/2 Î A and put
x_{1} = x+y/2, y_{1} = y and then
we will bisect the interval [x_{n}, y_{n}] and after will choose the
interval [x_{n+1}, y_{n+1}] with endpoints belonging different sets.
So we have the system of included intervals { [x_{n}, y_{n}] } _{n=1}^{¥}
with lengths tending to 0, and one endpoint of each interval belongs A,
the other belongs B. The intersection of all this intervals is one point.
It is boundary point of A and B, and belongs to A and B thanks to compactness.
But this contradicts to (ae,a) Ç AÇ B = Ø.
There is essential question about uniqueness of the solutions of equations from theorems 2 and 3. The equations describe the set where
derivative with respect to endpoints of interval of the functional F_{Q}^{x} vanishes. F_{Q}^{x} achieves its minimum and has vanish
derivation on the sets K
so that S_{µQx} K S^{µQx}. The next example shows that S^{µQx}\ S_{µQx} may be non
trivial, and by the same way one can construct very various sets.
In section 2 we have proved that set were constraint is not hit is continuous almost everywhere
that provides S_{µQx}=S^{µQx}= Ì S^{µQx+e}, i. e. the minimum of F_{Q}^{x} is unique a. e. and endpoints
of support or interval of equilibrium is isolated in the set of solutions. Continuity does not
mean that solution is unique a. e., there may be points where derivative is zero.
If the external field Q so that Q''>s
then we may guaranty that there is unique solution of systems from Theorem 2 and 3.
There are more complicated symptoms of uniqueness of the solutions of equations.
Example.
Let Q be
Q := 

ì í î 
= 0 
on 
[2,2] 
= g_{[2,2]}(y) 
on 
[2,3] 
= +¥ 
on 
(¥,2]È [3,¥),



the extremal measure with unique total mass µ_{Q}^{1} is the Chebyshev measure of [2,2] because
we know that U^{w[2,2]}(y)= g_{[2,2]}(y), it provide
U^{w[2,2]}(y)+Q(y) = 

ì í î 
= 0 
on 
[2,2] =: S_{µQ1} 
³ 0 
on 
R.



(It is the sufficient condition, see the section 1). S^{µQ1}=[2,3].
Easy to see that

([2,b]) =


( ln 

+ 
ó
õ 

g_{[2,2]}(y) 

p (y+2)(by) 

dy) º 0
on [2,3]. 
5 
Interval of equilibrium and the continuum limit of the Toda lattice. 

In this section we show that equations from theorem 3 locally satisfy the system of partial differential equations
which supplemented with the initial boundary conditions is wellknown Toda lattice.
Theorem 4 Let Q(l, t) := l t/2 + Q(l,t_{0}) for t from some neighborhood of t_{0} containing
[t_{0}T_{1},t_{0}+T_{2}] satisfies
a) Q(l,t_{0}) is continuous function on the interval [A,B] with respect to l, ( ¥ < A < B < +¥)
b) Q^{'}(l,t_{0}) is differentiated a. e. with respect to l and Q^{'}(l,t_{0}) Î L^{¥}([A,B]).
Let s be a constraint on the interval [A,B] so that
c) U^{s} is continuous,
d) ò_{ R}1/yg ds(y) < +¥ for an arbitrary g Î [A, B] .
e) the set of equilibrium for the extremal problem (??) is an interval for
(x,t) Î [x_{0}X_{1}, x_{0}+X_{2}]× [t_{0}T_{1},t_{0}+T_{2}]
, to wit S_{µ} Ç S_{s µ}=[a(x,t),b(x,t)] ,
A < a(x,t) < b (x,t) < B where µ = µ_{Q}^{s, x} is the equilibrium measure for external field
Q(l, t) and constraint s with total mass x £ s(R).
f) the map [x_{0}X_{1}, x_{0}+X_{2}]× [t_{0}T_{1},t_{0}+T_{2}] {( a(x,t),b (x,t))}
so that Jacobi matrix is welldefined, continuous
and nowhere degenerate in the neighborhood of (x_{0},t_{0}).
Then a(x,t) and b (x,t) satisfy PDE in some neighborhood of (x_{0},t_{0})
Proof.
Since Jacobian in the condition f) is nowhere degenerate in the
neighborhood of
(x_{0}
,t_{0}
) we can inverse the
PDE (3) by
Then the inversed system for (3) is
We have got equations for endpoints of interval of equilibrium (Theorem 3)



ó
õ 



ds (l)+



ó
õ 

Q^{'}(l) 

dl=
x+ 




ó
õ 



ds (l)



ó
õ 

Q^{'}(l) 

dl= x
 





(5) 
After addition and subtraction we arrive to





ó
õ 


(2lab) ds (l) 

(la)(l  b) 



ó
õ 


(2lab) ds (l) 

(la)(l  b) 

+ 




(6) 
By differentiating the first and the second
equations in (5) with respect to a and
b respectively we have


= 
ó
õ 



ds (l)
 
ó
õ 



ds (l) 








With the system (6) one can easily see that this is our
PDE (4)
because the sum of all rest members is equal to zero.
To complete the proof we have to check possibility of differentiating in
(5). We discuss only the
first equation. There is problematic place in the first integral. Like in the section 3 we define
functions
and h_{n}^{+}(l) = 0
when a < l < a + 1/n.
The functions h_{n}^{±}(l) converge pointwise to the derivative of
la/l  b with respect to a.
The absolute value of h_{n}^{+}(l) is dominated by
1/(la)(l  b).
By d) it is
integrable function for the measure s. The claim follows by applying Lebesgue's Dominated
Convergence Theorem.
By convexity of la/l  b with respect to
a the functions h_{n}^{}(l) are decreasing and they absolute values are
dominated by the absolute value of derivative of
la/l  b with respect to a, which has a finite
integral by d). The assertion follows by Lebesgue's Monotone Convergence Theorem.
Now we consider derivative of F(a) = ò_{a}^{b} Q'(l)la/bl dl.
= 
ó
õ 

Q'(l) 

dl

ó
õ 

Q'(l) 

dl 




+


ó
õ 

Q'(l) 

dl

ó
õ 

Q'(l) 

dl 




.

The limit of the last part when n ¥ is equal to zero because
 
ó
õ 

Q'(l) 

dl
£ Q'_{L¥}* 

* 
ó
õ 


dl.

Now we define functions h_{n}(l) and our task is to change integration and differentiation.
h_{n}(l) :=

ì
ï
ï
í
ï
ï
î 

Then

ó
õ 

Q'(l) 

dl

ó
õ 

Q'(l) 

dl 




=

ó
õ 

h_{n}(l) dl.

The absolute values of the functions h_{n}(l) are dominated by Q'_{L¥}/(la)(bl)
and by applying Lebesgue's Dominated Convergence Theorem we have considered the case of writehand derivative.
For the leftend derivative we can apply the above reasoning with the same dominator function for the

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This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.
