The Support of the Equilibrium Measure for a Class of External Fields on a Finite Interval

1. The Support of the Equilibrium Measure for a Class of External Fields on a Finite Interval • Introduction and Results • Potential Theory Preliminaries • Elasticity Interpretation • Proofs S. B. Damelin, Peter Dragnev*, and A. B. J. Kuijlaars

2. 1. Introduction and Results External field Let Q C( ), say  =[-1,1]. Weighted energy Define IQ(): Extremal measureQis defined as: IQ(Q)=min{IQ(): ( )}, where ( ) are all unit Borel measures supported on.

3. Applications of extremal measures: weighted approximation orthogonal polynomials integrable systems random matrix theory Remark: The support of the extremal measure is a main ingredient in the solution. If we know that the support consists of N intervals, then we set up a system of equations for theendpoints; once we know the endpoints, we find the extremal measure from a Riemann-Hilbert problem.

4. Two results on the nature of the support: • (Mhaskar-Saff, ‘85)If Q is convex, then supp(Q)is one interval. • (Deift-Kriecherbauer-McLaughlin, ‘98) If Q is real analytic, then supp(Q) consists of a finite number of intervals. The determination of the number of intervals is a nontrivial problem.

5. We shall study nonconvex and non real analytic external fields on  = [-1,1] of the type: Q,c(x)= -c sign(x) |x|(1.1) Main result: The support of Q forQ =Q,cconsists of at most two intervals. • For integer - considered in {DKM} and {DaK}; • For  = [0,1] - considered in {KDr}; Definition: Let Q be -Hölder continuous on [-1,1], i.e. Q C1+([-1,1])  |Q´(x)-Q´(y)|C|x-y|, for some >0 and some positive C.

6. Theorem 1. Let Q C1+([-1,1]). Suppose there exists a1[-1,1], such that a) Q is convex on [-1,a1]; b) for every a [-1,a1], there is t0 [a1 ,1], such that (1.2) decreases on [a1 , t0] and increases on [t0 ,1]. Then supp(Q) is the union of at most two intervals.

7. Theorem 2. For 1 and c>0, letQ,cbe given by (1.1). Then for every a[-1,0], there is t0 [0,1], such that (1.3) decreases on [0, t0] and increases on [t0 ,1]. As a result, the support of Q,c consists of at most two intervals. Remark:The result for any 1 is harder than the integer cases and required a new approach (Th1). This simplified the proofs in the other cases as well.

8. 2. Potential Theory Preliminaries • Frostman characterization of Qfor Q C1+([-1,1]): • U(x)+Q(x)=F for xsupp() • U(x)+Q(x)Ffor x[-1,1],(2.1) • whereU(x)= -log|x-t| d(t). • Let =supp(Q) and dQ = v(t) dt, then we have • log|x-t| v(t)dt = Q(x)-F, x(2.2) • v(t)dt = 1.

9. If =k [ak ,bk] we get a singular integral equation (2.3) The solution v(t) of the SIE in (2.3) has N (the number of intervals in ) free parameters, determined from the fact that the constant F is unique for all intervals and the total mass of v is one. In general, the solution is not positive. Denote the signed measure =v(t)dt. For=[a,1] (2.4)

10. Balayage of a measure: The balayage onto  of a nonnegative measure  is a measure that is supported on  and U (x)= U(x)+c, for q. e.x We write also =Bal(; ). For signed measure  = +-- Bal(;) = Bal(+; )-Bal(- ;). From the definition of  it is clear that if 12 then 1 = Bal(2; 1)

11. Two Lemmas on balayage and convexity: Lemma 1.Let  and n be finite unions of intervals with limnn= . Then limnn =  (in weak* sense). • Lemma 2.Let  be a finite union of closed intervals and  be the associated signed measure, and let v be its density. Suppose that [a,b] and: • (a) Q is convex on [a,b]; • (b) v(a)0 and v(b)0; • (c) v(t)0 on  \ [a,b]; • Then v(t)>0 for all t[a,b];

12. 3. Elasticity interpretation [KV] Let the lower half-plane be elastic and Q be the profile of a rigid punch (up to a constant). Suppose a force f is applied to Q. Let D be the displacement of Q. Denote the region of contact S and the pressure p(x)dx. For small D the profile of the plane is given by -U= log|x-y|p(y)dy. Thus log|x-y|p(y)dy= Q(x)+F on S  log|x-y|p(y)dy Q(x)+F on [-1,1]. (3.1) Then =p(t)dt/f is an extremal measure with external field Q/f (compare (3.1) with Frostman conditions (2.1)).

13. Now it is easy to illustrate our result. The rigid punch has a profile -x3. Applying force f corresponds to an external field Qc = -cx3 with c=1/f. Then we expect to have three critical numbers 0<c1<c2 <c3, such that the support is [-1,1] for c<c1; for c1<c<c2 it is [a,1] with a<0; for c2 <c<c3 it is [a,p][q,1]; and for c>c3 the support is [a,1] with a<0.

14. 4. Proofs Theorem 1. Let Q C1+([-1,1]). Suppose there exists a1[-1,1], such that a) Q is convex on [-1,a1]; b) for every a [-1,a1], there is t0 [a1 ,1], such that (1.2) decreases on [a1 , t0] and increases on [t0 ,1]. Then supp(Q) is the union of at most two intervals. Proof: Let a=min {x: x supp()}.

15. If supp()  [a1 ,1], then the problem reduces to {KDr}. Therefore, assume a< a1. For every pair (p,q), with a<pq1, let vp,qbe the density of the signed measure  with  =[a,p][q,1] (if q=1, then  =[a,p]); Introduce Z, consisting of all (p,q) such that a) a<pq1and qa1; b)supp()  [a,p][q,1]; c) If q<1, then vp,q(t) increases on (q,1). d) If p>a1, then vp,q(t) decreases on (a1,1).

16. First, Z. Indeed by formula (2.4) (recall) and condition b) of the theorem we have that (t0, t0)  Z. (2.4) Next, we show that Z is closed. Let (pn,qn)  Z and pnp, qnq. We have to show that (p,q)  Z. It is clear that (p,q) satisfies conditions a) and b).By Lemma 1 we obtain vpn ,qn vp,q from which we derive c) and d). Finally, we find a pair (p,q) in Z such that q-p maximal. For this choice vp,q is positive. Since supp()  [a,p][q,1], it followssupp() = [a,p][q,1]. QED

17. Theorem 2. For 1 and c>0, letQ,cbe given by (1.1). Then for every a[-1,0], there is t0 [0,1], such that decreases on [0, t0] and increases on [t0 ,1]. As a result, the support of Q,c consists of at most two intervals. Proof: Write Q=Q,c. Since Q is convex on [-1,1], it is left to show that (4.1) has the decreasing/increasing property for t[0,1].

18. Let (4.2) where I2 is p.v. integral. We establish the properties (i) G (0)  0; (ii) G (1) > 0; (iii) For every  >1, there is t  [0,1], such that G´ (t)<0 on [0,t ], G´ (t)>0 on [t ,1], and G´´ (t)  0 on [ t ,1].

19. Properties (i) and (ii) are straight forward. We prove (iii) by induction on k=[]. For =1 we explicitly find G1 (t)=t-(1+a)/2 and (iii) is true with t1=0. Suppose now 1< <2. Consider f(z)=z -1[(z-1)(z-a)]1/2 defined for z(-,1]. Then I2 may be written as (4.3) where the contours  and Raregiven below.

20. Using contour integration we get the representation: (4.4)

21. Differentiating (4.4) twice and letting R   we obtain: (4.5) We now conclude that G´´ (t) > 0 for t (0,1), in the case when 1< <2. Therefore G is strictly convex and (iii) follows easily from G (0) < 0 and G (1) > 0. Thus we have established (iii) for []=1.

22. Now let k 2, and suppose (iii) is true for all  with [] = k-1. (4.6) where

23. For the derivatives of F(t) we get for 0<t<1: (4.7) Differentiating (4.6) we get (4.8)

24. F´(t)<0 F´´(t)>0 (4.7) (4.8) By the inductive hypothesis there exists t -1, such that G´ -1(t)<0 on (0,t -1), and G´ -1(t)>0,G´´ -1 (t)  0 on (t -1,1). Consider first t -1>0 (the other case is similar). • On (0,t -1); G (0) 0,G´ -1(t) <0  G -1(t)  • On (0,t -1);G -1(t) < 0  G´ (t) < 0 on (0,t -1) • On (t -1,1); G´ -1(t)>0,G´´ -1(t)>0  G´´ (t)>0 (s.c.) • Then  t >t -1: G´ (t) is (+) on (0,t ) and(-)on (t ,1). • Of course G´´ (t)>0 on (t ,1). This proves (iii). QED