Integrable PDEs and Nonlinear Steepest
Descent
Spyros Kamvissis
Himeji, Japan

March 5, 2015

Rieman-Hilbert factorization
The so-called nonlinear stationary − phase − steepest − descentmethod for the asymptotic analysis of Rieman-Hilbert factorization
problems has been very successful in providing(i) rigorous results on long time, long range and semiclassicalasymptotics for solutions of completely integrable equations andcorrelation functions of exactly solvable models,(ii) asymptotics for orthogonal polynomials of large degree,(iii) the limiting eigenvalue distribution of random matrices of large
dimension (and thus universality results, i.e. independence of theexact distribution of the original entries, under some conditions),(iv) proofs of important results in combinatorial probability (e.g.
the limiting distribution of the length of longest increasing
subsequence of a permutation, under uniform distribution).

Stationary phase

Even though the stationary phase idea was first applied to a
Riemann-Hilbert problem and a nonlinear integrable equation by
Alexander Its (1982) the method became systematic and rigorousin the work of Deift and Zhou (1993).
In analogy to the linear stationary-phase and steepest-descentmethods, where one asymptotically reduces the given exponential
integral to an exactly solvable one, in the nonlinear case one
asymptotically reduces the given Riemann-Hilbert problem to an
exactly solvable one.

Non-self-adjointness
Our aim here is to comment on the distinction between thestationary-phase idea and the steepest-descent idea, stressing theimportance of actual steepest descent contours in some problems.
We claim that the distinction partly mirrors theself − adjoint/non − self − adjoint dichotomy of the underlying
Lax operator. To this aim we first have to review some of the mainideas appearing in the self-adjoint case. We mostly use thedefocusing /focusing nonlinear Schr ödinger equation as ourworking model, but we also digress to the KdV at some point.
We stress both here and in the main text that an extra featureappearing only in the nonlinear asymptotic theory is theLax − Levermore variational problem, discovered in 1979, before
the work of Its, Deift and Zhou, but reappearing here in the guiseof the so-called g − function which is catalytic in the process ofdeforming Riemann-Hilbert factorization problems to exactly
solvable ones.

THE LINEAR METHOD
Suppose one considers the Cauchy problem for, say, the linearized
KdV
ut − uxxx = 0.
It can off course be solved via Fourier transforms. The end resultof the Fourier method is an exponential integral. To understandthe long time asymptotic behavior of the integral one needs toapply the stationary-phase method (see e.g. Erdelyi). The
underlying principle, going back to Stokes and Kelvin, is that thedominating contribution comes from the vicinity of the stationary
phase points. Through a local change of variables at eachstationary phase point and using integration by parts we cancalculate each contributing integral asymptotically to all orders
with exponential error. It is essential here that the phase xξ − ξ 3 t
is real and that the stationary phase points are real.

Airy integral
On the other hand, suppose we have something like the Airy
exponential integral

Ai (z) = π1R∞0cos(s 3 /3 + zs)ds

and we are interested
in z → ∞. Set s = z 1/2 t and x = z 3/2 . SoR∞
exp(ix(t 3 /3 + t))dt.
−∞
3
The phase is h(t) = t3 + t and the zeros of h′ (t) = (t 2 + 1) are ±i .
1/3

Ai (x 2/3 ) = x2π

As they are not real, before we apply any stationary-phase method,
we have to deform the integral off the real line and along particularpaths: these are the steepest descent paths. They are given by thesimple characterization Imh(t) = constant. In our particular
example, the curves of steepest descent are the imaginary axis andthe two branches of a hyperbola. By deforming to one of thesebranches, we finally end up with Laplace type integrals and thenapply the same method as above (local change of variables plus
integration by parts) to recover valid asymptotics to all orders.

THE NONLINEAR METHOD
The nonlinear method generalizes the ideas above, but also
employs new ones.
Consider the initial value problem for the defocusing nonlinear
Schrödinger equationi ∂t ψ + ∂x2 ψ − |ψ|2 ψ = 0, underψ(x, 0) = ψ0 (x),where the initial data function is, say, Schwartz.
The analog of the Fourier transform is the scattering coefficientr (ξ) for the Dirac operator L =i ∂xi ψ0 (x).
∗−i ψ0 (x) −i ∂xSuppose we are now interested in the long time behavior of thesolution to (4). The inverse scattering problem can be posed interms of a Riemann-Hilbert factorization problem.

The Riemann-Hilbert problem for dNLS

THEOREM. There exists a 2x2 matrix Q with analytic entries in
the upper and lower open half-planes, such that the normal limits
Q+ , Q− , as ξ approaches the real line from above or below
respectively, existand satisfy
2

Q+ (ξ) = Q− (ξ)
and

1 − |r (ξ)|2−r ∗ (ξ)e −2i ξx−4i ξ t, ξ real
2i ξx+4i ξ 2t
r (ξ)e1limξ→∞Q(ξ) = I .

The solution to dNLS is recovered via ψ(x, t) = −2limξ→∞ ξQ12 .

The stationary phase idea

It was first realized by Alexander Its (1982) that the long time
asymptotics for the solution of dNLS can be extracted by reducing
the problem above to a ”local” Riemann-Hilbert problem locatedin a small neighborhood of the stationary phase point ξ0 such thatΘ′ (ξ0 ) = 0 where Θ = ξx + 2ξ 2 t. The deformation method has
been made rigorous and systematic by Deift-Zhou (1993). Here are
the basic ideas, which have been used in all works on thestationary-phase-steepest-descent-method since.

Equivalent singular integral operator

This goes back to Mushkelishvilli and the Georgian school andprovides a nice way to show that under some conditions, smallchanges in the jump data result in small changes in the solution.

R)−1 (I )(s)w (s)ds
Q = I + Σ (I −Cw2πi(s−z)

where w encodes the jump data, Cw is a sum of weighted Cauchy
operators and Σ is the jump contour.

Auxiliary Factorizationx
Suppose ξ0 = − 4t.
Consider the region ξ0 < M, some positive constant. Note thefollowing
upper/lower and lower/upper
respectively factorizations.


1 − |r (ξ)|2 −r ∗ (ξ)e −2i Θ2i Θ r (ξ)e ∗ −2i Θ  1
1
0
1 −r e=for01
re 2i Θ 1

ξ > ξ0 ,LU Factorizationand the lower/upper factorization=

1 − |r (ξ)|2 −r ∗ (ξ)e −2i Θr (ξ)e 2i Θ
1
!
−1
d−
0
d+

where

−1
rd−
exp(2i Θ)
1−|r |2d−0−r ∗ d+ exp(−2i Θ)
1−|r |2d+−1!

for ξ < ξ0 ,

Auxiliary Scalar Problem

d is a function analytic in C\(−∞, ξs ] such thatd+ (ξ) = d− (ξ)(1 − |r (ξ)|2)d+ (ξ) = d− (ξ)
d → 1 as ξ → ∞.

for − ∞ < ξ ≤ ξs ,for ξ > ξs ,

Conjugation
Note that both the JUMP MATRIX and the FACTORS are
CONSTANT (in x, t) matrices conjugated by the diagonal
22diag [e −i ξx−2i ξ t , e i ξx+2i ξ t ]. Note that because of the actual
conjugation, the exponential decay / increase of the off-diagonalterm depends on the triangularity! That is why different
factorizations (UL or LU) are appropriate in different areas of thecomplex spectral plane. These areas meet at the stationary phase
point.
Now consider a cross centered on the stationary phase point (seeFigure 1 below). The actual position of the branches is notimportant. The branches do not even have to be straighthalf-lines. All that matters is that the sign of the crucial quantity
Re(i Θ) is constant along each branch.

A stationary phase point centered on a cross
D2
+

+−

−

D

D

1

+

− +

−

D

4

Figure 1. A cross centered at a stationary phase point.

3Change of Variable

Definea new matrix byM = Q,

d −1
0


r ∗ de 2i Θ
1−|r |2

ξ ∈ D2 ∪ D4 ,Q, ξ ∈ D1 ∩ {Imξ > 0},
d −10, ξ ∈ D1 ∩ {Imξ < 0},
M =Q
rd −1 e −2i Θ
d
2
1−|r
|


1 −r ∗ e −2i ΘM =Q, ξ ∈ D3 ∩ {Imξ < 0},
0
1


1
0
M=
Q, ξ ∈ D3 ∩ {Imξ > 0}.
−re 2i Θ 1
M=dApproximation

One sees immediately that the off-diagonal terms of the resulting
jumps are exponentially small away from the center of the cross.
So, they can be neglected asymptotically. One ends up with a
Rieman-Hilbert problem on a cross centered at ξ0 . Apart from asmall cross centered at ξ0 the jumps are diagonal everywhere.
In this sense, the dominating contribution to the solution of the
Rieman-Hilbert problem comes from a small neighborhood of thestationary phase point. The Rieman-Hilbert problem can be solved
explicitly via parabolic cylinder functions (following Its) and theasymptotics are recovered.

Local RHP
More precisely, using the rescaling ξ → −ξ0 + ξ(−tξ0 )−1/2 , theRiemann-Hilbert problem is rescaled to a new problem on an
infinite cross. After deforming the components of the cross back to
the real line, it is equivalent to the following problem on the realline:

H+ (ξ) =2

H− (ξ)exp(−i ξ σ3 )H(ξ) ∼ ξ i νσ3 ,
1 − |r (ξ0 )|2 −r ∗ (ξ0 )exp(i ξ 2σ3 ),r (ξ0 )
1

where ν is a constant depending only on ξ0 and σ3 =is a Pauli matrix.




1 0
0 −1

Final RHP

After conjugating we end up with a jump that is constant.

H+ (ξ)exp(−i ξ 2σ3 ) =
2H− (ξ)exp(−i ξ σ3 )


1 − |r (ξ0 )|2 −r ∗ (ξ0 ),r (ξ0 )
1

H(ξ)exp(−i ξ 2 σ3 ) ∼ exp(−i ξ 2 σ3 )ξ i νσ3 ,The analogy with the linear case is clear.

ξ → ∞.

The g-function and a ”shock” phenomenon with no linear
analogue

The full force of the g-function idea and the connection to theLax − Levermore (1979) variational problem is explored in theanalysis of the KdV equation by Deift, Venakides Zhou (1997).

ut − 6uux + ǫ2 uxxx = 0, u(x, 0) = u0 (x), in the limit as ǫ → 0.
Assume for simplicity, that the initial data are real analytic,
positive and consist of a ”hump” of unit height.

The RHP for KdV

The associated
RH problem is S+ (z) =S− (z)

3
1 − |r (z)|2−r ∗ (z)e (−izx−4iz t)/ǫ, z real ,3r (z)e (izx+4iz t)/ǫ1

and limz→∞ S(z) = (1, 1).

The solution of KdV is given by∂ 1
S1 (x, t; ǫ).
u(x, t; ǫ) = −2i ǫ ∂x

Scattering data for small ǫ

The reflection coefficient r also depends on ǫ. In fact, the WKB
approximation isr (z) ∼ −ie −2i ρ(z)/ǫ χ[0,1] (z)1 − |r (z)|2 ∼ e −2τ (z)/ǫ ,R∞[z − (z 2 − u0 (x))1/2 ]dx,
τ (z) = Re (u0 (x) − z ) dx

where ρ(z) = x+ z +

R

x+
2 1/2

and x+ (z) is the largest solution of u0 (x+ ) = z 2 .

The Lax-Levermore Variational Problem
THEOREM [LL]. The weak limit of KdV exists and satisfies

w − limǫ→0 u(x, t, ǫ) = ∂xx [minA Q(ψ; x, t)],where A = {ψ ∈ L1 [0, 1] : 0 ≤ ψ ≤ φ}

R1
Q(ψ; x, t) = (4/π) 0 a(η, x, t)ψ(η)d η −
R1R1
η−µ 2
(1/π 2 ) 0 0 log ( η+µ) ψ(η)d ηψ(µ)d µ,R∞a(η, x, t) = ηx − 4η 3 t − ηx+ (η) − x+ (η) (η − (u(y ) + η 2 )1/2 )dy .
R x (η)φ(η, x, t) = x−+(η) (u(y ) + η 2 )−1/2 dy .

The g-function; the finite band ansatz

Let g (z) =Rlog (z − η)d µ(η)

where dµ = ψdx.
Further assume that supp(dµ) is supported in ∪N
j Ij , where
Ij = Ij (x, t), N = N(x, t).

The conditions for the g-function
Following Deift, Venakides, Zhou (1997) introduce thetransformation Ŝ(z) = S(z)exp(ig (z)σ3 /h)(i) g is analytic off the interval [0, 1] and vanishes at infinity.
(ii) ”Finite gap ansatz”. There exists a finite set of disjoint open
real intervals Ij ∈ [0, 1] such that the normal limits g+ , g− of gexist along these intervals and,denoting h(z) = g+ (z) + g− (z) − 2ρ + 4tz 3 + xz,(iia) For z ∈ ∪j Ij , we have −τ < (g+ − g− )/2i < 0 and h′ = 0.
(iib) For z ∈ [0, 1] \ ∪j Ij , we have 2i τ = g+ − g− and h′ < 0.

Justification of the finite band ansatz

In general (for any data u0 ) it is not true that the above conditions
can be satisfied. It is believed however that under the condition ofanalyticity a g-function satisfying the ”finite gap ansatz” exists.
(In fact Kuijlaars (2000) gave a proof of the ”finite gap ansatz” inthe analogous problem of the continuum Toda equations.)

Consequence of the finite band ansatz

Assuming the ”finite gap ansatz” one can show that the RH
problem reduces to one supported on the bands Ij with jumps ofthe form−ie −ih(z)/ǫ
−ie ih(z)/ǫ
0
and in fact, because of (iib), h(z) is a real constant on eachband Ij . This RH problem can be solved explicitly via theta
functions. The details in [DVZ97] involve the so-called”lens”-argument. Auxiliary contours are introduced near piecesof the real line, and appropriate factorizations and analytic
extensions are used, similarly to the discussion above.
0Steepest Descent Contours
Having reviewed some essential ideas in the previous sections, weare ready to consider the focusing NLS equation, following
Kamvissis, K .McLaughlin, P.Miller (2003).
2

i ~∂t ψ + ~2 ∂x2 ψ + |ψ|2 ψ = 0,
under ψ(x, 0) = ψ0 (x).

L=



ih∂x
−i ψ0 (x)−i ψ0∗ (x) −ih∂x

is a non-self-adjoint Lax operator.

We shall see that the deformation of the semiclassical RH problem
can be no more confined to a small neighborhood of the real axis
but is instead fully two-dimensional. A steepest descent contour
needs to be discovered!

Spectral Data for small ~By the way, in the long time asymptotics for the above with ~ = 1
a collisionless shock phenomenon is also present; for x, t in theshock region the deformed RH problem is supported on a verticalimaginary slit. (K. (1996)) But here, we rather focus on thesemiclassical problem ~ → 0 which is far more complicated.
For simplicity consider the very specific data ψ0 (x) = Asechx
where A > 0. Let x− (η) < x+ (η) be the two solutions of
sech2 (x) + η 2 = 0. Also assume that ~ = A/N and consider thelimit N → ∞. It is known that the reflection coefficient is
identically zero and that the eigenvalues of L lie uniformly placedon the imaginary segment [−iA, iA]. In fact the eigenvalues are thepoints λj = i ~(j + 1/2), j = 0, ..., N − 1 and their conjugates. The
norming constants oscillate between −1 and 1.

Riemann-Hilbert for focusing NLS
The associated RH problem is a meromorphic problem with nojump: to find a rational function with prescribed residues at thepoles λj and their conjugates. It can be turned into a holomorphic
problem by constructing two loops, one encircling the λj which wedenote by C and one C ∗ encircling their conjugates. We redefine
the unknown 2x2 matrix inside the loops so that the poles vanish(there is actually a discrete infinity of choices, corresponding to an
infinity of analytic interpolants of the norming constants, see
below) and thus arrive at a nontrivial jump across the two loops,
encircling the segments [0, iA] and [−iA, 0] respectively.
THEOREM. Let dµ = (ρ0 (η) + (ρ0 )∗ (η ∗ ))dη, where ρ0 = i is theasymptotic density of eigenvalues supported on the linear segment[0, iA]. Set X (λ) = π(λ − iA).

Riemann-Hilbert for focusing NLS
Letting M+ and M− denote the limits of M on Σ = C ∪ C ∗ fromleft and right respectively, we define the Riemann-Hilbert
factorization problem M+ (λ) = M− (λ)J(λ),
where J(λ) = v (λ), λ ∈ C , J(λ) = σ2 v (λ∗ )∗ σ2 , λ ∈ C ∗ ,limλ→∞ M(λ) = I ,and v (λ) =R
1 −i exp(1/h[ log (λ − η)d µ(η)) − (2i λx + 2i λ2t − X (λ))])01Then the solution of fNLS is given by ψ(x, t) = 2i limλ→∞ (λM12 ).

Asymptotic analysis of the Riemann-Hilbert
Our analysis in [KMM] makes use of all the ideas described in theprevious sections (factorization, lenses, the singular integral
operator, an auxiliary scalar problem), but it also takes care of thefact that while the loops can be deformed anywhere away from thepoles as long as h is not small, they have to be eventually locatedat a very specific position in order to asymptotically simplify the
RH problem, as h → 0. Appropriately, the definition of ag-function has to be generalized. Not only will it introduce thedivision of the loop into arcs, called ”bands” and ”gaps”, but it
must implicitly select a contour, the steepest descent contour .
Rather than giving the complicated set of equations andinequalities defining the g-function, we will rather focus on theassociated variational problem; it is not a maximization problem
but rather a maximin problem. Here’s the setting.

The variational problem
Let the complex upper-half plane {z : Imz > 0} be denoted by H
and its closure {z : Imz ≥ 0} ∪ {∞} be denoted by H̄. Let also{z : Imz > 0} \ {z : Rez = 0, 0 < Imz ≤ A} be denoted by K. In
the closure of this space, K̄, we consider the points ix+ and ix− ,where 0 ≤ x < A as distinct. In other words, we cut a slit in theupper half-plane along the segment (0, iA) and distinguish between
the two sides of the slit. The
point infinity belongs to K̄, but not|z−η∗ |
K. Let G (z; η) = log |z−η| be the Green’s function for the upper
half-plane and let dµ0 (η) be the nonnegative measure −id η on thesegment [0, iA] oriented from 0 to iA. The star denotes complex
conjugation. Let the ”external field” φ be defined byR
φ(z) = − G (z; η)d µ0 (η) − Re(π(iA − z) + 2i (zx + z 2 t)),where, wlog x > 0.

Free and weighted energy
Let M be the set of all positive Borel measures µ on K̄, such thatboth the free energy

E (µ) =RR

G (x, y )d µ(x)d µ(y )Rand φdµ are finite.
The weighted energy of the field φ isREφ (µ) = E (µ) + 2 φd µ, µ ∈ M.

Now, given any curve F in K̄, the equilibrium measure λF
supported in F is defined by Eφ (λF ) = minµ∈M(F ) Eφ (µ),
where M(F ) is the set of measures in M which are supported in
F , provided such a measure exists.

S-curves

The finite gap ansatz is equivalent to the existence of a so-called
S − curve joining the points 0+ and 0− and lying entirely in K̄. By
S-curve we mean an oriented curve F such that the equilibrium
measure λF exists, its support consists of a finite union of analytic
arcs and at any interior point of suppµ the so called S-property issatisfiedFFd(φ + V λ ) = dnd− (φ + V λ ).
dn+

The analytic arcs are actually trajectories of quadratic differentials

A maximin problem

The appropriate variational problem is: seek a a ”continuum” ( acompact connected set containing 0+ , 0−) C such thatEφ (λC ) = maxcontinua Eφ (λF ) = maxcontinua minµ∈M(F ) Eφ (µ).
The existence of a nice S-curve follows from the existence of acontinuum C maximizing the equilibrium measure, in particular the
associated Euler-Lagrange equations and inequalities.

Justification of the steepest descent method
EXISTENCE THEOREM [Kamvissis, Rakhmanov 2005]
For the external field φ, there exists a continuum F ∈ F such thatthe equilibrium measure λF exists andEφ [F ](= Eφ (λF )) = maxF ∈F minµ∈M(F ) Eφ (µ).
REGULARITY THEOREM [Kamvissis, Rakhmanov 2005]. The
continuum F is in fact an S-curve, so long as it does not touch the
spike [0, iA] at more than a finite number of points.
If F touches the spike [0, iA] at more than a finite number ofpoints, more work is required(K .2009, Kamvissis − Papadimitropoulos(2015)).
The S-curve lives in an infinite-sheeted Riemann surface!

Main ideas of the proofs: compactness

Let ρ0 be the distance between compact sets E , F in K̄:ρ0 (E , F ) = maxz∈E minζ∈F ρ0 (z, ζ). Introduce the Hausdorff metricon the set I ( K̄) of closed non-empty subsets of K̄:ρK (A, B) = sup(ρ0 (A, B), ρ0 (B, A)).
Compactness of F is the necessary first ingredient to proveexistence of a maximizing contour.

Main ideas of the proofs: upper semicontinuity

The second ingredient is semicontinuity of the energy functionalthat takes a given continuum F to the equilibrium energy on thiscontinuum:

E : F → Eψ [F ] = Eψ (λF ) = inf µ∈M(F ) Eψ (µ).
The proof involves balayage.

Main ideas of the proofs: regularity
For regularity, the crucial step is
THEOREM. Let F be the maximizing continuum of and λF be theequilibrium measure. Let µ be the extension of λF to the lower
complex plane via µ(z ∗ ) = −µ(z). ThenR V ′ (z)−V ′ (u)
R
′
2
′
2
Re( dµ(u)
+
V
(z))
=
Re(V
(z))
−
2Re
d µ(u)
u−z R
z−u
−2
′
+ Re [ z2(u + z)V (u) d µ(u)].

Here V is the logarithmic potential of µ.

The proof is achieved by taking Schiffer variations of the energyfunctional.

d µτ (z τ ) = d µ(z),
where

2

u
h(u) = u−z,z τ = z + τ h(z)h(u)−h(v )z2= 1 − (u−z)(vu−v−z)

Main ideas of the proofs: regularity
But the support of the equilibriumof the maximizing
R measure1
continuum is characterized by log |u−z| dµ(u) + VR (z) = 0. By
differentiating and using the identity above we see that it is alsocharacterized byReRz(Rµ )1/2 dz = 0, where

R
′
′ (u)
d µ(u)Rµ (z) = (V ′ (z))2 − 2 suppµ V (z)−Vz−uR+ z−2 ( suppµ 2(u + z)V ′ (u) d µ(u)).

The S-property follows easily and this proves the Regularity
Theorem.

The Theorem
Let x0 , t0 be given. The solution of NLS is described (locally) as aslowly modulated G + 1 phase wavetrain. Setting x = x0 + ~x̂ andt = t0 + ~t̂, so that x0 , t0 are ”slow” variables while x̂, t̂ are ”fast”
variables, there exist parametersa, U = (U0 , U1 , ...., UG )T , k = (k0 , k1 , ......, kG )T ,w = (w0 , w1 , ....., wG )T , Y = (Y0 , Y1 , ........., YG )T ,Z = (Z0 , Z1 , ......, ZG )T depending on the slow variables x0 and t0(but not x̂, t̂) such that

ψ(x, t) = ψ(x0 + ~x̂, t0 + ~t̂) ∼ A(X0 , t0 )e iU0 (x0 ,t0 )/~ e i (k0 (x0 ,t0 )x̂−w0 (x0 ,t0 )t̂)·

Θ(Y (x0 , t0 ) + iU(x0 , t0 )/~ + i (k(x0 , t0 )x̂ − w (x0 , t0 )t̂)).
Θ(Z (x0 , t0 ) + iU(x0 , t0 )/~ + i (k(x0 , t0 )x̂ − w (x0 , t0 )t̂))All parameters can be defined in terms of an underlying Riemannsurface X .

Conclusion

I expect that these results concerning the nonlinear steepest
contour method applied to integrable systems with anon-self-adjoint Lax operator, may be useful in the treatment of
Riemann-Hilbert problems arising in the analysis of generalcomplex or normal random matrices and the Hele-Shaw problem.
There is also a conjecture of Kuijlaars that some recent problems
where a vector equilibrium measure problem appears may bejustified by proving existence and regularity for a solution of a
maximin variational problem in two dimensions.

References
Spyridon Kamvissis, Kenneth D. T.-R. McLaughlin, Peter D.
Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear
Schrödinger Equation, Annals of Mathematics Study 154,
Princeton University Press, Princeton, NJ, 2003.
S. Kamvissis, E. A. Rakhmanov, Existence and Regularity for an
Energy Maximization Problem in Two Dimensions, Journal of
Mathematical Physics, v.46, n.8, 2005;
S.Kamvissis, Journal of Mathematical Physics, v.50, n.10, 2009.
S.Kamvissis, C.Papadimitropoulos, Existence and Regularity for an
Energy Maximization Problem in an infinite sheeted Riemannsurface, in preparation.