# An overdetermined problem in potential theory

###### Abstract.

We investigate a problem posed by L. Hauswirth, F. Hélein, and F. Pacard [HHP2011], namely, to characterize all the domains in the plane that admit a “roof function”, i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. As they suggested, we show, under some a priori assumptions, that there are only three exceptional domains: the exterior of a disk, a halfplane, and a nontrivial example found in [HHP2011] that is the image of the strip under . We show that in this example does not have any axially symmetric analog containing its own axis of symmetry.

###### Key words and phrases:

roof function, exceptional domain, null quadrature domains, Schwarz function, free boundary###### 2010 Mathematics Subject Classification:

Primary: 35N25, 35R35, 31A25, Secondary: 30E25, 30C20## 1. Introduction

In [HHP2011], the authors have posed the following problem:
find a smooth bounded domain in a Riemannian manifold with metric ,
such that the first eigenvalue of the Laplace-Beltrami operator on has a corresponding real,
positive eigenfunction satisfying on the boundary of .
Any such domain is called *extremal* because it provides a local minimum for the first eigenvalue of the Laplace-Beltrami operator,
under the constraint of fixed total volume of (see [HHP2011] and references therein).

In special cases one can find a sequence of extremal domains with increasing volumes,
such that the limit domain is unbounded, and its first eigenvalue vanishes as .
This limit extremal domain is then called *exceptional*, and the corresponding limit function is a positive,
harmonic function on which solves simultaneously the overdetermined boundary value problem with null Dirichlet data and constant Neumann data.

The problem of finding exceptional domains in and their corresponding functions (called “roof” functions by the authors of [HHP2011]) is a nontrivial problem of potential theory. There is no obvious variational principle to use, on the one hand because is unbounded (so the Dirichlet energy of [Astala, Ch. 1] will diverge), and, on the other hand, because the constant Neumann data constraint is not conformally invariant.

In the absence of a suitable variational formulation, we may interpret the scaling described above as a dynamical process, in which the pair evolves so that the limit solves the overdetermined problem. In other words, we can turn this observation into a constructive method for finding (building) exceptional domains. In order to do this, it is helpful to note that, upon compactification of the boundary (with metric ), the pair with flat metric becomes conformal to the half-cylinder , with metric

Under this reformulation, scaling of becomes equivalent to scaling of the metric structure given above, defined over the fixed space . This is reminiscent of the Ricci flow, in which the metric structure evolves with respect to a deformation parameter according to the equation

with the right side of the equation given by the covariant Ricci tensor. It is known [Top] that for the case of a two-dimensional manifold, with metric given by , the Ricci flow equations reduce to a single nonlinear equation

(since in two dimensions the Riemann tensor has only one independent component). This is a heat equation with the generator given by the Laplace-Beltrami operator corresponding to the metric . Therefore, if there is a stationary solution as , it will correspond to the scaling of the first eigenvalue and, by conformally mapping back using the solution , we will obtain the solution .

In other words, we can summarize this constructive method for finding exceptional domains in as follows: starting from a 2-dimensional Riemannian manifold with finite volume and metric encoded through the positive real function , and boundary set defined via , consider the time evolution given by the Ricci flow, without volume renormalization. Then [Top] the manifold will remain Riemannian at all times, and in the limit the function will become a solution of the nonlinear Laplace-Beltrami equation. Furthermore, if remains finite everywhere in the domain, then it is harmonic and satisfies both Dirichlet and Neumann conditions at all finite boundary components, so it is a solution for the overdetermined potential problem. Considered together with the (boundary) point at infinity, the manifold is equivalent [Poincare] to a pseudosphere (flat everywhere except at the infinity point, with overall positive curvature). (We wish to emphasize that there is no reason to assume that such constructive methods would be exhaustive.)

Thus, so motivated, it is natural to try to characterize exceptional domains in flat Euclidean spaces. The authors in [HHP2011] suggested that in two dimensions there are only three examples: a complement of a disk, a halfplane, and a nontrivial example obtained as the image of the strip under the mapping . They posed as an open problem to determine if these are the only examples [HHP2011, Section 7]. (They gave some evidence by characterizing the halfplane under a global assumption on the gradient of the roof function [HHP2011, Prop. 6.1].) They also posed the problem of finding nontrivial examples in higher dimensions and suggested the possibility of axially symmetric examples similar to the nontrivial example in the plane [HHP2011, Remark 2.1].

We address both of these problems.
The paper is organized as follows. In Section 2, we review the theory of Hardy spaces in order to address a subtlety that arises
in connection with the regularity of the boundary of an exceptional domain.
This leads us to assume in our theorems that the domain is Smirnov.
In Section 3, we characterize exteriors of disks as being the only exceptional domain whose boundary is compact.
In Section 4, we establish a connection between the ”roof function” of an exceptional domain and the so-called *Schwarz function* of its boundary,
and we also show that the boundary of a simply connected exceptional domain can pass either (i) once or (ii) twice through infinity.
In Section 5, we show that Case (i) implies that is a halfplane.
In Section 6, we show that Case (ii) implies that is the nontrivial example found in [HHP2011, Section 2].
In each of these theorems we assume that is Smirnov, but we allow the roof function to be a weak solution
merely satisfying the boundary conditions almost everywhere.

In Section 7, we extend the result of Section 3 to higher dimensions. In Section 8, we show that the nontrivial example from Section LABEL:sec:nontrivial does not allow an extension to axially symmetric domains in four dimensions, contrary to what was suggested in [HHP2011, Remark 2.1] (and we conjecture that this example has no analogues in any number of dimensions greater than two).

Sections 3 through 6 together partially confirm what was suggested in [HHP2011, Section 7]
under some assumptions on the topology of and assuming that is Smirnov.
In Section 9, we give concluding remarks including a conjecture that, up to similarity,
there are only three *finite genus* exceptional domains.
The additional assumption of finite genus is due to a remarkable example of an infinite-genus exceptional domains that appeared in the fluid dynamics literature
[BSS76].
See Section 9 for discussion.

###### Remark 1.

After this paper was submitted, Martin Traizet announced a more complete classification of exceptional domains [MT2013] after developing an exciting new connection to minimal surfaces. He characterized the three examples among those having finitely many boundary components. M. Traizet’s preprint that appeared while we have been revising the previous version of our paper finds a new beautiful connection of the problem with the theory of minimal surfaces. From this point of view, he noticed the above-mentioned family of infinite genus examples [BSS76] and also characterized them among periodic domains for which the quotient by the period has finitely many boundary components [MT2013, Theorem 13]. For this latter result, he invokes a powerful theorem of W. H. Meeks and M. Wolf. Our methods mostly rely on classical function theory ( spaces) and potential theory and in most parts are different from Traizet’s. Interestingly, as Traizet notes in his preprint [MT2013, Remark 5], if one could prove his Theorem 13 by only invoking pure function theory this would give (via Traizet’s results) a new and independent proof of the Meeks-Wolf result from minimal surfaces. An attractive challenge!

Acknowledgement: The authors are indebted to Dimiter Vassilev for bringing the article [HHP2011] to their attention. We wish to thank Alexandre Eremenko for sharing an improved proof of Theorem LABEL:thm:Martin and Arshak Petrosyan and Koushik Ramachandran for pointing out the example of a cone as an exceptional domain. We also wish to thank Martin Traizet for helpful discussion regarding his preprint. The two first named authors acknowledge partial support from the NSF under the grant DMS-0855597.

## 2. Classical vs. Weak Solutions, Regularity of the Boundary, and Hardy Spaces

From the rigidity of the Cauchy problem, one might expect to obtain, “for free”, regularity of the boundary of an exceptional domain (as is often the case for solutions of free boundary problems). Unfortunately, the problem at hand is complicated by a remarkable family of examples with rectifiable but non-smooth boundaries, a.k.a. non-Smirnov domains - cf. [Duren, Ch. 10]. This results in adding a Smirnov condition to the assumptions on the domains if we desire to consider ”weak solutions”, i.e., harmonic ”roof functions” satisfying the Dirichlet and Neumann boundary conditions almost everywhere with respect to the Lebesgue measure.

In order to address this subtlety, we first give some background from theory - cf. [Duren] for details.

An analytic function is said to belong to the Hardy class , , if the integrals:

remain bounded as .

Recall that a *Blaschke product* is a function of the form

where is a nonnegative integer and . The latter condition ensures convergence of the product (See Theorem 2.4 in [Duren]).

A function analytic in is called an *inner function* if its modulus is bounded by and its modulus has radial limit almost everywhere on the boundary.
If is an inner function without zeros, then is called a *singular inner function*.

An *outer function* for the class is a function of the form

(2.1) |

where is a real number, , , and .

The following theorem [Duren, Ch. 2, Ch. 5] (also cf. [fisher]) provides the parametrization of functions in Hardy classes by their zero sets, associated singular measures, and moduli of their boundary values.

###### Theorem 2.1.

Every function of class () has a unique (up to a unimodular constant factor) factorization of the form , where is a Blaschke product, is a singular inner function, and is an outer function for the class .

Suppose is a Jordan domain with rectifiable boundary and is a conformal map.
Then by Theorem 3.12 in [Duren].
By Theorem 2.1, has a canonical factorization ,
and since is a conformal map does not vanish, so .
Then is called a *Smirnov domain* if so that is purely an outer function.
This definition is independent of the choice of conformal map.

There are examples of non-Smirnov domains with, as above, , but now and the singular inner function is not constant. Such examples were first constructed by M. Keldysh and M. Lavrentiev [KeldLavr] using complicated geometric arguments. Their existence was somewhat demystified by an analytic proof provided by P. Duren, H. S. Shapiro, and A. L. Shields [DSS]. Like the disk, such a domain has harmonic measure at zero (assuming ) proportional to arc-length. Thus, its boundary is sometimes called a “pseudocircle”.

Similarly, there are “exterior pseudocircles”,
arising as the boundary of an unbounded non-Smirnov domain [JonesSmirnov] for
which the harmonic measure at infinity is proportional to arclength,
and thus Green’s function with singularity at infinity
provides a roof function that is a weak solution satisfying the boundary conditions almost everywhere.
Thus, this provides a pathological example of an exceptional domain in a weak sense.
In order to construct such an unbounded non-Smirnov domain, let us follow the method in the above mentioned [DSS],
which is presented in Duren’s book [Duren, Section 10.4].
We recall that the construction is carried out by “working backwards”, first writing down a singular inner function
as a candidate for the derivative of the conformal map .
The difficulty is then to show that is not only analytic, but is also *univalent* so that it actually gives a conformal map from to some domain .
Univalence is established using a criterion of Nehari which states that the following growth condition on the Schwarzian derivative
is sufficient for univalence:

(2.2) |

Let us follow this procedure, indicating the step that needs to be modified. Start with a measure , singular with respect to Lebesgue measure on the circle, yet sufficiently smooth, so that it belongs to the Zygmund class (cf. [Duren, Section 10.4]).

We will also require to have the first moment zero, i.e.,

(2.3) |

This can always be acheived by symmetrizing around the origin and replacing by . Then the center of mass is at the origin, which is (2.3).

As in [Duren], let be the Schwarz integral of

Let be the exponential of a constant (to be chosen later) times ,

Here is where we depart slightly from [Duren] in order to get an unbounded domain as the image of . Instead of taking as a candidate for , we take

Note that the residue of is zero (from having made the first moment of zero) so that its antiderivative is analytic in except for a simple pole at . Also, a.e. on .

A calculation shows that the Schwarzian derivative of is:

As explicitly stated in [Duren, Section 10.4], , , and are each

Moreover, by the vanishing of the first moment of , , so that is also . Thus, for a small enough choice of , satisfies the Nehari criterion for univalence (2.2).

Hence, is a conformal map mapping onto the complement of a Jordan domain with rectifiable boundary. To see why the boundary is rectifiabile, note that, as stated in [Duren, Section 10.4]), , and so is in in an annulus .

This seemingly excessive construction of an exterior pseudocircle cannot be avoided by
simply taking an inversion of an interior pseudocircle; the result will be non-Smirnov, but it will *not* be an exterior pseudocircle.
Nor can one simply take the complement.
As P. Jones and S. Smirnov proved in [JonesSmirnov], the complement of a non-Smirnov domain is often Smirnov!
(This unexpected resolution of a long standing problem
put to rest all hopes to characterize the Smirnov property in terms of a boundary curve.)