Recurrence and Resonance in the Cubic Klein-Gordon Equation

In a number of models for coupled oscillators and nonlinear wave equations primary resonances dominate the phase-space phenomena. A new feature is that in a Hamiltonian framework, the interaction of primary and higher order resonances is shown to be important and can be signaled by using recurrence properties. The interaction may involve embedded double resonance. We will demonstrate these phenomena for the cubic Klein-Gordon equation on a square with Dirichlet boundary conditions using normal form techniques. The results are qualitatively and quantitatively very different from the one-dimensional spatial case.


Introduction
Boundary value problems for nonlinear wave equation produce in a natural way problems with various resonances. We will consider these problems for a typical case, the cubic Klein-Gordon equation on a square. Galerkin projection and truncation will in this case lead to finite-dimensional Hamiltonian systems.
Consider the two-dimensional cubic Klein-Gordon equation as formulated in [9]: with smooth initial conditions u(x, y, 0) = Φ(x, y), u t (x, y, 0) = Θ(x, y), (2) and homogenous Dirichlet boundary conditions zero at the sides of the spatial domain. In a sense the subsequent analysis will be a continuation of [9] with attention to higher order B F. Verhulst f.verhulst@uu.nl 1 Mathematisch Instituut, Utrecht University, PO Box 80.010, 3508TA Utrecht, Netherlands resonance and qualitatively new phenomena. In [9] a rectangle is considered instead of a square producing different constant coefficients in Eq. (1). If we would repeat our analysis for a rectangle, the resonances will be different but the analysis runs along the same lines. After summarizing the approximation theory for nonlinear wave equations we will indicate the resonances in the case of the 2-dimensional cubic Klein-Gordon equation in Sect. 2. The 1:1 resonances are a basic feature of this problem, they are analyzed in various combinations. Recurrence or lack of it will signal the presence or absence of resonance zones that may complicate the dynamics. Detuned resonances will produce embedded double resonance, see Sect. 2.7. Section 3 outlines the asymptotics of detuning.

Approximation of Wave Equations
We will employ two approximation steps for the cubic Klein-Gordon equation: Galerkin truncation of the system and averaging the resulting finite-dimensional system; they produce an asymptotic estimate for the solution of the initial-boundary value problem.
Suppose that the eigenfunctions of the linearized equation (ε = 0) are φ kl (x, y), k, l = 1, 2, . . .. The solution of Eq. (1) can be written as: u(x, y, t) = ∞ k,l=1 u kl (t)φ kl (x, y). ( Substituting this Fourier series into Eq. (1) and taking inner products with the eigenfunction expansion (3) produces with corresponding expansion of the initial values an initial value problem for an infinite system of ODEs. The infinite system is equivalent to the original PDE problem. Suppose that Fourier analysis of the initial conditions (smooth functions of x, y) produces a finite series of N terms or N terms with a rest term that can be neglected. A natural Galerkin approximation u N of the solution of Eq. (1) is a truncation of the series (3) with slightly more than N terms. There are many publications using this method but most of them are concerned with formal approximations, see for an example and more references [7]. The analysis of Krol [6] gives the mathematical approximation theory of both the one-and more-dimensional cases. See also the papers of Bambusi [1,2] and Fečkan [4] for analysis in the same spirit. The cubic Klein-Gordon equation on a rectangle was studied by Galerkin-averaging in [9]. The resulting system of ODEs obtained for u N produces an approximationũ N . The proof by Pals [9] uses suitable Sobolev spaces and gives the error estimate in the sup norm. Explicitly: valid on an interval of time O(1/ε). In [9] interesting differences with the one-dimensional spatial case are pointed out; many more resonances may arise like detuned and double resonances.
The frequencies ω = (ω 1 , ω 2 , . . . , ω n ) are chosen positive; we can approximate them by rational numbers as the rationals are dense in the set of real numbers but its consequence is that we have to discuss detuned resonances. The Hamiltonian terms H j (p, q), j = 3, 4, . . . are homogeneous polynomials in p, q of degree j . We assume that at least two of the frequencies are close to a first or second order resonance, or frequency ratios 1:2, 1:1 or 1:3. We may have detuning effects to allow for small frequency perturbations. For an exhaustive list of first and second order resonances of three dof Hamiltonians see [12], Tables 10.3-10.4. In many applications a combination of low and higher order resonances takes place. To avoid this one usually concentrates on the low order resonances neglecting the higher order ones. The purpose of this paper is a more complete theory by exploring the cases of combined low and higher order resonance. As we shall see, the tools will be averaging-normal form theory and the use of the Poincaré recurrence theorem to characterize the dynamics in resonance zones.
A powerful theorem on the stability of Hamiltonian systems in the sense of exponentiallylong time invariance of the actions was formulated and proved by Nekhoroshev [8]. This theorem presupposes steepness of the Hamiltonian and so the absence of first or second order resonances in the system. We cannot use the theorem in our case.
In Eq. (1) a small parameter is present but for general H (p, q) it is convenient to scale the coordinates near the stable origin of the system by putting p, q → εp, εq and dividing by ε 2 . This leads to the Hamiltonian So ε 2 is a measure for the energy with respect to stable equilibrium at the origin. Often we introduce action-angle coordinates I, φ by the transformation: leading with (6) to We will also use amplitude-phase coordinates r, ψ with transformations q i ,q i → r i , ψ i : So, the new variables are r i (t), ψ i (t) where we have to exclude a neighborhood of r i = 0. We will introduce near-identity transformations producing normal forms; see [12] for theory and background literature. Prominent terms in the normal forms are produced by the resonances induced by the frequencies ω.
A two dof Hamiltonian system in first or second order resonance has an integrable normal form. The treatment of higher order resonance in [11] is quite general for two dof, for more than two dof the complexity of higher order resonance increases enormously. However, in the case of combined lower and higher order resonance we will, as in [16], consider for interactions the so-called resonance zones where the periodic solutions of the (primary) lower resonance are located.

Low and Higher Order Resonance
We consider the theory from [11] and extend it following [13], see also [12]. Consider first a two dof Hamiltonian system with frequencies k, l ∈ N near stable equilibrium. If k + l > 4 its Birkhoff-Gustavson normal form is in action-angle coordinates: where A, B, C, α are constants, the dots stand for Birkhoff normal form terms (dependent on I 1 , I 2 only), χ = lφ 1 − kφ 2 . The angle χ plays no part in the normal form to O(ε 2 ). Considered as an isolated two dof system, it can be shown that the actions of the corresponding modes I 1 , I 2 are constant to O(ε) on the timescale 1/ε 2 ; with some effort the error is reduced to O(ε 2 ) on the timescale 1/ε 2 . The combination angle χ may vary locally in a resonance zone as follows: The resonance manifold N embedded in the compact energy manifold E is defined by dχ/dt = 0 or: If Eq. (10) has no solution, the resonance manifold N does not exist; in this case the combination angle χ is timelike, we can average over χ . If N exists, small exchanges of energy will take place between the two dof in a resonance zone located in an O(ε (k+l−4)/2 ) neighborhood of N (this is an improved estimate based on [13]). The resonance zone contains stable and unstable periodic solutions, the exchange of energy takes place on tori in the resonance zone with timescale 1/ε −(k+l)/2 . In [13] it is also proved that for potential problems α = 0. Assuming now more than two dof and that the frequency spectrum contains also first and/or second order frequency ratios. The corresponding low order frequency modes will dominate the phase-flow of higher order resonance except in primary resonance zones where the low order actions do not vary; in these zones the low order short-periodic solutions are located.
In the case of many dof our strategy will be to locate the low order resonance zones (small neighborhoods of the resonance manifolds) and find out whether higher order resonance manifolds exist embedded in these zones; they will be called secondary resonance zones. This can be done analytically using normalization. The phenomenon will be called 'embedded double resonance', see [16]. For the general theory of double resonance see [3,5] and more references there.

The Recurrence Theorem
Consider a dynamical system defined on a compact set in R n with the property that the flow induced by the system is measure-preserving. Poincaré uses the term volume-preserving for the phase-flow induced by a time-independent Hamiltonian system without singularities on a compact domain, see [10], vol. 3, Chap. 26. Using the invariance of the volume of phase-elements under the flow, it is proved that most orbits return an infinite number of times arbitrarily close to their initial position; this is called recurrence. The recurrence time depends on the specific dynamical system considered, the initial condition chosen and the size of the neighborhood to be revisited. Consider for instance an initial point P 0 and a ball with radius d > 0 centered around P 0 . The recurrence theorem states that after a finite time T d an orbit starting in this ball will enter the ball again; there are exceptions for certain initial conditions but the exceptional initial conditions have measure zero.
It is easy to obtain an upper limit L for recurrence times, dependent on the Euclidean distance d(0) = d 0 to the initial condition. Consider time-independent Hamiltonian (6). Assume that H 2 (p, q) is Morse at (p, q) = (0, 0) and that the quadratic part is definite, so the origin is a stable equilibrium of the equations of motion. In [15] it is argued that: To illustrate recurrence we present in Fig. 1 the Euclidean distance to an initial state for the Hénon-Heiles system that has been shown to be non-integrable and an integrable system, both for 2 dof; see for a detailed analysis [14]. The Hamiltonian is: Doubling the integration time for the Hénon-Heiles system, a = −1, produces a similar picture. In the integrable case a = 1 we have periodic solutions and tori foliating the energy manifold; in this case the recurrence is called regular. We might also call recurrence regular if we have a non-integrable system with complex regions that are relatively small (a precise definition of "regular" is difficult as there are so many different cases).
In the sequel we have often n = 3, d 0 = 0.1 producing L = 10 5 , if n = 4, d 0 = 0.1, L = 10 7 . The actual Poincaré recurrence times are lower than L but passage of resonance zones can delay recurrence as the orbits will wind around the tori embedded in the resonance zones.
A preliminary test for embedded double resonance can be carried out using the recurrence theorem. Constructing numerical solutions of orbits passing the resonance zones, we expect fairly regular recurrent behavior if these zones contain no or very small resonance manifolds. Complicated and long time recurrent behavior points at motion around tori and other invariant manifolds during passage. In [16] the 1:1:4 resonance and the Fermi-Pasta-Ulam α-chain were discussed as examples. Recurrence will be tested by computing the Euclidean distance d(t) to the initial conditions as a function of time.

The Cubic Klein-Gordon Equation
In the sequel we will consider problems derived from the cubic Klein-Gordon equation (1). The eigenfunctions of the linearized equation on a square are φ kl (x, y) = sin kx sin ly, k, l = 1, 2, . . . .

Asymptotic Approximations
The eigenfunction expansion (3) that satisfies the boundary conditions is: The eigenfunctions sin kx sin ly of the linearized Eq. (1) correspond with the eigenvalues Fourier expansion turns the partial differential equation into an equivalent system of an infinite number of coupled ordinary differential equations of the form with f kl (u) cubic in u kl , k, l = 1, 2, . . .. As announced in the Introduction we will employ two approximation steps: Galerkin truncation of the system and averaging the resulting finite-dimensional system. The asymptotic approximation will have a validity on an interval of time of O(ω kl /ε); as we shall see, in practice this estimate is often too pessimistic. Part of our interest will be on the interaction of the resonant and non-resonant part of the spectrum. The analysis in [9] is quite general, here we illustrate the resonances for the 69 modes with ω 2 kl ≤ 100. Note that all cases with k = l produce a 1:1 resonance; the 1:1 resonances turn out to be basic for this problem. We find in addition: The 3:7 resonance turns out to have no resonance manifold in a Galerkin projection as the corresponding combination angle is timelike, so we leave out this case. Note also that the larger k + l is, the smaller the resonance zones become and the larger the interaction timescales are, see Sect. 1.3.
We will start with a sketch of the basic 1:1 resonances; after this we study the dynamics and weak interactions with other resonances. Our focus will be on the interesting case where recurrence highlights more complicated dynamics involving higher order resonance.

The Basic 1:1 Resonances
Restricting to two modes u kl , u lk , k = l we find an infinite number of 1:1 resonances. Galerkin projection produces with ω 2 kl = ω 2 lk = ω 2 the system: As the size of the parameter ω is still free it is natural to rescale time t → τ = ωt and call the rescaled time again t . The system becomes: The dynamics of this system was analyzed in [14] and [9]; it is characterized by an integrable normal form, two unstable normal modes and two resonance zones with periodic solutions. We add some new elements. We exclude the normal modes in the next approximation procedure as we use polar coordinates. Putting, u kl = r 1 cos(t + ψ 1 ), u lk = r 2 cos(t + ψ 2 ), the equations from first order averaging-normalization become (see Sect. 3): with χ 1 = ψ 1 − ψ 2 . We find: An integral of the normal form (17) is with constant E 1 ≥ 0; so the normal form is integrable. Note (again) that increasing k, l and so ω = ω kl , we increase the timescale of validity that characterizes the dynamics. Phaselocked periodic solutions are found in the resonance manifolds and are determined by: Because of the symmetry of the equations we can obtain exact solutions. The periodic solutions correspond with the solutions u kl (t) = u lk (t), u kl (t) = −u lk (t). They satisfy the equation:ü Substitution in (1) produces a Galerkin two-mode projection of periodic solutions. An additional feature of the Galerkin projection for the cubic Klein-Gordon equation is that there exist other phase-locked solutions if cos 2χ 1 = 1/4. Substituting this value into system (17) we find heteroclinic invariant manifolds connecting the unstable normal modes. Both the periodic solutions in the resonance zones and the heteroclinic solutions will be a returning feature in what follows.
We can introduce the variable angular momentum J (t) for the u kl , u lk interaction: For J we find from system (16) the equation We conclude that in the resonance manifolds, where u 2 kl = u 2 lk , the angular momentum of the u kl , u lk interaction is conserved. The resonance zones correspond with critical points of the angular momentum equation (22).

Combining Independent 1:1 Resonances
Suppose we have a Galerkin truncation with a finite number M of basic 1:1 resonances k i , l i , k i = l i for certain indices k i , l i . We exclude the special resonance cases as presented in Sect. With this assumption the M basic 1:1 resonances will be independent of each other. The resulting approximation will be a superposition of the individual basic resonances.
Note that for values of ω such that 1/ω 2 ≤ ε the contribution of such a 1:1 resonance will be of order ε 2 .

The First Three Modes
Apart from the basic 1:1 resonances it is natural to consider the first three modes. The eigenvalues are ω 2 11 = 3, ω 2 12 = ω 2 21 = 6. With the corresponding u 11 (t), u 12 (t), u 21 (t) we propose the three-terms Galerkin truncation: Substituting this truncation into Eq. (1) and taking inner products with the eigenfunctions we find the 1:1 resonance imbedded in a three dof system. Replacing √ 6t by t we find: The three normal modes are solutions of system (24). Putting u 11 (t) = 0, the system has a two dof (4-dimensional) invariant manifold, the dynamics of which was analyzed before (Sect. 2.2). It is simple to repeat part of the calculation to assess the role of u 11 ; the frequency ratios are 1 2 √ 2:1:1 so the first mode is not close to resonance with the other two modes. Excluding the normal modes, averaging-normalization for amplitudes and phases r, ψ from The solutions of system (25) are O(ε) approximations of the amplitudes and phases from system (24) valid on the timescale 1/ε; note that ε plays here the part of ε 2 in expansion (6). It was proved in [9] that using these approximations in the Galerkin expansion (13) with corresponding initial conditions produces an asymptotic approximation as ε → 0 of Eq. (1) on the same timescale 1/ε. System (25) has the integrals r 2 2 + r 2 3 = 2E 1 and r 1 (t) = r 1 (0). For the combination angle χ 1 we have: Apart from the normal modes of system (24), four families of iso-energetic quasi-periodic solutions parametrized by r 1 (0) arise of system (25) if sin 2χ 1 = 0, r 2 = r 3 . The two periods of each family depend on the initial conditions i.e. the energy level. They correspond with approximate quasi-periodic standing waves of Eq. (1). As in the case of Sect. 2.2 there exist other phase-locked solutions if cos 2χ 1 = 1/4. Substituting this value into system (25) we find heteroclinic invariant manifolds connecting the u 2 , u 3 normal modes. For instance for r 2 from the equatioṅ with cos 2χ 1 = 1/4, c = sin 2χ 1 .
The resonance zones M on the energy manifold are neighborhoods of the quasi-periodic solutions given by sin 2χ 1 = 0, r 2 = r 3 corresponding with sin 2χ 1 = 0, u 12 = u 21 . Consider as an example the zone r 2 2 = r 2 3 = E 1 , χ 1 = 0 and r 1 (0) to be chosen. Passage of the resonance zone is shown in Fig. 2; I 1 (t) shows variations of order 0.01. The dynamics in the resonance zones shows regular quasi-periodic behavior near the periodic solutions as follows from our analysis, there is hardly any exchange of energy with mode u 11 ; the numerics shows that details of recurrence are slightly affected by the choice of u 11 (0). Choosing as recurrence criterion d 0 = 0.1, the recurrence times are nearly 8000 timesteps, the interval of time to let d approach zero again and again.
The equations for the phases in M are: It turns out that the variations of the three actions when starting in M are of size 0.01 on an interval of 40000 timesteps. As the frequency ratio ω 11 : ω 12 is close to 7:10, one could look for the 7:10 higher order resonance in the resonance zones. As we find from Eq. (27) that the combination angle (10ψ 1 − 7ψ 2 ) is timelike, this higher order resonance does not arise.
We can introduce the variable angular momentum J (t) for the u 12 , u 21 interaction by (21).
For dJ /dt we find from system (24) again Eq. (22). It is remarkable that the equation for J does not depend on u 11 , we draw conclusions similar to those in Sect. 2.2. In the resonance zones, where u 2 12 = u 2 21 , angular momentum of the u 12 , u 21 interaction is conserved. The resonance zones correspond with critical points of the angular momentum equation (22). In addition we can derive from system (24) the equation for u 11 (t) in M: In M we expect u 11 (t) to vary very little.

Coupled 1:1 Resonances, the 1:1:1:1 Resonance
Consider now a truncation to 4 modes generated by k 1 l 1 , k 2 l 2 (k 1 = l 1 , k 2 = l 2 ) with ω k 1 l 1 = ω k 2 l 2 = ω. In Sect. 2.1 we gave examples with ω 2 = 66 and 86. Substituting the corresponding 4-mode truncation into Eq. (1) and taking inner products with the eigenfunctions we obtain 4 coupled equations of motion. We divide the equations by ω 2 , replacing ωt by t and abbreviate u k 1 l 1 = u 1 , u l 1 k 1 = u 2 , u k 2 l 2 = u 3 , u l 2 k 2 = u 4 ; we find the system: Note that 3/(4ω 2 ) is a small number, ε need not be very small. The four normal modes are solutions of system (29); we will show that they are unstable. The system (32) is very symmetric and we can find other exact solutions and invariant manifolds, for instance the 8 periodic solutions given by: with for instance in M 123 , u = u 1 = u 2 = u 3 . Averaging- The equations for the angles contain the combination angles (ψ 1 − ψ 2 ), (ψ 1 − ψ 3 ) etc. but we do not need them as in this case we have explicit expressions for the periodic solutions. An integral of the normal form system (35) is: with constant E 1 ≥ 0. Consider an orbit starting near an unstable normal mode. We expect transitions through several resonance zones. Assume a fixed, positive value E 1 in Eq. (36). For instance starting near the normal mode plane of the 1st and 3rd mode we have that r 1 (0) 2 + r 3 (0) 2 will be close to 2E 1 , the other initial conditions are small. The instability will move the orbit on the 7-sphere described by the integral (36) away from the u 1 , u 3 normal mode plane. The first resonance zone to encounter will be given by r 2 i = E 1 /2, i = 1, . . . , 4, the second one are possibly 4 resonance zones with r 2 i = 2E 1 /3, then follow 2 dof resonance zones with r 2 i = E 1 after which the recurrence can start. See for illustration Figs. 3 and 4.
We find again the integral r 2 2 + r 2 3 = 2E 1 , E 1 ≥ 0 and similar to the case of the first three modes: Putting cos χ 1 = 1/4 we find stable and unstable invariant manifolds, in fact heteroclinics, of the u 57 , u 75 normal modes given bẏ In the primary resonance zones M 1 , M 2 where by r 2 = r 3 , sin 2χ 1 = 0 we have for the periodic solutions: In Fig. 5 we have chosen the initial conditions near the unstable u 75 normal mode. This causes repeated passage through the primary resonance zones. In Fig. 5 we have left the case of nearly pure 1:1 resonance (choosing u 33 = 0 would eliminate the mode u 33 completely).
The middle and right figure shows the influence of secondary resonances (embedded double resonance) that complicate passage through the zones.
The constants a 1 , . . . , a 21 are positive. System (44) has the integral It is clear from system (44) that the 1:1 resonance of the 2nd and 3rd modes dominates the flow outside the resonance zones M 1 , M 2 and outside the invariant manifolds M 12 , M 13 .
For M 13 we have analogous results. If sin χ 2 = 0 and remains zero in time, the amplitudes r 1 , r 2 are constant in time. We have in M 12 : if δ ≤ 0 there are no such solutions. Assume δ > 0; this produces from the necessary condition a relation between δ, ε, r 2 and r 2 . The second condition is thatχ 2 = 0 at next order. This produces with cos χ 2 = ±1 another equation between the same quantities. If, together with the energy integral, we can solve these 3 equations, we have determined a resonance manifold in M 12 where a 4:2 resonant periodic solution between the first two modes in M 12 can be found. For a choice of parameters such a solution is shown in Fig. 6 (left).

Normalization in a General Position Resonance Zone
In the primary resonance zones we have obtained to first order r 2 2 = r 2 3 = E 1 , sin2χ 1 = 0. For the amplitudes we find to second order from system (44) with sin χ 2 + sin χ 3 = 2 sin χ 4 cos 2χ 1 , with the dots representing higher order terms. We consider the combination angle χ 4 characterizing a possible resonance manifold in general position (all modes non-zero). In a domain in a primary resonance zone where sin χ 4 = 0 andχ 4 = 0 we expect to find a 1:2:2 periodic solution. Note that it suffices to require sin χ 2 = sin χ 3 = 0. We find from system (44) to first order in 2 resonance zones: In the case of Fig. 5 and putting ε = 0.5 in (39) as the coefficients are very small we have from (49) zeros of the righthand side producing secondary resonances in the primary resonance zones. We find cos 2χ 1 = 1, r 1 (0) = 0.87 and cos 2χ 1 = −1, r 1 (0) = 0.78. To show more evidence for the existence of secondary resonance and the presence of tori that influence the recurrence as in fig, 5, consider Fig. 6. The theory of higher order resonance predicts two stable 1:2:2 periodic solutions and two unstable ones. In Fig. 6  Again, the critical points of the angular momentum equation correspond with the resonance zones.
Averaging over the common period T we find for i = 1, . . . , n, r = r 1 , . . . , r n , ψ = ψ 1 , . . . , ψ n :ṙ We can apply near-identity transformation (51) to system (55) to obtain a second order approximation; the resulting system to solve will be of the form (52) with terms added of size δ i and δ 2 i . Applying higher order normalization to a detuned resonance like system (38) we find at O(ε 2 ) terms of the form r 3 1 r 2 2 sin(4ψ 1 − 2ψ 2 ) and similar terms involving r 3 . These terms introduce the 4:2 resonances with corresponding periodic solutions discussed in the preceding section.

Conclusions
1. In mathematical physics PDEs have often been analyzed in the case of one space dimension. We have shown that when allowing more space dimensions, the results may change remarkably. We discuss a typical case of mathematical physics, the cubic Klein-Gordon equation. 2. The validity of asymptotic approximations holds on intervals of time proportional to 1/ε and ω kl . Our analysis yields some conclusions using the truncation procedure of series (3). First, if ω kl is large enough the tail of the series will take an extremely long time to become effective. Secondly, the modes in 1:1 resonance are ubiquitous but combination of independent 1:1 systems produces no new phenomena. Thirdly, interesting phenomena like embedded double resonance arise from detuning with new resonant interactions. There will exist an infinite number of such detuned systems, but they arise for large values of ω kl . For the complicated dynamics described in the preceding sections to be observed one has to choose the corresponding initial conditions producing the resonant modes. Exciting for instance only one mode, there will be nontrivial evolution if this mode is unstable in a resonant setting and if it is slightly perturbed. 3. The most interesting dynamics is described for the 1:1:1:1 resonance in Sect. 2.6 and the detuned resonance in Sect. 2.7. The resonance zones vanish (have size o(1)) as ε → 0 and so are free boundary layers in the sense of singular perturbation theory. In these zones the resonances produce locally stable and unstable periodic solutions with corresponding stable and unstable manifolds. Intersection of invariant manifolds associated with the periodic solutions in the resonance zones are in Hamiltonian mechanics the main source of chaos. For small values of ε this will enable the possibility of boundary layer chaos in the cubic Klein-Gordon equation, it may be more prominent if ε increases. 4. Changing the boundary conditions or the shape of the domain will of course change our results. However, the set-up of our analysis is typical for such new problems. Symmetries, for instance considering circle or ring domains, may simplify the analysis. New phenomena may arrive when studying non-convex domains.