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Research article Special Issues

The infinity-Laplacian in smooth convex domains and in a square

  • We extend some theorems for the infinity-ground state and for the infinity-potential, known for convex polygons, to other domains in the plane, by applying Alexandroff's method to the curved boundary. A recent explicit solution disproves a conjecture.

    Citation: Karl K. Brustad, Erik Lindgren, Peter Lindqvist. The infinity-Laplacian in smooth convex domains and in a square[J]. Mathematics in Engineering, 2023, 5(4): 1-16. doi: 10.3934/mine.2023080

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  • We extend some theorems for the infinity-ground state and for the infinity-potential, known for convex polygons, to other domains in the plane, by applying Alexandroff's method to the curved boundary. A recent explicit solution disproves a conjecture.



    Dedicated to Giuseppe Mingione on his fiftieth birthday.

    The - Laplace operator

    Δui,juxiuxj2uxixj

    is formally the limit of the p - Laplace operator

    Δpu(|u|p2u)asp.

    The two-dimensional equation

    (ux1)22ux21+2ux1ux22ux1x2+(ux2)22ux22=0

    was introduced by G. Aronsson in 1967 as a tool to provide optimal Lipschitz extensions, cf. [1,2]. It has been intensively studied ever since; some highlights are

    ● Viscosity solutions for Δ were introduced by T. Bhattacharya, E. DiBenedetto, and J. Manfredi in [3].

    ● R. Jensen proved uniqueness in [12].

    ● Differentiability was proved by O. Savin and L. Evans. See [8,24].

    ● The connexion with stochastic game theory ("Tug-of-War") was discovered by Y. Peres, O. Schramm, S. Sheffield, and D. Wilson, cf. [23].

    By examples we shall shed some light on two problems in the plane. The problems are related but not identical. In convex domains so similar methods often work for both problems that it is optimal to treat them simultaneously. —We remark that the main difficulty is the lack of second derivatives. The solutions are to be interpreted as viscositysolutions. The reader may consult [7,17].

    The first one is the boundary value problem

    {Δu=0inGu=0onΩu=1onK (1.1)

    in a convex ring G=ΩK, where Ω is a bounded convex domain in R2 and KΩ is a closed convex set. The unique solution, say u, belongs to C(¯G) and always takes the boundary values; K is often only an isolated point. We say that u is the -potential. In [18] the term "capacitary function" is used.

    The second object is the - Eigenvalue problem

    {max{Λ|v|v,Δv}=0inΩvC(¯Ω),v|Ω=0,v>0. (1.2)

    Solutions are called - Ground States. Problem (1.2) is the asymptotic limit as p of the equation

    (|vp|p2vp)+λp|vp|p2vp=0, (1.3)

    where vpW1,p0(Ω),vp>0. Problem (1.2) has a solution if and only if

    Λ=limppλp,

    but uniqueness ( = simplicity of Λ) is not known to hold even in convex domains.*

    *In more general domains this fails, cf. [10].

    We shall restrict ourselves to those - Ground States that come as limits of sequences of solutions to (1.3). Such a limit v has the advantage that logv is concave. This valuable property is the reason for why we prefer these so-called variational - Ground States. See [13,28].

    The main achievement of this paper is to complement our study in [20,21]. There the crucial assumption that |u| (or |v|) has only a finite number of maxima and minima on the boundary Ω was properly verified merely for convex polygons. Smooth domains were out of reach. Our contribution now is to provide a class of explicit smooth domains having the desired property: for example, the ellipse is included. We are grateful to B. Kawohl, who informed us about [15] and suggested Alexandroff's moving plane method.

    We think that every bounded convex domain with C3 - boundary will do.

    Despite sharing similar properties, solutions of (1.1) and (1.2) may not coincide even under the (necessary) condition that K be chosen as the High Ridge

    K={xΩ|dist(x,Ω)=R},R=maxxΩdist(x,Ω). (1.4)

    (Here R is the radius of the largest inscribed ball in Ω.) Nevertheless, it is shown in Theorem 3.3 in [28], that in a certain class of domains, which includes the stadium–like domains, the distance function is the unique solution to both (1.1) and (1.2). In general, coincidence is a difficult problem.

    We conclude the work by noticing a recent result in a punctured square: the ring domain is a square with its center removed. It has been predicted that the -Potential would coincide with the - Ground State, cf. [14]. Brustad's explicit formula in [5] reveals that the functions do not coincide. However, the maximal difference between the functions is <103 for a square of area 4, according to numerical calculations in [4].

    We use standard notation. Here ΩR2 will always denote a bounded convex domain with smooth boundary, say at least of class C2,α. We shall denote the - Ground State by v, and the solution of (1.3) by vp. Analogously, the solution of (1.1) is u, and up will denote the solution of problem (3.1), see Section 3.1. Thus

    v=limpvp,u=limpup,

    perhaps via a subsequence. We will also use the normalization

    maxxΩv=1. (2.1)

    According to Theorem 2.4 in [28], the High Ridge (see (1.4)) is also the set where v attains its maximum 1.

    The contact set The contact set Υ plays a central role. To define it, following Y. Yu we use the operator

    S(x)=limr0{minyB(x,r)v(y)v(x)r}.

    According to Theorem 3.6 in [28], S is continuous in Ω and at points of differentiability S(x)=|v(x)|. The contact set

    Υ={xΩ|S(x)=Λv(x)}

    is closed and has zero area, see Corollary 3.7 in [28]. In addition, by Corollary 7 in [21], Υ does not reach Ω. The set where v is not differentiable is contained in Υ, cf. Lemma 3.5 in [28]. In the open set ΩΥ, v is C1 and Δv=0, by Theorem 3.1 in [28].

    Streamlines For the benefit of the reader, we describe the role of the streamlines. All this can be found in [20,21]. We begin with the -potential u. From each boundary point ξΩ a unique streamline α=α(t) starts and reaches K in finite time:

    dα(t)dt=+u(α(t)),α(0)=ξ;0tT.

    It may meet and join other streamlines, but streamlines do not cross. Suppose now that

    We do not know of any convex plane domain for which this is not valid!

    AlongΩthespeed|u(ξ)|hasonlyafinitenumberoflocalminima.

    More exactly we allow a finite number of strict local minimum points and a finite number of boundary arcs along which strict minima are obtained. It is problematic to deduce this from the shape of the domain. Polygons were treated in [20], and for a family of smooth domains the above assumption will be verified in Section 5. — This is our achievement in the present work.

    The streamline starting at a strict local minimum point is called an attracting streamline. If a local minimum is attained along a whole closed boundary arc (there |u| is constant), then this minimum produces two attracting streamlines: the two streamlines emerging from the endpoints of the arc. The attracting streamlines are special, indeed. We cite the main theorem from [20].

    Theorem 1. Let α be a streamline of u that is not an attracting one. Then it cannot meet any other streamline before it either meets (and joins) an attracting streamline or reaches K. The speed |u(α(t))| is constant along α until it joins an attracting streamline, after which the speed is non-decreasing.

    The corresponding theorem for the streamlines of v is similar; we only have to replace K by the High Ridge of Ω; see (1.4). The High Ridge is also the set of points at which v attains its maximum. Suppose that γ1,γ2,,γN are the attracting streamlines. It is spectacular that outside the closed set

    Γ=γ1γ2γN

    the - Ground State is - harmonic. This explains how the two problems are connected! (The corresponding statement is false for a finite p.) Indeed, from [21] we have:

    Theorem 2. Let β be a streamline of v that is not an attracting one. Then it cannot meet any other streamline before it either meets (and joins) an attracting streamline or reaches the High Ridge. The speed |v(β(t))| is constant along β until it joins an attracting streamline.

    In the open set ΩΓ the - Ground State satisfies the -Laplace Equation Δv=0.

    We mention that area(Γ)=0.

    For our purposes, we need the convergence of the modulus of the gradient vp up to the boundary of Ω, as p. A similar result is needed for the p-Potential. We split the proof for the two problems (1.1) and (1.2).

    We shall use the p-harmonic approximation upu where

    {Δpup=0inG=ΩKup=0onΩup=1onK. (3.1)

    As usual, Δpu=(|u|p2u). For p>2 (in two dimensions) it is known that upC(¯G) and that it takes the correct boundary values at each point. (This valuable property holds in arbitrary domains, be they convex or not.) We recall the following results of J. Lewis in [18]; see also [11]:

    upu in ¯G.

    up0 in G.

    up is real-analytic in G.

    up has convex level curves.

    up is superharmonic in G.

    We shall need continuous second derivatives on the boundary Ω. If Ω is of class C2, then |up|cp>0 in ΩK according to Lemma 2 in [18]. It is known that upC1(¯ΩK). By classical theory for the equation

    i,j(δij|up|2+(p2)upxiupxj)2wxixj=0

    with "frozen coefficients" we can conclude, using the Calderon-Zygmund theory, that the particular solution w=upC2,α(¯ΩK) provided that Ω is of class C2,α. See Theorem 6.14 in [9]. This is sufficient for our purpose.

    When p= we have that uC1(¯ΩK) if Ω is of class C2, according to Theorem 1.1 in [27]. By results in [16], the convergence |up||u| holds locally uniformly in G. Notice the absolute values! See Section 5 and Theorem 7 in [20] for a clarification. We shall need the convergence also at the outer boundary.

    Lemma 3. Let Ω be of class C2. Then

    limp|up(ξ)|=|u(ξ)|,ξΩ.

    Proof. Let n denote the outer unit normal at ξ. By Theorem 1.1 in [27] uC(¯ΩK). We see that

    up(ξ)n=|up(ξ)|

    for 2<p. Since upu and up(ξ)=u(ξ), we can for h>0 write

    u(ξhn)u(ξ)hup(ξhn)up(ξ)h

    and let h0 to obtain

    u(ξ)nup(ξ)n,u(ξ)nlimp(up(ξ)n).

    The limit exists by monotonicity.

    For the reverse inequality, choose an interior disk |(ξεn)x|<ε tangent to Ω at the point ξ. By the comparison principle in the ring

    0<|(ξεn)x|<ε

    we have

    up(x)ε(p2)/(p1)|x(ξεn)|(p2)/(p1)ε(p2)/(p1)up(ξεn),

    where the fraction in the minorant is the fundamental solution of the p-Laplace equation. At the point ξ this inequality can be differentiated in the normal direction. It follows that

    up(ξ)np2p1up(ξεn)ε

    and hence

    limp(up(ξ)n)u(ξεn)ε=u(ξεn)u(ξ)εu(ξ)n,

    as ε0. This concludes the proof.

    The boundary convergence for |vp|, where vp is the p-eigenfunction in Eq (1.3) has a slightly different proof.

    We shall later need continuous second derivatives on the boundary. Again, in a boundary zone, say 0<dist(x,Ω)<δ, there holds |vp|˜cp>0, when p is large. Indeed, since 2vpup on K for large p by (2.1), where up is the p-Potential with K chosen to be the High Ridge (see (1.4)), the comparison principle implies that 2|vp||up|cp at the boundary. By continuity it follows that near the boundary |vp|˜cp>0 for some ˜cp. Therefore, we may as in Section 3.1 conclude that vp has continuous second order derivatives on the boundary, provided that Ω is of class C2,α.

    Near the boundary, v is a solution to Δv=0, see Corollary 7 in [21]. In particular, v is continuous up to the boundary in this zone, see Theorem 1.1 in [27].

    Lemma 4. Let Ω be of class C2. Then

    limp|vp(ξ)|=|v(ξ)|,ξΩ.

    Proof. The inequality

    lim infp|vp(ξ)||vp(ξ)| (3.2)

    comes from a similar comparison as in Lemma 3.1. Now

    Δpvp=λpvp1p<0

    so that vp is a supersolution of the p-Laplace equation. Let n denote the outer unit normal at ξ. Using the comparison principle in the interior ring 0<|x(ξ(εn)|<ε we have

    vp(x)ε(p2)/(p1)|x(ξεn)|(p2)/(p1)ε(p2)/(p1)vp(ξεn),

    where the fraction in the minorant is the fundamental solution of the p-Laplace equation. Now the inequality (3.2) comes as in the previous subsection.

    The reverse inequality requires some tinkering, because we do not know whether vvp. Consider the ascending streamline βp=βp(t) for the p-eigenfunction vp starting at ξ:

    ddtβp(t)=+vp(βp(t)),βp(0)=ξ.

    (The ascending streamlines are unique.) According to the end of Section 4 in [21] we have

    ddt(|vp(βp(t))|2+12κpt2)0,t>0,

    where the constant κp0+ as p. It follows that

    |vp(βp(t))|2+12κpt2|vp(βp(0))|2=|vp(ξ)|2.

    By the results in [21] (see the proof of Theorem 10) we know that vpv locally uniformly and that βpβ pointwise§, where β is the streamline of v emerging at ξ. We conclude that

    §In particular, the bound for vp in Lemma 5 in the arXiv version of [21] yields
    |βp(t2)βp(0)|=|t20vp(βp(t))dt|(λpdiamΩ)1p1vp(t20)
    and it follows that
    |β(t)ξ|Λvt.

    |v(β(t))|lim supp|vp(ξ)|,t>0.

    Since v is continuous up to the boundary, by sending t to 0+ we finally arrive at

    |v(ξ)|lim supp|vp(ξ)|.

    Thus the lemma is proved.

    In the plane, Alexandroff's method is about a moving line, across which solutions are reflected. For simplicity, we immediately make the following assumptions.

    Assumptions: Suppose from now on that

    1) Ω is a bounded convex domain in the xy-plane.

    2) Ω is symmetric with respect to the x-axis and y-axis.

    3) Ω is of class C3,α.

    4) The curvature of Ω is non-decreasing in the first quadrant when x increases.

    5) K is the origin.

    See [6,15,25]. We note that the above assumptions are valid for the case when Ω is an ellipse in proper position.

    We restrict our description to the first quadrant and consider a non-horizontal line . The line divides the plane in two open half-planes T+ and T, where T is chosen so that T lies to the right of . Let x denote the reflexion of the point xT across the line . The above assumptions are designed to guarantee that

    (ΩT)Ω,

    when is a normal to Ω. In other words, reflexion in the normals is possible. See Figure 1.

    Figure 1.  The reflection illustrated in the case when Ω is an ellipse.

    Lemma 5. If the above assumptions are valid, then reflexion in the normals is possible. In the first quadrant, the orientation is chosen so that ΩT is to the right of the normal through a boundary point.

    Proof. See Lemma 4.2 in [6].

    We define the reflected function of fC(¯Ω) as

    f:(¯ΩT)R,f(x)=f(x).

    The p-eigenvalue problem. We use the p-eigenfunctions vp in Eq (1.3) and recall that they are continuous up to the boundary.

    Lemma 6. The reflected function vp satisfies vpvp in (ΩT).

    Proof. Obviously vp satisfies the same Eq (1.3) as vp. Now vpvp on the boundary (ΩT). Indeed, vp=vp on the line and vp0=vp on (Ω). By the comparison principle vpvp in (ΩT). The proof in [22] works for the comparison principle.

    Proposition 7. Let n denote the outer unit normal at the boundary point ξΩ. Let t be the unit vector orthogonal to n and pointing from T to T+. Then we have at all points η=ξεn lying on the normal line in Ω that

    vp(η)t0.

    Proof. Take h>0. By the previous lemma

    vp(η+ht)vp(η)hvp(η+ht)vp(η)h=vp(ηht)vp(η)h

    and as h0 we see that

    vp(η)tvp(η)t.

    This proves the desired inequality.

    For the next theorem we recall that vp is of class C1(¯Ω) and that vp is of class C2 in a boundary zone ¯Ω{x|0<dist(x,Ω)<δ}. See Section 3.2. At boundary points ξΩ we naturally have

    vp(ξ)t=0andvp(ξ)n=|vp(ξ)|.

    Theorem 8. At the boundary point ξ

    t|vp(ξ)|0,

    where the tangent t at ξ points from T to T+.

    Proof. We first claim that

    n(vpt)0

    at ξ. To see this, let ε0+ in the difference quotient

    vp(ξεn)tvp(ξ)tε0,

    which follows from Proposition 7, since vpt=0 at the boundary. To conclude the proof, use

    t|vp|=t(vpn)=n(vpt).

    The mixed partial derivatives do not commute in general, but here they do, again due to vpt=0, so that the last term disappears from the general formula

    t(vpn)=n(vpt)+κ(ξ)vpt

    in differential geometry. See Eq (10) in [15] or page 45 in [26].

    The p - potential function. For the p – potential function up solving problem (3.1) we encounter an extra problem caused by the inner boundary K, which might hinder reflexion in some normals. Even if (ΩT)Ω, it is difficult to control K: often it happens that (KT) contains points in ΩK, when the line of reflexion is a normal to Ω. In order to avoid a detailed geometric description, we have therefore chosen to assume that K is the origin.

    Theorem 9. Assume that Ω satisfies the assumptions in Section 4 and that K={(0,0)}. Then the inequality

    t|up(ξ)|0

    is valid when ξΩ, where the tangent t at ξ points from T to T+. (That is, in the direction of decreasing curvature.)

    Proof. We follow the same steps as in the proof for vp. First, the counterparts to Lemmas 5 and 6 follow as before, when one notices that the presence of K does not spoil the comparison upup. Indeed, if ξ2=f(ξ1) is the equation of the boundary Ω in the first quadrant ξ1>0,ξ2>0, then the normal through (ξ1,ξ2)Ω intersects the x-axis at the point

    (f(ξ1)f(ξ1)+ξ1,0).

    By Lemma 4.3 in [6], the assumptions imply that f(ξ1)f(ξ1)+ξ10. In other words, K (the origin) is not reflected at all. Thus nothing hinders the comparison upup. This yields Lemma 6 for up. The counterpart to Proposition 7 follows. So does Theorem 8 for up.

    Assume again that the assumptions on the domain in Section 5 are fulfilled and that K is the origin. By Theorems 8 and 9

    ddt|up(ξ)|0,ddt|vp(ξ)|0

    where the tangent t points in the direction of non-increasing curvature. That is to the left in the first quadrant. If now ξ and ζ belong to Ω in the first quadrant and are ordered so that ξ is to the left of ζ, then

    |up(ξ)||up(ζ)|and|vp(ξ)||vp(ζ)|

    i.e., |up| and |vp| increase when the curvature decreases.

    On the boundary, Lemmas 3 and 4 assure that we can proceed to the limits. We arrive at

    |u(ξ)||u(ζ)|and|v(ξ)||v(ζ)|. (5.1)

    This monotonicity enables us to conclude that |u| has only two maxima and two minima on the boundary Ω, viz. at the four intersections with the coordinate axes: the maxima are on the y-axis, the minima on the x-axis. —The same goes for |v|.

    We want to show that this monotonicity is strict. To see this, assume that |u(ξ)|=|u(ζ)| for two different boundary points in the first quadrant. Then |u| would be constant along the boundary arc between the points. According to Lemma 12 and Lemma 16 in [20] this means that all the streamlines emerging from this arc are straight line segments that cannot intersect each others, except at K, which now is the origin. But this forces the boundary arc to be an arc of a circle centered at the origin.

    The minimum of |u| on the boundary is attained at the x-axis. We claim that the minimum is strict. If not, we would have |u|=c on a circular arc. The above mentioned Lemmas 12 and 16 in [20] also imply that the eikonal equation |u(x,y)|=c is valid in the whole closed circular sector. Assuming that Ω is not a disk, which case is trivial, we choose a boundary point ξ not on the circular arc. Then |u(α(0)|=C>c.

    Recall that the (ascending) streamlines α=α(t) are defined through

    dα(t)dt=+u(α(t)),α(0)=ξ.

    They start at the boundary and reach the origin. Always, the speed |u(α(t)| is non-decreasing. We see that

    lim sup(x,y)(0,0)|u(x,y)|C,lim inf(x,y)(0,0)|u(x,y)|c.

    This contradicts Proposition 10 in [19] according to which the full limit exists at the origin. Therefore the minimum is strict.

    The two streamlines starting at the intersection of Ω with the x-axis are attracting streamlines in the terminology of [20]. By symmetry, they are line segments on the x-axis. Now Theorem 1 can be stated in the following form.

    Proposition 10. Suppose that the assumptions in Section 4 are valid and assume that the domain is not a disk. Let α be a streamline whose initial point is not on the x-axis. It cannot meet any other streamline before it meets and joins the x-axis. The speed |u(α(t)| is constant along α until it meets the x-axis, after which the speed is non-decreasing.

    Proof. Equation (5.1) and the above discussion allows us to conclude this from Theorem 3 in [20].

    A similar version of Theorem 2 holds for v. Now circular boundary arcs where |v| is constant are not excluded. In addition, we can infer the following interesting property.

    Proposition 11. Suppose that the assumptions in Section 4 are valid. Then the - Ground State satisfies the equation

    Δv=0inΩexcept possibly on thexaxis.

    Streamlines cannot meet outside the x-axis.

    Proof. This essentially follows from Theorem 2. To see this, we first claim that the minimum boundary speed |v|, which is attained at the x-axis, is either strict or is attained along a circular arc. The strict case is immediately clear by Theorem 2.

    In order to treat the other case, we need the contact set Υ, see Section 2. It is closed, has zero area and v is C1 and satisfies Δv=0 outside Υ. Moreover, Υ does not touch Ω.

    Suppose now that we have a closed boundary arc C which is symmetric about the x-axis and that the speed is constant along it. We can assume that it is of maximal length: the boundary speed outside is strictly larger. By symmetry, the two streamlines γ1 and γ2, starting at its endpoints, intersect at a point P on the x-axis. We shall now argue that P is the only point of the contact set Υ lying in the closed region bounded by γ1, γ2 and the boundary arc C. Suppose, towards a contradiction, that the lowest level curve ω that in this closed region reaches Υ does not contain P. (Thus P is at a higher level.) So v is of class C1 and satisfies Δv=0 in the open region bounded by ω, γ1, γ2 and C. Therefore, we can apply Lemma 12 in [20] to conclude that the

    eikonal equation|v|=c

    is valid in this region.

    Along any lower level curve, say ω, we now have that

    |v(ω)||v(ω)|=cv(ω)=constant.

    The approximation ωω of the level curve from below implies, by the continuity of S operator, that the whole arc of the level curve ω between γ1 and γ2 belongs to Υ. By Lemma 9 in [21], the whole sector between ω, γ1 and γ2 (with apex at P) belongs to Υ. This contradicts the fact that Υ has zero area.

    Hence, the first point in Υ is P. In particular, the streamlines γ1 and γ2 do not contain any points of Υ below P. Hence, v is of class C1 and satisfies Δv=0 in the whole region bounded by γ1, γ2 and C. Again by Lemma 12 in [20], the eikonal equation holds here. It follows from Lemma 1 in [2] that the streamlines emerging from the arc C are non-intersecting straight lines intersecting only at the point P. This implies that the arc C has to be circular.

    Moreover, since |v| is constant along γ1 and γ2 until they meet at P, no streamline emerging from the part of Ω that is outside of C can meet γ1 or γ2 below P, because the emerging streamline has too high an initial speed.

    In conclusion, if the minimum is not strict we have a circular boundary arc with constant speed, and the streamlines are rays joining at a point on the x-axis.

    In fact, one can extract more, but we are content to provide one good example.

    Example: The ellipse

    x2a2+y2b2=1,0<b<a,

    fulfills all our assumptions. Given the standard parametrization x(t)=acost and y(t)=bsint, the curvature at (x(t),y(t)) is given by

    ab(a2sin2t+b2cos2t)32.

    Clearly the curvature is decreasing from t=0 to t=π/2. The rest of the assumptions in Section 4 are obviously satisfied. —See also page 215 in[6].

    The remarkable formula

    u(x,y)=min0ψπ2max0ρ1{xρcos(ψ)+yρsin(ψ)W(ρ,ψ)}

    where the function W(ρ,ψ) has the explicit representation

    W(ρ,ψ)=8π(ρ46sin(2ψ)+ρ36210sin(6ψ)+ρ100990sin(10ψ)+)

    was discovered in [5] for the -Potential u of the punctured square 0<|x1|<1,0<|y1|<1. Here the center (1,1) is removed from the square. The formula is valid in the subsquare 0x1,0y1 and is extended by symmetry. The resulting function is of class C1 up to the sides (but not at the center) and it is real-analytic outside the diagonals y1=±(x1). See Figure 2.

    Figure 2.  The -Potential in the square.

    This function is not equal to the -Ground State v. See Section 5 in [5] for the original proof of this fact. Below, we briefly explain why they do not coincide.

    Observe that if the functions would coincide, then also v should be of class C1. By (1.2), this implies that

    |v|v1and hence|u|u1

    in the punctured square. (Here Λ=1.) In particular, the last inequality should hold on the diagonal. A numerical calculation below will show that this is not the case.

    From [5] we have, using the variables

    t=8πn=1(1)n1mnm2n1ρm2n1,mn=4n2, (6.1)

    that on the diagonal the so-called Rayleigh quotient takes the form

    R(t)=|u(t/2,t/2)|u(t/2,t/2)=ρρWρ(ρ,π4)W(ρ,π4).

    Here 0t2 and 0ρ1. The quantities involved are

    W(ρ,π4)=8πn=1(1)n1(m2n1)mnρm2nρWρ(ρ,π4)=8πn=1(1)n1mnm2n1ρm2nρWρ(ρ,π4)W(ρ,π4)=8πn=1(1)n1mnρm2n

    and so we arrive at the simple expression

    1R=8πn=1(1)n1mnρm2n1 (6.2)

    for the reciprocal value of the Rayleigh quotient. The series converges for 0ρ1, and its sum is 1 when ρ=1. To conclude, we only have to exhibit a value of ρ for which R>1. When ρ=0.97 we have by (6.2)

    1R=8π(120.973160.9735+1100.9799)>8π(120.973160.9735)1.01590>1.

    Here we have used Leibniz's rule for alternating series. By (6.1), ρ=0.97 corresponds to t1.4112. The corresponding point (0.9979,.0.9979) on the diagonal is at the distance 0.0030 from the center.

    We can extract further information about the "unknown" variational - Ground State v, be it unique or not, using the fact that it is not the - Potential. First, it cannot be of class C1 in the whole punctured square, because Theorem 3.1 in [28] would then imply that the functions coincide. Second, using Theorem 2 we can deduce that the variational - Ground State v is -harmonic except on a portion of the diagonals lying in a symmetric neighbourhood around the center. In other words, the contact set looks like the letter X, where the crossing line segments have length at most 2(21); probably much shorter.

    Recall that this is obtained as the limit of vp. It is not known to be unique, but it inherits the symmetries of vp.

    The described results are based on a fairly recent theory. So several immediate questions seem to be open problems. A few of them are

    ● For which convex domains do |u(ξ)| or |v(ξ)| have only a finite number of minima on the boundary?

    ● Are u and v twice differentiable or even real-analytic outside the attracting streamlines?

    ● Is log(v) concave for all solutions v of the - Eigenvalue problem (1.2), be they variational or not?

    ● Are there other domains than the stadiums in which we have uv?

    ● How do the streamlines run in non-convex domains?

    ● What about several dimensions?

    There are many more interesting questions, but we must stop here.

    We thank Bernd Kawohl for his valuable piece of advice. This work was done while the authors during the fall of 2022 were participating in the research program "Geometric Aspects of Nonlinear Partial Differential Equations" at Institut Mittag-Leffler. It was supported by the Swedish Research Council under grant no. 2016-06596. E. L. is supported by the Swedish Research Council, grant no. 2017-03736 and 2016-03639.

    The authors declare no conflict of interest.



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