Counterexamples to Strichartz estimates for the kinetic transport equation based on Besicovitch sets

Nonlinear Analysis 74 (2011) 2515–2522

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Counterexamples to Strichartz estimates for the kinetic transport equation based on Besicovitch sets Evgeni Y. Ovcharov Angewandte Mathematik und Bioquant, Universität Heidelberg, INF 267, Heidelberg 69120, Germany

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Article history: Received 22 June 2010 Accepted 7 December 2010

abstract We show that the Strichartz estimates for the kinetic transport equation containing L∞ x -norms in the spatial variables x fail on the characteristic functions of Besicovitch sets. © 2010 Elsevier Ltd. All rights reserved.

MSC: Primary 35B45 Secondary 35Q20 Keywords: Strichartz estimates Besicovitch Kinetic transport Counterexamples

1. Introduction Let us consider the Strichartz estimates for the kinetic transport equation

∂t u(t , x, v) + v · ∇x u(t , x, v) = F (t , x, v), u(0, x, v) = f (x, v).

(t , x, v) ∈ (0, ∞) × Rn × Rn ,

The solution u to the Cauchy problem above is given by u(t ) = U (t )f + W (t )F , where U (t )f = f (x − t v, v) denotes the kinetic transport propagator and W (t )F =

t

∫

U (t − s)F (s)ds 0

denotes the associated Duhamel’s operator. Castella and Perthame [1] (1996) and Keel and Tao [2] (1998) proved all homogeneous Strichartz estimates for that equation apart for the L2t -endpoints for which the latter authors remark that their methods are not powerful enough to resolve. The endpoint that also contains the L∞ x -norm in the spatial variables x occurs in one spatial dimension for this equation and has the form

‖U (t )f ‖L2 L∞ 1 . ‖f ‖L2 . x Lv x,v t

(1)

As in the context of the wave and the Schrödinger equation one expects this estimate to fail. Indeed, that was shown by Guo and Peng [3] (2007), who give an example of a function f ∈ L2x,v for which the norm in the left-hand side of (1) is infinite.

E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.12.007

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E.Y. Ovcharov / Nonlinear Analysis 74 (2011) 2515–2522

Later on, we carried out a detailed study [4] on the range of validity of the Strichartz estimates for the kinetic transport equation. This raised the question of the validity of all estimates of the form p . ‖f ‖ b c ‖U (t )f ‖Lq L∞ Lx Lv x Lv

(2)

t

for 1 ≤ q, p, b, c ≤ ∞ in any spatial dimension n. Related to it is the question of validity of all inhomogeneous estimates of the form p . ‖F ‖ q˜ ′ ′ p˜ ′ , ‖W (t )F ‖Lq L∞ x Lv L Lr˜ L t

(3)

v

x

t

for 1 ≤ q, p, q˜ , r˜ , p˜ ≤ ∞ in any spatial dimension n. The latter question contains the former one as a special case as it can be shown e.g. see [4] that the validity of the homogeneous estimate (2) is equivalent to that of the inhomogeneous estimate (3) with q˜ ′ = 1, r˜ ′ = b, and p˜ ′ = c. Note that by duality, see [4] or in a more general context [2,5], the question of validity of estimate (3), where r = ∞, is equivalent to the question of validity of estimate

‖W (t )F ‖Lq Lrx Lpv . ‖F ‖Lq˜ ′ L1 Lp˜ ′ , t

t

x

1 ≤ q, r , p, q˜ , p˜ ≤ ∞,

where r˜ = ∞. Therefore, the main question we address in our work is that of the validity or failure of the Strichartz estimates

‖W (t )F ‖Lq Lrx Lpv . ‖F ‖Lq˜ ′ Lr˜′ Lp˜ ′ , t

t

x

v

˜′

′

′

∀F ∈ Lqt Lrx˜ Lpv˜ ,

(4)

where either r or r˜ is equal to infinity. Note that generally these estimates cannot be proved in any straightforward way by the standard perturbation techniques since the L∞ x -norm is an endpoint at which they break down. The main new idea to answer this question is to consider the estimates on the characteristic functions of Besicovitch sets in the plane (or their cartesian products in higher dimensions) which will provide counterexamples to (4). It was quite surprising to us that such a simple and geometrically appealing type of counterexample can be given to treat both the homogeneous and the inhomogeneous estimates of the considered types. The paper is organized as follows. We begin with an informal discussion, then in Section 3 we introduce the notation and give some basic facts about Besicovitch sets that will be needed in what follows. In Section 4 we present the answers to the raised questions together with their proofs. In the Appendix we expose the proof of the well-known existence of Besicovitch sets with the required properties in order to make the presentation self-contained and more accessible. 2. Informal discussion The Strichartz estimates involving the L∞ -spatial norm have attracted attention in the context of other equations as well. The case of the wave and, more generally, that of the Klein–Gordon equation is quite interesting. In three spatial dimensions Escobedo and Vega [6] proved the estimate . ‖f ‖H s(p) , ‖K˙ m (t )f ‖Lp L∞ x

p > 2, s(p) =

t

3 2

1

− , p

(5)

where

 ˙ m (t )(ξ ) = cos t ⟨ξ ⟩m , K

⟨ξ ⟩m =



m2 + |ξ |2 ,

m ≥ 0.

This estimate has been generalized to all other dimensions by Fang and Wang [7], and to the inhomogeneous setting via the TT ∗ method by us in our thesis [8]. The validity of such an estimate is not in anyway obvious since in the context of the wave equation one typically uses Littlewood–Paley spectral decompositions which preclude the case of the L∞ x -norm. However, Strichartz estimates in the L∞ -spatial norm are of particular interest e.g. in the study of semilinear equations with nonlinearities of a power type, see e.g. [6] for an application in the context of a nonlinear Dirac equation. But even in the best studied context of the wave equation the full range of estimates containing the L∞ x -norm is not yet known, see e.g. the discussion in [7]. In our paper we shall show that unlike the wave equation in the context of the kinetic transport equation essentially 1 all estimates containing the L∞ x -norm (and by duality, the Lx -norm) fail. The only valid such inhomogeneous estimates are somewhat trivial, they are direct consequences of the dispersive estimate

‖U (t )f ‖Lrx Lpv .

1

|t |n

‖f ‖Lpx Lrv ,

1 ≤ p ≤ r ≤ ∞,

(6)

1 with r = ∞ in the case of the L∞ x -norm (and similarly with p = 1 for the Lx -norm) applied to the operator W (t ) and the Hardy–Littlewood–Sobolev inequality. Their proof is given in [4]. We were led to the idea of using Besicovitch sets by the following observation. All counterexamples that we tried initially on the estimate (4) could not distinguish between the cases p˜ ′ = ∞ and p˜ ′ < ∞. But it is exactly the case p˜ ′ = ∞ when the trivial estimates discussed above hold! So any argument that is delicate enough to fail the estimates with p˜ ′ < ∞ (the bulk case) should break down at p˜ ′ = ∞. After we interpreted the kinetic transport propagator U (t ) geometrically it became clear that the characteristic functions to Besicovitch sets provided the perfect match since they are small in size only when measured in terms of Lebesgue norms not having infinite exponents.

E.Y. Ovcharov / Nonlinear Analysis 74 (2011) 2515–2522

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3. Notation and preliminaries q

By Lt Lrx Lpv and Lbx Lcv we denote the space Lq ([0, ∞); Lr (Rn ; Lp (Rn ))) and Lb (Rn ; Lc (Rn )) respectively. The spatial dimension n should be clear from the context. Let us recall several necessary conditions on the Lebesgue exponents for the validity of a given Strichartz estimate, see [4] in the present context or [2,5] in the general context. By scaling, the estimate

‖U (t )f ‖Lq Lrx Lpv . ‖f ‖Lbx Lcv t

can only hold for all initial data f ∈ Lbx Lcv if 1 q

+

n r

=

n b

1

,

r

+

1 p

=

1 b

1

+ . c

(7)

The exponent triplet (q, r , p) in the estimate above must also satisfy 1 q




1 p

−

1



r

,

(8)

or q = ∞, p = r. Such triplets are called KT-acceptable in the present context. For the inhomogeneous estimate (4) the scaling conditions are

  1 1 1 1 1 1 =n 1− − , + + + = 2. (9) q q˜ r r˜ r p r˜ p˜ The two exponent triplets (q, r , p) and (˜q, r˜ , p˜ ) must be KT-acceptable. By translation invariance in time, we need to have also q ≥ q˜ ′ , or in symmetric notation 1

1

+

1

1

≤ 1. q q˜ In 1919 Besicovitch, see [9,10], constructed a plane set of (Lebesgue) measure zero with the interesting property that it contains a unit line segment in every direction. Besicovitch’s original construction was simplified significantly by Perron (1928) and later by others. The basic idea behind all such constructions is to form a ‘‘Perron tree’’ – a figure obtained by splitting a given triangle of height one into smaller elementary triangles of height one by partitioning the base of the original triangle into subintervals which provide the bases of the new triangles, and then by sliding the elementary triangles thus obtained along the base. The construction of a Besicovitch set on the plane is schematically shown in Fig. 1. Note that all directions along line segments that start at the top and end at the base of the original triangle are retained in the resulting Perron tree as we are only translating the elementary triangles along a fixed direction. It can be shown that the area of the tree can be made arbitrary small if the number of elementary triangles is big enough. Taking congruent copies of the tree rotated through suitable angles we obtain a figure containing a unit line segment in every direction of arbitrary small positive measure. We shall call any set on the plane containing a unit line segment in every direction a Besicovitch set. For our purposes it will suffice to use the following well-known fact. +

Lemma 3.1 (Falconer [9], Besicovitch [10]). There exists a sequence of uniformly bounded Besicovitch sets on the plane whose measure is decaying to zero. To make our paper self-contained and more accessible we have included a short proof of that lemma in the Appendix following the exposition in Falconer [9]. 4. Main results Consider the velocity averages A[f ](t , x) =

∫

∞

f (x − t v, v)dv −∞

of the kinetic transport propagator U (t ) in one spatial dimension (n = 1). This can be interpreted as the line integral ∫ 1 Af = √ f (l)dl 1 + t 2 γx,t along the straight line γx,t through the point (x, 0) in the x–v -plane of gradient −1/t. Let χQ be the characteristic function of some measurable set Q in the x–v -plane. Then, we have ∫ 1 ‖U (t )χQ ‖L∞ = ess sup χQ (l)dl. √ 1 x Lv x∈R γx,t 1 + t2 The latter identity expresses that ‖U (t )χQ ‖L∞ L1v is equal to the (essential) supremum of the line measure of the intersections x of Q with straight lines of gradient −1/t multiplied by a fixed factor in t. As t swipes through [0, ∞) the direction of the lines swipes through a right angle starting at the vertical direction and moving anticlockwise ends at the horizontal direction.

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E.Y. Ovcharov / Nonlinear Analysis 74 (2011) 2515–2522

Fig. 1. Construction of Perron tree.

As we have already mentioned in the Introduction our main objective is to give counterexamples to the inhomogeneous estimates for W (t ). However, for the sake of clarity we first illustrate the crux of our idea in the simpler homogeneous setting. Theorem 4.1. All Strichartz estimates for the kinetic transport equation on Rn of the form p . ‖f ‖ b c , ‖U (t )f ‖Lq L∞ Lx Lv x Lv

0 < q, p, b, c ≤ ∞,

t

(10)

fail, except for the trivial identity ∞ ∞ = ‖f ‖L∞ . ‖U (t )f ‖L∞ x,v t Lx Lv

Proof. We shall only need to consider functions of the product type f (x1 , . . . , xn , v1 , . . . , vn ) = f˜ (x1 , v1 ) . . . f˜ (xn , vn ). This reduces the case of n spatial dimensions to one spatial dimension, since we have the identity





U (t ) f˜ (x1 , v1 ) · · · f˜ (xn , vn ) = U (t )f˜ (x1 , v1 ) · · · U (t )f˜ (xn , vn ). So we can assume that n = 1. Suppose first that c < ∞. Let ϵ > 0 be an arbitrary small number and choose fϵ to be the characteristic function of a Besicovitch set Bϵ of measure not exceeding ϵ . From the construction of Bϵ we know that there exist a fixed polygon Q ⊃ Bϵ for all ϵ . Thus by Hölder’s inequality ‖fϵ ‖Lb Lcv . ‖fϵ ‖Lcx Lcv . ϵ 1/c . However, x

1/p

p = ‖U (t )fϵ ‖ ‖U (t )fϵ ‖L∞ 1 & √ x Lv L∞ x Lv

1

1 + t2

1/p

.

Hence, the left-hand side in (10) is always larger than a fixed positive constant, while the right-hand side can be made arbitrary small, showing that the estimate fails. Suppose now that c = ∞. In view of (7) we have that p = b. However, conditions (7) and (8) imply that q = ∞ and that p = r = ∞. We are left with a trivial estimate that holds.  Consider now a function F of the form F (s, x, v) = χQs (x, v), where Qs , for s ∈ [0, ∞), is a family of plane sets. Then, similarly to the homogeneous setting, we can give a suitable geometric interpretation of the term

‖W (t )F ‖L∞ 1 x Lv

(11)

in one spatial dimension. For a fixed time t ≥ 0 we consider the sector on the plane between the angles π /2 and π − arctan 1/t centered at (x, 0). When s runs the interval [0, t ], the line γx,t −s swipes through this sector clockwise. For each s we take the line measure of the intersection between γx,t −s and the set Qs , and then take the integral of all such values against the weight 1



1 + (t − s)2

.

E.Y. Ovcharov / Nonlinear Analysis 74 (2011) 2515–2522

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The supremum in x of the latter gives us the term in (11). The problem has an interesting interpretation from a 3d-perspective. If R is the set {(x, v, t ) | t ∈ [0, ∞), (x, v) ∈ Qt } the term ‖W (t )F ‖L1v gives us the surface area of the cut in R (weighed by a factor) that will be obtained by a horizontal line rotating along the t-axis and moving vertically at the same time, i.e. some sort of a generalized helicoid. Similarly to the 2d case, we would like to construct a solid R whose volume can be arbitrary small but which contains a family of these helicoid-like surfaces with surface areas that do not decay when the measure of R goes to zero and remain big in the sense explained above. Theorem 4.2. All Strichartz estimates for the kinetic transport equation on Rn of the form p . ‖F ‖ q˜ ′ ′ p˜ ′ , ‖W (t )F ‖Lq L∞ x Lv L Lr˜ L t

x

t

(12)

v

fail in the range 1 ≤ q, q˜ , p, r˜ , p˜ ≤ ∞ if p˜ > 1 (p˜ ′ < ∞). In the remaining cases when p˜ = 1 the estimate p . ‖F ‖ q˜ ′ p ‖W (t )F ‖Lq L∞ x Lv L L L∞ t

(13)

x v

t

q˜ ′ p

holds for all F ∈ Lt Lx L∞ v if and only if 1 q

1

+

=

q˜

n p

< 1,

1 < q, q˜ < ∞.

Proof. We remark that in the statement of the theorem three distinct cases are concealed. The first one is the bulk case p˜ > 1 which we prove by giving counterexamples in terms of Besicovitch sets. The second and third cases concern the remaining endpoint in p˜ estimates for which p˜ = 1. Note that, by scaling and translation invariance in time, we must have 1 q

1

+

=

q˜

n p

≤ 1.

We discern between the non-endpoint subcase q > q˜ ′ , q ̸= ∞, q˜ ̸= ∞ and the endpoint subcase q = q˜ ′ , or q = ∞, or q˜ = ∞. In the former case the estimates hold for reasons explained in the Introduction, see also [4] for a complete proof, while in the latter case the estimates fail due to Lemma 4.3 and Corollary 4.4 that follow below. We remark that we see Lemma 4.3 as an extension of Guo and Peng’s original counterexample [3] to the inhomogeneous setting. We mention in advance that the crucial point for the lemma to work is that the estimates are also endpoint in q and that the method used there does not discern between the cases p˜ ′ < ∞ and p˜ ′ = ∞ and therefore, as explained in the Introduction, could not be used to resolve the bulk case. This, again, was our original motivation to seek for another type of counterexamples which we now present. Let us construct our counterexamples by specifying F explicitly. For simplicity let us begin with the case n = 1. Consider the transformation of points on the plane Ts (x, v) = (x − sv, v),

s ∈ R.

Fix ϵ > 0 and let Q be a Besicovitch set of measure not exceeding ϵ . We set Qs = Ts (Q ) for s ∈ [0, ∞). Note that the Jacobian of Ts is equal to one and that Ts−1 = T−s . Therefore, the measure m(Qs ) = m(Q ) for all s. Set F (t , x, v) = χ[0,1] (t )χQt (x, v). For t ≥ 1 we have the estimate

∫

∞

W (t )F dv = −∞

∫ t∫ 0

∞

= 0

χ[0,1] (s)χQs (x − (t − s)v, v)dv ds

−∞ ∞

∫ t∫

χQ (x − t v, v)dv ds = √

−∞

1 1 + t2

∫ γx,t

χQ (l)dl.

In view of Hölder’s inequality and the fact that the support of W (t )F is uniformly bounded in v (with respect to ϵ ), for t ≥ 1 we obtain the key estimate p & ‖W (t )F ‖ ∞ 1 & √ ‖W (t )F ‖L∞ Lx Lv x Lv

1 1 + t2

.

Hence, the term on the left-hand side of (12) does not depend on the measure of the Besicovitch set Q and is always bigger than some fixed positive constant. For the right-hand side we have

‖F ‖Lq˜ ′ Lr˜′ Lp˜ ′ . ‖χ[0,1] (t )χQt (x, v)‖Lq˜ ′ Lp˜ ′ Lp˜ ′ t

x

v

t

. ‖χ[0,1] (t )χQ (x, v)‖

x

v

. ϵ 1/˜p . ′

q˜ ′ p˜ ′ p˜ ′ Lt Lx Lv

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E.Y. Ovcharov / Nonlinear Analysis 74 (2011) 2515–2522

Note that here we have used Hölder’s inequality in the x-variables and the fact that all Qt , for 0 ≤ t ≤ 1, are uniformly bounded in x. Provided that p˜ ′ < ∞, we have that the term above can be made arbitrary small. For the general case of n-spatial dimensions for F we take the product function F (t , x, v) = χ[0,1] (t )χQt (x1 , v1 ) . . . χQt (xn , vn ). The proof proceeds along the same lines.



Lemma 4.3. All Strichartz estimates for the kinetic transport equation on Rn of the form p . ‖F ‖ q ′ p˜ ′ , ‖W (t )F ‖Lq L∞ x Lv L Lr˜ L t

t x

v

1 ≤ q, p, r˜ , p˜ ≤ ∞,

(14)

q ′ ′ fail for some F ∈ Lt Lrx˜ Lpv˜ . By duality, all estimates of the form

‖W (t )F ‖Lq Lrx Lpv . ‖F ‖Lq L1 Lp˜ ′ ,

1 ≤ q, r , p, p˜ ≤ ∞

t x v

t

also fail. Proof. We shall find a function F explicitly for which (14) fails. Suppose that F (t , x, v) = φ(t )ψ(x)ξ (v) ≥ 0,

′

′

φ ∈ Lq (R), ψ ∈ Lr˜ (Rn ), ψ ∈ Lp˜ (Rn ).

We choose φ(t ), ψ(x), and ξ (v) as follows

φ(t ) = χ (0 ≤ t ≤ 1), 1

ψ(x) = ξ (v) =

n/˜r ′

| x|

(− ln |x|)1/˜r ′ +ϵ 1

|v|n/˜p′ (− ln |x|)1/˜p′ +ϵ

χ (|x| < 1/2), χ (|x| < 1/2),

p˜ ′ < ∞, ′

or ξ (v) = χ (|v| < 1/2) if p˜ ′ = ∞. We take also the function g ∈ Lp (Rn ), g (v) =

1

|v|n/p′ (− ln |x|)1/p′ +ϵ

χ (|x| < 1/2).

By scaling, we may assume that 1/p′ + 1/˜r ′ + 1/˜p′ = 1. We have the following simple inequality

∫  p &  ‖W (t )F ‖L∞  x Lv

 

Rn

W (t )Fg (v)dv  

L∞ x

.

The latter integral can be rewritten as

∫ Rn

W (t )Fg (v)dv =

∫ t∫ 0

Rn

φ(s)ψ(x − (t − s)v)ξ (v)g (v)dv ds.

The function h(t , x) =

∫ Rn

ψ(x − t v)ξ (v)g (v)dv

is continuous in x, for t > 0. Indeed, consider the inequality

|h(t , x1 ) − h(t , x2 )| ≤ ‖ψ(x1 − t v) − ψ(x2 − t v)‖Lr˜′ ‖ξ g ‖Lrv˜ . v

We can assume that r < ∞, since otherwise we are reduced to a trivial case, more precisely estimate (16). Then, we have that the first norm in the above inequality tends to zero as |x2 − x1 | → 0 which furnishes the claim. Hence,

˜′

t

∫

p ≥ ‖W (t )F ‖L∞ x Lv

φ(s)h(t − s, 0)ds.

(15)

0

For 0 < t < 1/2, we have that h(t , 0) ≥

∫

1 t n/˜r

′

|v|≤t

1

1

|v|n (− ln |v|)1+3ϵ

dv &

1 t n/˜r (ln t )3ϵ ′

.

Again, due to scaling we may assume that 1/q + 1/q′ = n/˜r ′ . Hence, n/˜r ′ = 1 and the norm in (15) is infinite.



E.Y. Ovcharov / Nonlinear Analysis 74 (2011) 2515–2522

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Fig. 2. Reduction in area at each step.

Corollary 4.4. All Strichartz estimates for the kinetic transport equation on Rn of the form p . ‖F ‖ ‖W (t )F ‖Lq L∞ ′ p˜ ′ , x Lv L1 Lr˜ L t

1 ≤ q, r , p, r˜ , p˜ ≤ ∞,

v

t x

p . ‖F ‖ q˜ ′ ′ p˜ ′ , ‖W (t )F ‖L∞ L∞ x Lv L Lr˜ L t

x

t

v

1 ≤ r , p, q˜ , r˜ , p˜ ≤ ∞

fail apart from the trivial identity ∞ ∞ = ‖F ‖ 1 ∞ ∞ . ‖W (t )F ‖L∞ L Lx Lv t Lx Lv

(16)

t

Proof. The claim for the first estimate follows directly from the fact that its validity is equivalent to that of the homogeneous estimates considered in Theorem 4.1 which we showed to fail, see again [4]. The second estimate fails for more trivial reasons. By scaling, we must have

  1 =n 1− q˜ r˜

1

but under such an assumption it is not possible to find a KT-acceptable exponent triplet (˜q, r˜ , p˜ ), unless we have the identity (16).  Corollary 4.5. All Strichartz estimates for the kinetic transport equation on Rn of the form

‖W (t )F ‖Lq Lrx Lpv . ‖F ‖Lq˜ ′ L1 Lp˜ ′ , t

t

x v

fail in the range 1 ≤ q, q˜ , p, r , p˜ ≤ ∞ if p > 1. In the remaining cases when p = 1 the estimate

‖W (t )F ‖Lq Lrx L1v . ‖F ‖Lq˜ ′ L1 Lr t

t

holds for all F ∈

q˜ ′ Lt L1x Lrv

1



q

+

1 q˜

x v

if and only if

=n 1−

1 r



< 1,

1 < q, q˜ < ∞.

Proof. It follows immediately from Theorem 4.2 by duality.



Acknowledgement The author would like to thank Francisco Villarroya for the useful discussions they held on the present work. Appendix Proof of Lemma 3.1. Let us consider in Fig. 2 two adjacent elementary triangles T1 and T2 of bases b in the partition of the original triangle S. Suppose that one of them is shifted along the line L lying on the base of S so that the overlapped interval along L has length 2(1 − α)b for some 1/2 < α < 1. The resulting figure 6 consists of a triangle T and two smaller auxiliary triangles. Let us note that triangle T is a homothetic copy of triangle T1 ∪ T2 dilated by factor α . Indeed, that follows from the fact that their corresponding sides are parallel and that the length of the base of T is shrunk by factor α from the length of the base of triangle T1 ∪ T2 . Therefore, the ratio of the areas of the two triangles is equal to α 2 . The reduction in area obtained by replacing T1 ∪ T2 by 6 is given by m(T1 ∪ T2 ) − m(6 ) = m(T1 ∪ T2 )(3α − 1)(1 − α). Here m is plane area or Lebesgue measure. Indeed, the overlapping region consists of a homothetic triangle to T1 ∪ T2 of factor 1 − α plus a parallelogram of area (4α − 2)(1 − α)m(T1 ∪ T2 ). Suppose now that we partition the original triangle S into 2k elementary triangles and let us denote the resulting Perron tree by Sk . The reduction in area in the first step has factor (3α − 1)(1 − α). Analogously, the reduction in area in the j-th step, for j = 1, . . . , k, will be of factor at least

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E.Y. Ovcharov / Nonlinear Analysis 74 (2011) 2515–2522

(3α − 1)(1 − α)α j−1 , considering only the homothetic triangles obtained in the previous step. Hence,   m(Sk ) ≤ m(S ) − (3α − 1)(1 − α) 1 + α 2 + α 4 + · · · + α 2(k−1) m(S )   (3α − 1)(1 − α 2k ) = m(S ) 1 − . 1+α We choose α close enough to one so that (3α − 1)/(1 + α) is close to one, and k big enough, so that α 2k is close enough to zero. Thus the area of the Perron tree Sk can be made arbitrary small. Note that in this construction no elementary triangle was moved by more than the length of the base of S as the left most elementary triangle might be kept fixed, and all others be moved to the left. Therefore, all Sk , k = 1, 2, . . . , ∞, lie inside a fixed polygon in the plane.  References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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