aa r X i v : . [ m a t h . O C ] J a n Irrigable Measures for Weighted Irrigation Plans
Qing SunDepartment of Mathematics, Penn State Universitye-mails: [email protected] 15, 2020
Abstract
A model of irrigation network, where lower branches must be thicker in order to supportthe weight of the higher ones, was recently introduced in [4]. This leads to a countablefamily of ODEs, describing the thickness of every branch, solved by backward induction.The present paper determines what kind of measures can be irrigated with a finite weightedcost. Indeed, the boundedness of the cost depends on the dimension of the support of theirrigated measure, and also on the asymptotic properties of the ODE which determinesthe thickness of branches.
In a ramified transport network [1, 2, 10, 12, 13], the Gilbert transport cost along each arc iscomputed by [length] × [flux] α (1.1)for some given α ∈ [0 , α <
1, this account for an economy of scale: transporting thesame amount of particles is cheaper if these particles travel together along the same arc.In the recent paper [4], the authors considered an irrigation plan where the cost per unit lengthis determined by a weight function W . The main motivation behind this model is that, for afree standing structure like a tree, the lower portion of each branch needs to bear the weightof the upper part. Hence, even if the flux of water and nutrients is constant along a branch,the thickness (and hence the cost per unit length) grows as one moves from the tip towardthe root.In this model, the weights are constructed inductively, starting from the outermost branchesand proceeding toward the root. Along each branch, the weight W is determined by solvinga suitable ODE, possibly with measure-valued right hand side. This is more convenientlywritten in the integral form W ( s ) = Z ℓs f ( W ( σ )) dσ + m ( s ) , (1.2)1here s ∈ [0 , ℓ ] is the arc-length parameter along the branch, s m ( s ) is a non-increasingfunction describing the flux, and f is a non-negative, continuous function. A natural set ofassumptions on f is (A1) The function f : R + R + is continuous on [0 , + ∞ [ , twice continuously differentiablefor z > , and satisfies f (0) = 0 , f ′ ( z ) > , f ′′ ( z ) ≤ for all z > . (1.3)The main result in [4] established the lower semicontinuity of the weighted irrigation cost,w.r.t. the pointwise convergence of irrigation plans. In particular, for any positive, boundedRadon measure µ , if there is an admissible irrigation plan whose weighted cost is finite, thenthere exists an irrigation plan for µ with minimum cost.The goal of the present paper is to understand whether a given Radon measure µ irrigableor not, with the weighted irrigation cost. That is, whether there exists an irrigation plan for µ whose weighted irrigation cost is finite. In the case without weights, i.e., with the classicalGilbert cost, this problem has been studied in [6]. The authors in [6] proved that if a measure µ is α -irrigable, then it must be concentrated on a set with Hausdorff dimension ≤ − α . Onthe other hand, if α > − d , then every bounded Radon measure with bounded support in R d has finite irrigation cost [1, 6].As shown by our analysis, in the presence of weights the irrigability of a measure µ dependson the dimension of the set where µ is concentrated, on the exponent α , and also on theasymptotic behavior of the function f ( z ) as z → To illustrate the basic idea of the weighted irrigation model, we first consider a network withfinitely many branches. As shown on the left of Fig. 1, each directed branch will be denotedby γ i : [ a i , b i ] R d , i = 1 , . . . , N , oriented from the root toward the tip and parameterized byarc-length. Call P i = γ i ( b i ) the ending node of the branch γ i .On each branch γ i , we first prescribe a left-continuous, non-increasing function m i : [ a i , b i ] R + , which can be interpreted as the ”flux” along the branch. Roughly speaking, m i ( t ) is theamount of mass transported through the point γ i ( t ).Call O ( i ) the set of branches originating from the node P i = γ i ( b i ), that is O ( i ) = n j ∈ { , . . . , N } ; γ j ( a j ) = P i o . (2.1)2 P P P P W W W W W γ γ γ γ γ Figure 1:
Left: A free standing tree with 5 branches. In this example, O (1) = { , } , O (3) = { , } , O (2) = O (4) = O (5) = ∅ . Right: On each branch, the weight decreases as one moves from thelower portion to the tip. Moreover, consider the sets of indices inductively defined by I . = ∅ , I . = { i ∈ { , . . . , N } ; O ( i ) = ∅} , I k +1 . = { i ∈ { , . . . , N } ; O ( i ) ⊆ I ∪ · · · ∪ I k } \ ( I ∪ · · · ∪ I k ) . (2.2)From [4] the weight function W i ( · ) on each branch γ i is defined inductively on I k , k ≥ k = 1, on each branch γ i : [ a i , b i ] R d with i ∈ I , the weight W i : [ a i , b i ] R + isdefined to be the solution of ω ( t ) = Z b i t f ( ω ( s )) ds + m i ( t ) , t ∈ ] a i , b i ] , (2.3)where f is a given function, satisfying (A1) .(ii) Assume the weight functions W i ( t ) have already been constructed along all branches γ j : [ a j , b j ] R d with j ∈ I ∪ . . . ∪ I k − .For i ∈ I k , the weight W i ( t ) along the i -th branch is defined to be the solution of ω ( t ) = Z b i t f ( ω ( s )) ds + m i ( t ) + ω i , t ∈ ] a i , b i ] , (2.4)where ω i . = X j ∈O ( i ) W j ( a j +) − X j ∈O ( i ) m j ( a j +) . (2.5) Following Maddalena, Morel, and Solimini [10], the transport network for general Radonmeasure can be described in a Lagrangian way. Let µ be a Radon measure on R d with µ ( R d ) = M and let Θ = [0 , M ]. We think of θ ∈ Θ as a Lagrangian variable, labeling a waterparticle. An irrigation plan for µ is a function χ : Θ × R + R d , θ and continuous w.r.t. t , which satisfies the following conditions: • All particles initially lie at the origin: χ ( θ,
0) = 0 for all θ ∈ Θ. • For a.e. θ ∈ Θ the map t χ ( θ, t ) is 1-Lipschitz and constant for t large. Namely, thereexists τ ( θ ) ≥ ( | χ ( θ, t ) − χ ( θ, s ) | ≤ | t − s | for all t, s ≥ ,χ ( θ, t ) = χ ( θ, τ ( θ )) for every t ≥ τ ( θ ) . Throughout the following, τ ( θ ) will denote the smallest time τ such that χ ( θ, · ) is con-stant for t ≥ τ . • χ irrigates the measure µ . That is, for each Borel set V ⊆ R d , µ ( V ) = meas ( { θ ∈ Θ; χ ( θ, τ ( θ )) ∈ V } ) . One can think of χ ( θ, t ) as the position of particle θ at time t .To define the flux on χ , which measures the total amount of particles travel along the samepath, we first need an equivalence relation between two Lipschitz maps. Definition 2.1
We say that two 1-Lipschitz maps γ : [0 , t ] R d and e γ : [0 , e t ] R d are equivalent if they are parametrizations of the same curve, and write it as γ ≃ e γ . When weuse the arc-length re-parametrization σ γ ( s ( σ )) , where Z s ( σ )0 | ˙ γ ( t ) | dt = σ, then two 1-Lipschitz maps are equivalent means their arc-length re-parametrizations coincide. Throughout the following, we denote by γ (cid:12)(cid:12)(cid:12) [0 ,t ] the restriction of a map γ to the interval [0 , t ]. Definition 2.2
Let χ : Θ × R + R d be an irrigation plan for the measure µ . On the set Θ × R + , we write ( θ, t ) ∼ ( θ ′ , t ′ ) whenever χ ( θ, · ) (cid:12)(cid:12)(cid:12) [0 ,t ] ≃ χ ( θ ′ , · ) (cid:12)(cid:12)(cid:12) [0 ,t ′ ] . This means that themaps s χ ( θ, s ) , s ∈ [0 , t ] and s χ ( θ ′ , s ) , s ∈ [0 , t ′ ] are equivalent in the sense of Definition 2.1.The multiplicity at ( θ, t ) is then defined as m ( θ, t ) . = meas (cid:16) { θ ′ ∈ Θ ; ( θ ′ , t ′ ) ∼ ( θ, t ) for some t ′ > } (cid:17) . (2.6)Given an irrigation plan χ : Θ × R + R d , in order to have finite weighted irrigation costconstructed in the next section, we should always assume the following conditon. (A2) For a.e. θ ∈ Θ , one has m ( θ, t ) > for every ≤ t < τ ( θ ) . .3 Weight functions for an irrigation plan. Given an irrigation plan χ : Θ × R + R d for a general Radon measure, in this section wereview the construction of the weight function W = W ( θ, t ) on the irrigation plan. Noticethat for an irrigation plan χ of a general Radon measure, for each particle θ ∈ Θ, the map χ ( θ, · ) : R + R d describes a continuous curve in R d . Thus χ may contain infinitely manybranches. To construct the weight function on each branch, the idea is to first compute theweights W ε on χ ε , which is the truncation of χ on the branches with multiplicity ≥ ε . It turnsout that χ ε only consists of finitely many branches, so that we can compute W ε as in Section2.1 . The weight W is then constructed by taking the limit of W ε , as ε → Definition 2.3
Given an irrigation plan χ , a path γ : [0 , ℓ ] R d , parameterized by arc-length, is ε -good if and only ifmeas (cid:18)n θ ∈ Θ ; χ ( θ, · ) (cid:12)(cid:12)(cid:12) [0 ,t ] ≃ γ for some t = t ( θ ) > o(cid:19) ≥ ε, (2.7) where the equivalence relation ≃ is given in Definition 2.1. In other words, γ is ε -good if there is an amount ≥ ε of particles whose trajectory contains γ as initial portion.For any given ε >
0, following [4] we define the ε -stopping time τ ε : Θ R + by setting τ ε ( θ ) . = max { t ≥ m ( θ, t ) ≥ ε } . (2.8)Define the ε -truncation χ ε of irrigation plan χ as χ ε ( θ, t ) . = ( χ ( θ, t ) if t < τ ε ( θ ) χ ( θ, τ ε ( θ )) if t ≥ τ ε ( θ ) (2.9)In other words, in the ε -truncation χ ε , only those paths in χ with multiplicity ≥ ε are kept.For any θ ∈ Θ, if τ ε ( θ ) >
0, the ε -good portion χ ( θ, · ) (cid:12)(cid:12)(cid:12) [0 ,τ ε ( θ )] of the path t χ ( θ, · ) is includedin χ ε .Notice that the family of all curves parameterized by arc-length comes with a natural partialorder. Namely, given two maps γ : [0 , ℓ ] R d , e γ : [0 , e ℓ ] R d , we write γ (cid:22) e γ if ℓ ≤ e ℓ and γ ( s ) = e γ ( s ) for all s ∈ [0 , ℓ ]. In the family of all ε -good paths in the irrigation plan χ , wecan thus find the maximal ε -good paths, w.r.t the above partial order. As shown in [4], thetotal number of maximal ε -good paths in the irrigation plan χ is bounded by Mε , where M isthe total mass of µ . Therefore, the ε -truncation χ ε is a network with finitely many branches,consisting of all maximal ε -good paths in χ .For a fixed ε >
0, to compute the weight functions on the ε -truncation χ ε , we now let { b γ , . . . , b γ ν } be the set of all maximal ε -good paths. Along each path b γ i : [0 , ˆ ℓ i ] R d wedefine the multiplicity b m i : [0 , ˆ ℓ i ] R + by setting b m i ( s ) = meas (cid:18) (cid:26) θ ∈ Θ ; there exists t ≥ χ ( θ, · ) (cid:12)(cid:12)(cid:12) [0 ,t ] ≃ b γ i (cid:12)(cid:12)(cid:12) [0 ,s ] (cid:27) (cid:19) . (2.10)5ince two maximal paths may coincide on the initial portion and bifurcate later, we considerthe bifurcation times τ ij = τ ji . = max n t ≥ b γ i ( s ) = b γ j ( s ) for all s ∈ [0 , t ] o . (2.11)For each maximal path b γ i , we split it into several elementary branches γ k , by the followingPath Splitting Algorithm (PSA) , which is first introduced in [4]. (PSA) For each i ∈ { , . . . , ν } , consider the set { τ i , . . . , τ iν } = { t i, , . . . , t i,N ( i ) } , where the times 0 < t i, < t i, < · · · < t i,N ( i ) = ˆ ℓ j (2.12)provide an increasing arrangement of the set of times τ ij where the path b γ i splits apartfrom other maximal paths. For each k = 1 , . . . , N ( i ), let γ i,k be the restriction of themaximal path b γ i to the subinterval [ t i,k − , t i,k ]. The multiplicity function m i,k along thispath is defined simply as m i,k ( t ) = b m i ( t ) t ∈ [ t i,k − , t i,k ] . (2.13)If τ ij >
0, i.e. if the two maximal paths b γ i and b γ j partially overlap, it is clear that someof the elementary branches γ i,k will coincide with some γ j,l . To avoid listing multipletimes the same branch, we thus remove from our list all branches γ j,l : [ t j,l − , t j,l ] R d such that t j,l ≤ τ ij for some i < j . After relabeling all the remaining branches, thealgorithm yields a family of elementary branches and corresponding multiplicities γ i : [ a i , b i ] R d , m i : [ a i , b i ] R + , i = 1 , . . . , N. (2.14) γ1 γ γ γ γ γ γ γ Figure 2:
Left: Two finite truncation plans, showing three maximal ε -good paths (thick lines) andsix maximal ε ′ -good paths (thin lines), for 0 < ε ′ < ε . Right: The three maximal ε -good paths can bepartitioned into five elementary branches, by the Path Splitting Algorithm. On these elementary branches γ i , i ≥
1, we can compute the weight function W i on each γ i inductively, as in Section 2.1. 6n each maximal ε -good path b γ j with 1 ≤ j ≤ ν , the above construction yields a weight c W j,k on the restriction of b γ j to each subinterval [ t j,k − , t j,k ]. Along the maximal path b γ j , the weight c W j : [0 , ˆ ℓ j ] R + is then defined simply by setting c W j ( t ) = c W j,k ( t ) if t ∈ [ t j,k − , t j,k ] . (2.15)Next, on the ε -truncation χ ε we define the weight function W ε : Θ × R + R + by setting W ε ( θ, t ) . = c W i ( s ) if t ≤ τ ε ( θ ) , χ ( θ, · ) (cid:12)(cid:12)(cid:12) [0 ,t ] ≃ b γ i (cid:12)(cid:12)(cid:12) [0 ,s ] , t > τ ε ( θ ) . (2.16)As proved in [4], the map ε W ε ( θ, t ) is nondecreasing for each ( θ, t ). This leads to: Definition 2.4
Let χ : Θ × R + R d be an irrigation plan satisfying (A2) . The weightfunction W = W ( θ, t ) for χ is defined as W ( θ, t ) . = sup ε> W ε ( θ, t ) . (2.17)Once we computed the weight functions on the irrigation plan χ , its weighted irrigation cost E W,α is defined as follows:
Definition 2.5
Let f : R + R + be a continuous function, satisfying all the assumptions in (A1) . Let χ be an irrigation plan satisfying (A2) and let W = W ( θ, t ) be the correspondingweight function, as in (2.17). The weighted cost E W,α for some α ∈ [0 , is E W,α ( χ ) . = Z M Z τ ( θ )0 ( W ( θ, t )) α m ( ξ, t ) | ˙ χ ( θ, t ) | dt dθ . (2.18)In the special case where χ consists of only finitely many branches, let W i be the correspondingweight functions on the branch γ i : [ a i , b i ] R + , by applying the change of variable formula,we have the following identities for the weighted irrigation costs[4]: E W,α ( χ ) = N X i =1 Z b i a i [ W i ( s )] α ds , (2.19)where N is the total number of branches. In this section we recall the main theorems on the lower semicontinuity of weighted irrigationcost, proved in [4]. Given a sequence of irrigation plans χ n : Θ × R + R d , we say that χ n converges to χ pointwise if, for every κ > θ ∈ Θ,lim n →∞ k χ n ( θ, · ) − χ ( θ, · ) k L ∞ ([0 ,κ ]) = 0 . (2.20)7 heorem 2.6 Let ( χ n ) n ≥ be a sequence of irrigation plans, all satisfying (A2) , pointwiseconverging to an irrigation plan χ . Assume that the function f satisfies the conditons in (A1) .Then E W,α ( χ ) ≤ lim inf n →∞ E W,α ( χ n ) . (2.21)Given a positive, bounded Radon measure µ on R d , we define the weighted irrigation cost I W,α ( µ ) of µ as I W,α ( µ ) . = inf χ E W,α ( χ ) , (2.22)where the infimum is taken over all irrigation plans for the measure µ , and E W,α is defined asin (2.18). By Theorem 2.6, if there is an irrigation plan for µ with finite weighted irrigationcost, then the infimum in (2.22) is actually a minimum. That is, there exists an optimalirrigation plan χ ∗ of µ , such that the weighted irrigation cost E W,α ( χ ∗ ) is minimum among alladmissible irrigation plans, and I W,α ( µ ) = E W,α ( χ ∗ ).The next result states the lower semicontinuity of the weighted irrigation cost, w.r.t. weakconvergence of measures. For a proof, see Theorem 6.2 in [4]. Theorem 2.7
Let f satisfies the conditons in (A1) . Let ( µ n ) n ≥ be a sequence of boundedpositive Radon measures, with uniformly bounded supports, such that weakly converges to some µ . Then I W,α ( µ ) ≤ lim inf n →∞ I W,α ( µ n ) . (2.23) When f = 0, α > − d , it is well known that all measures with bounded support and finitemass in R d are α -irrigable [1, 6]. Here is a formal computation in this direction. It is obtainedby modifying the estimates at p. 113 of [1].Let X ⊂ R d be the compact support of µ , whose total mass is 1. For each j = 1 , . . . , n , let P j be the set of centers of balls of radius r j = 2 − j that cover X . In dimension d , we can assumethat the cardinality of this set is P j ≤ C jd We can define a map γ j : P j
7→ P j − such that | x − γ j ( x ) | ≤ · − j for every x ∈ P j .Consider a probability measure µ n , supported on P n . The cost of transporting this measurefrom P n to another measure supported on P n − is E α ( P n , P n − ) ≤ [number of arcs] × [flow] α × [length] ≤ C nd · (cid:0) C nd (cid:1) α · · − n = 3 C − α · (2 αd − d +1 ) − n . (3.1)Notice that we are here considering the worst possible case, where all arcs carry equal flow.8umming over j = 1 , , . . . , n we obtain that the total transportation cost is bounded by E α ≤ C − α · n X j =1 (2 αd − d +1 ) − j ≤ C − α αd − d +1 − . (3.2)The series P k [( d − − αd ]( k +1) converges provided that( d − − αd < , hence α > − d To understand what happens in the case where weights are present, we first make an explicitcomputation in the case of a dyadic irrigation plan [1, 12]. More precisely, as shown in Fig. 3,we now assume µ = Radon measure with total mass M , concentrated on a cube Q in R d . Q is centered at theorigin and with edge size L > n ≥
1, we divide Q into 2 nd smaller cubes of equal size, with edge size L/ n . Take { Q ni } nd i =1 the set of all these closed smaller cubes, call P n . = { x ni } nd i =1 the set of centers of thesesmaller cubes of edge size L/ n . For each n ≥
1, define the dyadic approximated measure µ n µ n . = X x ni ∈P n m ni δ x ni , (3.3)where δ x ni is the Dirac measure at x ni , and m ni is determined as b Q ni . = Q ni \ [ j
Figure 3:
Left: The dyadic approxmiated measure µ is supported on the four centers x , . . . , x ofsmall cubes. Right: Dyadic approximated measures corresponding to a family of partitions into dyadiccubes in R . It is not hard to show that µ n weakly converges to µ , see for example [1, 12]. That is, forany bounded continuous function φ : R d R , one has R φ dµ n → R φ dµ. For each µ n , weconstruct an irrigation plan χ n as follows: 9 First, move the particles from the origin(center of Q ) to the centers in P , with 2 d straight paths connecting the origin and the centers in P = { x , x , . . . , x d } . Eachpath has length √ dL , on the path that connecting x i , ≤ i ≤ d , the multiplicity isconstant m i . • By induction, at the level k, < k ≤ n , for the particles arriving at each center x k − i in P k − , where x k − i is the center of the cube Q k − i , we transport them to the 2 d neighboringcenters in P k , which are all contained in the cube Q k − i . Without loss of generality, fixed x k − i in P k − , let { x k , . . . , x k d } be the 2 d neighboring centers around x k − i . For each x kj , ≤ j ≤ d , we build a straight path connecting x k − i to x kj , with length √ dL k +1 andconstant multiplicity m kj . Q (cid:127) (cid:127)
123 4 (cid:127) (cid:127) Q Figure 4:
The dyadic irrigation plans in R . Left: The dyadic irrigation plan χ . The multiplicityon each branch equals to the mass on the terminal point. Right: The dyadic irrigation plan χ . Theparticles are first transported to the 4 centers in P , then on each center in P , the particles aretransported to the neighboring 4 centers in P . Since the dyadic measure µ n is supported on the centers in P n , after n steps we build anirrigation plan for µ n , which we call the dyadic irrigation plan χ n .For example, in the case R , Fig. 4 shows two dyadic irrigation plans constructed by thepreceding procedure.Given an irrigation plan with finite branches as in Section 2.1, consider the case f ( z ) . = cz β ,with some constant c > , < β <
1. It is readily to check that f satisfies (A1) . With thenotions in Section 2.1, consider a measure µ consisting of finitely many point masses m i ≥ P i , where P i is the ending node of branch γ i ( s ) : [0 , ℓ i ] R d . In this case, themultiplicity function on each branch is constant. Then the computation of weights (2.3)-(2.5)becomes W i ( s ) = (cid:16) W − βj + c (1 − β )( ℓ i − s ) (cid:17) − β ,W i = m i + P j ∈O ( i ) (cid:16) W − βj + c (1 − β ) ℓ j (cid:17) − β . (3.4)If O ( i ) = ∅ , that is i ∈ I , from (3.4) we have W i = m i .The following two lemmas proved that under suitable conditions, the weighted irrigation costsof the dyadic irrigation plans { χ n } n ≥ are uniformly bounded. Utilizing this fact and Theorem10.7, since the dyadic approximated measures µ n weakly converges to µ , we can conclude theirrigability of µ with weighted cost.To fix the ideas, we first consider the case that µ is the Lebesgue measure on the unit cube Q . Lemma 3.1
Suppose > β > − d , ≥ α > − d , while µ is the Lebesgue measureon the unit cube Q in R d . Then, in the dyadic irrigation plans χ n , the weight function W n remains uniformly bounded on all branches. Moreover, the irrigation cost E W,α ( χ n ) isuniformly bounded. That is, there exists an uniform constant C > , such that for all dyadicirrigation plan χ n , W n ≤ C, E W,α ( χ n ) ≤ C . (3.5)
Proof.
For the dyadic irrigation plan χ n , since each dyadic irrigation plan has finite branchesand µ n is supported on the centers in P n , we can use formula (3.4) to compute the weights W n . We start from the centers in P n . From P n to P n − , for each x n − i ∈ P n − , by the construction of the dyadic irrigation plan χ n , there are straight paths connecting x n − i to the 2 d neighboring centers in P n . Since µ is theLebesgue measure on unit cube Q , mass on each center in P n is nd . The branches connecting x n − i to centers in P n are identical, with branch length √ d/ n +1 and constant multiplicity1 / nd . We only need to compute the weight on one such branch, and write it as W nn , wherethe superindex n means it is the weight for irrigation plan χ n , and the subindex n means from P n to P n − .By formula (3.4), for s ∈ [0 , √ d n +1 ], W nn ( s ) = (cid:16) ( 12 nd ) − β + c (1 − β )( √ d n +1 − s ) (cid:17) − β , (3.6) W nn (0) = (cid:16) ( 12 nd ) − β + c (1 − β ) √ d n +1 (cid:17) − β . (3.7) From P n − to P n − , using formula (3.4), on each branch we need to first compute theweights W nn − at the tip. For the dyadic approximated measure µ n , it is supported on P n ,thus the mass on each center in P k , k = n is 0. Since each center in P n − connects 2 d identicalcenters in P n − , we therefore have W nn − = 2 d W nn (0) = 2 d (cid:16) ( 12 nd ) − β + c (1 − β ) √ d n +1 (cid:17) − β . (3.8)Each branch between P n − and P n − has length √ d n . By formula (3.4), for s ∈ [0 , √ d n ], W nn − ( s ) = (cid:16) d (1 − β ) h ( 12 nd ) − β + c (1 − β ) √ d n +1 i + c (1 − β )( √ d n − s ) (cid:17) − β , (3.9) W nn − (0) = (cid:16) ( 12 ( n − d ) − β + c (1 − β ) h √ d n +1 − d (1 − β ) + √ d n i(cid:17) − β . (3.10) From P n − to P n − . At the tip of each branch, W nn − = 2 d W nn − (0) . (3.11)11he branches have length √ d n − , by formula (3.4), for s ∈ [0 , √ d n − ], W nn − ( s ) = (cid:16) d (1 − β ) " ( 12 ( n − d ) − β + c (1 − β ) h √ d n +1 − d (1 − β ) + √ d n i + c (1 − β )( √ d n − − s ) (cid:17) − β = (cid:16) ( 12 ( n − d ) − β + c (1 − β ) h √ d n +1 − d (1 − β ) + √ d n − d (1 − β ) + ( √ d n − − s ) i(cid:17) − β , (3.12) W nn − (0) = (cid:16) ( 12 ( n − d ) − β + c (1 − β ) h √ d n +1 − d (1 − β ) + √ d n − d (1 − β ) + √ d n − i(cid:17) − β . From P n − k to P n − k − , each branch has length √ d n +1 − k . Similarly for s ∈ [0 , √ d n +1 − k ], W nn − k ( s ) = (cid:16) ( 12 ( n − k ) d ) − β + c (1 − β ) √ d n +1 k X j =1 ( k − j )+ jd (1 − β ) + c (1 − β )( √ d n +1 − k − s ) (cid:17) − β ,W nn − k (0) = (cid:16) ( 12 ( n − k ) d ) − β + c (1 − β ) √ d n +1 k X j =0 ( k − j )+ jd (1 − β ) (cid:17) − β = (cid:16) ( 12 ( n − k ) d ) − β + c (1 − β ) √ d n +1 − k k X j =0 [1 − d (1 − β )] j (cid:17) − β . (3.13) Since W nn − k ( s ) ≤ W nn − k (0), we only need to estimate W nn − k (0), for each 0 ≤ k ≤ n − . When β > − /d , one has 1 − d (1 − β ) >
0. By (3.13), for each k , W nn − k (0) ≤ (cid:16) ( 12 ( n − k ) d ) − β + c (1 − β ) √ d n +1 − k · − ( ) − d (1 − β ) (cid:17) − β . (3.14)Therefore, we have an uniform bound for the weight function W n ≤ c (1 − β ) √ d − ( ) − d (1 − β ) ! − β , (3.15)which is independent of n . We now estimate the irrigation cost E W,α ( χ n ) by the formula (2.19). Fixed the dyadicirrigation plan χ n , call E nn the cost from P n to P n − . There are 2 nd branches from centers in P n to centers in P n − . On each branch, the weight W nn is given by (3.6). Therefore, E nn = 2 nd Z √ d n +1 (cid:16) ( 12 nd ) − β + c (1 − β )( √ d n +1 − s ) (cid:17) α − β ds = 2 nd c (1 + α − β ) " ( 12 nd ) − β + c (1 − β ) √ d n +1 α − β − β − (cid:20) ( 12 nd ) − β (cid:21) α − β − β . (3.16)12imilarly, denote E nn − k the cost from P n − k to P n − k − . There are 2 ( n − k ) d branches from centersin P n − k to centers in P n − k − . E nn − k = 2 ( n − k ) d Z √ d n +1 − k (cid:16) (cid:0) W nn − k (cid:1) − β + c (1 − β )( √ d n +1 − k − s ) (cid:17) α − β ds = 2 ( n − k ) d c (1 + α − β ) (cid:16)h (cid:0) W nn − k (cid:1) − β + c (1 − β ) √ d n +1 − k i α − β − β − (cid:0) W nn − k (cid:1) α − β (cid:17) . (3.17)In the following, we use the same C to denote different constants which only depend on c, α, β and the dimension d . From (3.14) and the fact that (1 − β ) d <
1, for each n and k , (cid:0) W nn − k (cid:1) − β + c (1 − β ) √ d n +1 − k ≤ C ( n − k )(1 − β ) d . (3.18)Consider x, y ≥ g ( x, y ) . = ( x + y ) α − β − β − x α − β − β , x + y ≤ C ( n − k )(1 − β ) d . (3.19)then, by a first order Taylor expansion, g ( x, y ) ≤ α − β − β (cid:18) C ( n − k )(1 − β ) d (cid:19) α − β · y ≤ Cy ( n − k ) αd . (3.20)Applying (3.18) and (3.20) in (3.17), we obtain E nn − k ≤ ( n − k ) d C ( n − k )( αd +1) = C ( n − k )[( α − d +1] . (3.21)When α > − /d , one has ( α − d + 1 >
0. Then by (3.21), E W,α ( χ n ) = n − X k =0 E nn − k ≤ n − X k =0 C ( n − k )[( α − d +1] ≤ C − ( ) ( α − d +1 , (3.22)where C is some constant independent of n . Combining the estimates (3.15) and (3.22), weobtain the existence of a constant C , independent of n , such that (3.5) holds.Under the same conditons on α, β , this uniform boundedness result holds for general positive,finite Radon measures. Lemma 3.2
Suppose − d < β < , − d < α ≤ . µ is a finite measure on the cube Q withedge size L in R d , denote M the total mass of µ . Then in the dyadic irrigation plan χ n , theweight function W n on each branch remains uniformly bounded, W n ≤ C (cid:16) M − β + L (cid:17) − β , (3.23) Moreover, the irrigation cost E W,α ( χ n ) is uniformly bounded, namely E W,α ( χ n ) ≤ C (cid:16) M α L + L α − β (cid:17) (3.24) where C is some constant independent of n . roof. For the dyadic irrigation plan χ n , to compute the weights W n , we start from thecenters in P n . From P n to P n − . Let m ni be the mass of µ n on the center x ni in P n . On the branch from x ni to center in P n − , the arc-length of the branch is √ dL n +1 and the multiplicity is constant m ni .Let W nn,i be the corresponding weights, where the superindex n means we consider the weightfunction on irrigation plan χ n , the subindex ( n, i ) means we consider the weight on the i -thbranch from P n to P n − . Then by formula (3.4), for s ∈ [0 , √ dL n +1 ], W nn,i ( s ) = (cid:16) ( m ni ) − β + c (1 − β )( √ dL n +1 − s ) (cid:17) − β , (3.25) W nn,i (0) = (cid:16) ( m ni ) − β + c (1 − β ) √ dL n +1 (cid:17) − β . (3.26) From P n − to P n − . For each center x n − i in P n − , to compute the weight W nn − ,i from x n − i to center in P n − , we first estimate W nn − ,i . Each x n − i in P n − connects 2 d nearbycenters in P n . By (3.4) and (3.26) one has, W nn − ,i = X j ∈O ( i ) W nn,j (0) = X j ∈O ( i ) (cid:16) (cid:0) m nj (cid:1) − β + c (1 − β ) √ dL n +1 (cid:17) − β . (3.27)Notice for fixed b ≥ g ( x ) . = ( x − β + b ) − β is a concave function of x on R + . Thus for any N , 1 N N X j =1 ( x − βj + b ) − β ≤ (cid:16) ( P Nj =1 x j N ) − β + b (cid:17) − β . (3.28)For each i , the cardinality of the set O ( i ) is 2 d . from (3.27)-(3.28), W nn − ,i ≤ d h(cid:16) P j ∈O ( i ) m nj d (cid:17) − β + c (1 − β ) √ dL n +1 i − β . (3.29)Each branch from x n − i to P n − has length √ dL n . By the formula (3.4), for s ∈ [0 , √ dL n ], W nn − ,i ( s ) = (cid:16) (cid:0) W nn − ,i (cid:1) − β + c (1 − β )( √ dL n − s ) (cid:17) − β ≤ (cid:16) ( X j ∈O ( i ) m nj ) − β + c (1 − β ) h √ dL n +1 − d (1 − β ) + ( √ dL n − s ) i(cid:17) − β , (3.30) W nn − ,i (0) ≤ (cid:16) ( X j ∈O ( i ) m nj ) − β + c (1 − β ) h √ dL n +1 − d (1 − β ) + √ dL n i(cid:17) − β . (3.31) From P n − to P n − . For each center x n − i in P n − , according to (3.31), W nn − ,i = X k ∈O ( i ) W nn − ,k (0) ≤ X k ∈O ( i ) (cid:16) ( X j ∈O ( k ) m nj ) − β + c (1 − β ) h √ dL n +1 − d (1 − β ) + √ dL n i(cid:17) − β . (3.32)14sing the concavity inequality (3.28), W nn − ,i ≤ d "(cid:16) P k ∈O ( i ) ,j ∈O ( k ) m nj d (cid:17) − β + c (1 − β ) h √ dL n +1 − d (1 − β ) + √ dL n i − β . (3.33)In the following, for each center x ki in P k , if there is a concatenated path from x ki to center x nj ∈ P n in the dyadic irrigation plan χ n , we say i ≺ j . With this notation, (3.33) can bewritten as W nn − ,i ≤ d "(cid:16) P i ≺ j m nj d (cid:17) − β + c (1 − β ) h √ dL n +1 − d (1 − β ) + √ dL n i − β . (3.34)Each branch from x n − i to P n − has length √ dL n − . By the formula (3.4), for s ∈ [0 , √ dL n − ], W nn − ,i ( s ) = (cid:16) (cid:0) W nn − ,i (cid:1) − β + c (1 − β )( √ dL n − − s ) (cid:17) − β ≤ (cid:16) ( X i ≺ j m nj ) − β + c (1 − β ) h √ dL n +1 − d (1 − β ) + √ dL n − d (1 − β ) + ( √ dL n − − s ) i(cid:17) − β . (3.35) W nn − ,i (0) ≤ (cid:16) ( X i ≺ j m nj ) − β + c (1 − β ) h √ dL n +1 − d (1 − β ) + √ dL n − d (1 − β ) + √ dL n − i(cid:17) − β . (3.36) From P n − k to P n − k − . Similarly we have, W nn − k,i ≤ d (cid:16) ( P i ≺ j m nj d ) − β + c (1 − β ) √ dL n +2 − k k − X l =0 [1 − d (1 − β )] l (cid:17) − β , (3.37) W nn − k,i ( s ) ≤ (cid:16) ( X i ≺ j m nj ) − β + c (1 − β ) √ dL n +1 − k k X l =1 [1 − d (1 − β )] l + c (1 − β )( √ dL n +1 − k − s ) (cid:17) − β (3.38) W nn − k,i (0) ≤ (cid:16) ( X i ≺ j m nj ) − β + c (1 − β ) √ dL n +1 − k k X l =0 [1 − d (1 − β )] l (cid:17) − β . (3.39) Since W nn − k,i ( s ) ≤ W nn − k,i (0), we only need to estimate W nn − k,i (0), for each 0 ≤ k ≤ n − , ≤ i ≤ d ( n − k ) . When β > − /d , one has 1 − d (1 − β ) >
0. From formula (3.39), W nn − k,i (0) ≤ (cid:16) ( X i ≺ j m nj ) − β + c (1 − β ) √ dL n +1 − k − − d (1 − β ) (cid:17) − β . (3.40)Since P i ≺ j m nj ≤ M , if denote W n the weights on dyadic irrigation plan χ n , from (3.40) thereis an uniform bound for the weight function W n ≤ (cid:16) M − β + c (1 − β ) √ dL − − d (1 − β ) (cid:17) − β ≤ C (cid:16) M − β + L (cid:17) − β (3.41)15here we use the same C to denote all constants that independent of n . This completes theproof of (3.23). We now estimate the irrigation cost E W,α ( χ n ). In the dyadic irrigation plan χ n , let E nn bethe cost from P n to P n − , by (3.25), E nn = X x ni ∈P n Z √ dL n +1 (cid:0) W nn,i ( s ) (cid:1) α ds = X x ni ∈P n Z √ dL n +1 (cid:16) ( m ni ) − β + c (1 − β )( √ dL n +1 − s ) (cid:17) α − β ds . (3.42)Similarly, denote E nn − k the cost from P n − k to P n − k − , E nn − k = X x n − ki ∈P n − k Z √ dL n +1 − k (cid:0) W nn − k,i ( s ) (cid:1) α ds (3.43)From (3.39) and the non-decreasing of W nn − k,i ( s ), E nn − k ≤ X x n − ki ∈P n − k √ dL n +1 − k (cid:16) ( X i ≺ j m nj ) − β + c (1 − β ) √ dL n +1 − k (1 − − d (1 − β ) ) (cid:17) α − β ≤ X x n − ki ∈P n − k h CL ( P i ≺ j m nj ) α n +1 − k + CL α − β ( n +1 − k )( α − β +1) i = I n − k + J n − k (3.44)where C is some constant that only depends on α, β, c and on the dimension d . The cardinalityof P n − k is 2 ( n − k ) d . Therefore J n − k . = X x n − ki ∈P n − k CL α − β ( n +1 − k )( α − β +1) ≤ CL α − β ( n − k )(1+ α − β − d ) . (3.45)On the other hand, 1 ≥ α >
0, by elementary concavity inequality, I n − k . = X x n − ki ∈P n − k CL ( P i ≺ j m nj ) α n +1 − k ≤ ( n − k ) d (cid:16) P x n − ki ∈P n − k P i ≺ j m nj ( n − k ) d (cid:17) α CL n +1 − k ≤ CM α L ( n − k )[1 − d (1 − α )] . (3.46)When 1 ≥ α > − /d and 1 > β > − /d , one has1 − d (1 − α ) > , α − β − d > . (3.47)Therefore, using (3.44)-(3.46), E W,α ( χ n ) = n − X k =0 E nn − k ≤ n − X k =0 [ I n − k + J n − k ] ≤ C n X j =0 h LM α [1 − d (1 − α )] j + L α − β (1+ α − β − d ) j i ≤ C (cid:16) LM α + L α − β (cid:17) (3.48)where C is some constant independent of n . This completes the proof of (3.24).16y the previous results, when f ( z ) . = cz β in (2.3)-(2.5), with the conditions in Lemma 3.2, wehave the uniform bounds (3.24) for the dyadic irrigation plan sequence { χ n } n ≥ . Since each χ n is an admissible irrigation plan for µ n , by the definition (2.22), we have a uniform boundon all the irrigation costs I W,α ( µ n ), n ≥
1. By the weak convergence µ n ⇀ µ and the lowersemicontinuity of the irrigation cost, stated in Theorem 2.7, we conclude I W,α ( µ ) < + ∞ .By a comparison argument we can now prove the irrigability for a wide class of functions f and measures µ , with the weighted irrigation cost I W,α in (2.22).
Theorem 3.3
Let µ be a positive, bounded Radon measure in R d , with total mass M > andsupported in the cube Q of edge size L > . Assume α > − d , f satisfies (A1) and lim sup z → z − β f ( z ) < + ∞ (3.49) for some β > − d . Then I W,α ( µ ) < + ∞ . Proof.
The assumptions (3.49) and (A1) together imply that f ( z ) ≤ cz β for all z ∈ [0 , − β z ] ,f ( z ) ≤ cz for all z ∈ [ z , ∞ ) , (3.50)with some constants c, z >
0. We will prove that the weighted irrigation costs of the dyadicapproximated measures µ n , defined as in (3.3), are uniformly bounded. By Theorem 2.7, since µ n weakly converges to µ , this uniform bound implies the boundedness of I W,α ( µ ) . It suffices to prove the uniform bound for dyadic approximated measures µ n = P x ni ∈P n m ni δ x ni with n ≥ n , where n is some fixed integer. Choose n large enough such that in (3.40), c (1 − β ) √ dL n · (1 − − d (1 − β ) ) < z − β . (3.51)In the following, we construct the irrigation plan for µ n with uniformly bounded weightedcost. Consider first from P n to P n − . For those x ni such that m ni ≥ z , we transport the particlesat x ni along a straight path directly to the origin. Let S n be the set of all such pathes. Foreach path in S n , the multiplicity is larger than z and bounded by M . The length of path isbounded by √ dL . Let W ( t ) be the weight function on these pathes, then clearly W ( t ) ≥ z .By formula (2.3)-(2.5) and (3.50) the weight satisfies W ( t ) ≤ Z √ dLt f ( W ( s )) ds + M ≤ Z √ dLt cW ( s ) ds + M ≤ e c √ dL M . (3.52)On the other hand, for the remaining centers x ni , we transport the particles from P n to P n − ,using the branches of the dyadic irrigation plan χ n , defined as in Lemma 3.2. Notice on eachsuch branch, m ni < z . Then from (3.26) and (3.51), the weight W nn,i : [0 , √ dL n +1 ] R + on thebranch γ i from P n to P n − satisfies W nn,i ( s ) = (cid:16) ( m ni ) − β + c (1 − β ) √ dL n +1 (cid:17) − β ≤ ( z − β + z − β ) − β = 2 − β z , (3.53)17here we compute the weight W nn,i as ˙ W nn,i = c ( W nn,i ) β . Let W i be the corresponding solutionof (2.3), by (3.50) and comparision principle from ODE theory, W i ( s ) ≤ W nn,i ( s ) . (3.54)Then clearly the total cost on these dyadic branches from P n to P n − is bounded by E nn , givenin (3.42). From P n − to P n − . After removing the point masses transported by branches in S n , westill denote the remaining measure as µ n , and transport µ n to the centers in P n − , using thebranches from P n to P n − of the dyadic irrigation plan χ n . Notice that after removing themasses transported by branches in S n , m ni ≤ z for each 1 ≤ i ≤ nd , with some m ni = 0.For each center x n − i in P n − , when X j ∈O ( i ) m nj ≥ z (3.55)we then connect x n − i to the origin directly by a straight branch. Let S n − be the set ofall such branches. Similarly as in (3.52), the weight on each branch in S n − is bounded by e c √ dL M . For the remaining x n − i , we transport the flux from P n − to P n − , by the branchesof dyadic irrigation plan χ n . From (3.40) and (3.50)-(3.51), on each dyadic branch γ i from P n − to P n − , W i ( s ) ≤ W nn − ,i ( s ) < − β z , s ∈ [0 , √ dL n ] . (3.56)Then clearly the total cost on these dyadic branches from P n − to P n − is bounded by E nn − ,defined in (3.43). By backward induction we construct the irrigation plan until to the level P n . For each k > n , from P k to P k − , there are two types of paths, one is the branches in S k , and theother one is the dyadic branches of χ n . Clearly we have (cid:16) n [ k>n S k (cid:17) ≤ Mz (3.57)where M is the total mass of µ . Indeed, from our construction, each branch in ∪ nk>n S k willtransport distinct groups of particles with mass ≥ z , the total mass of µ n is M , thus wehave the upper bound in (3.57). For each branch in S k , there is an uniform bound (3.52) onthe weight W ( t ), and the length of each branch is bounded by √ dL , thus the total cost J onbranches in S k , k > n is bounded by J ≤ Mz · (cid:16) e c √ dLM (cid:17) α √ dL . = κ (3.58)On the other hand, the total cost I on the dyadic branches is bounded by I ≤ n X k>n E nk ≤ C (cid:16) M α L + L α − β (cid:17) . = κ (3.59)18here the last inequality comes from (3.24).Notice the bounds in (3.58)-(3.59) are independent of n , therefore, there exists a uniformconstant C >
0, such that for each dyadic approximation µ n , we have I W,α ( µ n ) ≤ C . Thanksto Theorem 2.7, we conclude that I W,α ( µ ) ≤ C . In the following we show some cases for measures µ with infinite weighted irrigation cost I W,α . Definition 3.4
Let µ be a positive, bounded measure in R d . If there exists γ > and aconstant C ≥ such that C r γ ≤ µ ( B ( x, r )) ≤ Cr γ , for all x ∈ supp ( µ ) , r ∈ [0 , , (3.60) then we say µ is Ahlfors regular in dimension γ . Here supp ( µ ) is the support of µ , B ( x, r ) isthe ball of radius r that centered at x . Remark 3.5
If a measure µ is Ahlfors regular in dimension γ , then one can prove supp ( µ ) has Hausdorff dimension γ . Indeed, consider any covering ∪ ∞ i =1 B ( x i , r i ) of supp ( µ ) , consistsof closed balls with radius less than 1. From the second inequality in (3.60) one has X i ≥ ( r i ) γ ≥ X i ≥ µ ( B ( x i , r i )) C ≥ MC > , which implies supp ( µ ) has Hausdorff dimension ≥ γ . On the other hand, by the Vitali’sConvering Theorem[7], there exists a countable subcollection of disjoint B ( x i , r i ) , which we stilldenote as P ∞ i =1 B ( x i , r i ) , such that supp ( µ ) ⊆ ∪ ∞ i =1 B ( x i , r i ) . Then from the first inequalityin (3.60), since B ( x i , r i ) are disjoint, X i ≥ (5 r i ) γ = 5 γ X i ≥ r γi ≤ γ C X i ≥ µ ( B ( x i , r i )) ≤ β CM, and it implies the Hausdorff dimension of supp ( µ ) ≤ γ . For the irrigation cost I α ( · ) without weights that defined in [10], we recall the followingtheorem. For a proof, see Theorem 1.2 in [10]. Theorem 3.6
Let µ be a finite α -irrigable measure, with α ∈ (0 , . That is, I α ( µ ) < ∞ .Then there is a Borel set E ⊆ R d , µ ( R d \ E ) = 0 , such that for any s > − α , H s ( E ) = 0 , where H s ( E ) is the s -Hausdorff measure of the set E . In other words, if µ is α -irrigable, then µ is concentrated on a set E with Hausdorff dimension ≤ − α . Remark 3.7
As mentioned in [4], for any bounded Radon measure µ , we always have I W,α ( µ ) ≥I α ( µ ) . Therefore, if I W,α ( µ ) < + ∞ , from Theorem 3.6, µ is concentrated on a set E withHausdorff dimension ≤ − α . emma 3.8 Let χ be an irrigation plan of µ with finite weighted irrigation cost E W,α ( χ ) < ∞ .Then for any r > , µ ( R d \ B (0 , r )) ≤ (cid:18) E W,α ( χ ) r (cid:19) α . (3.61) Proof.
The function x (cid:16) x − β + c (1 − β )( r − t ) (cid:17) − β , x ∈ R + is concave. Let m r . = µ ( R d \ B (0 , r )), then by definition we have Z r (cid:16) m − βr + c (1 − β )( r − t ) (cid:17) α − β dt ≤ E W,α ( χ ) . (3.62)Since r − t ≥
0, it implies that (cid:16) m − βr (cid:17) α − β · r = m αr · r ≤ E W,α ( χ ) , which completes the proof of (3.61). Theorem 3.9
Let µ be a positive, bounded Radon measure in R d and Ahlfors regular indimension d . Let f satisfy (A1) .If either α < − d or lim inf s → s − β f ( s ) > for some β < − d − , then I W,α ( µ ) = + ∞ . Proof.
CASE 1: If α < − d , then − α < d . Suppose I W,α ( µ ) < + ∞ , by Remark 3.7, µ isconcentrated on a set E with Hausdorff dimension ≤ − α < d , which is a contradiction to theassumption that µ is Ahlfors regular in dimension d . Thus, we have I W,α ( µ ) = + ∞ .CASE 2: The assumption (3.63) implies that, for some constants c, s > f ( s ) ≥ cs β for all s ∈ [0 , s ] . (3.64)Since µ is Ahlfors regular in dimension d , then for each irrigation plan χ , there are O ( δ d )disjoint cubes with diameter δ and each of them has measure ≈ δ d . In each cube, the lowerbound for the cost is Z δ (cid:16) δ d (1 − β ) + c (1 − β )( δ − t ) (cid:17) α − β dt (3.65)and the total number of such disjoint cubes is δ d . E W,α ( χ ) ≥ δ d Z δ (cid:16) δ d (1 − β ) + c (1 − β )( δ − t ) (cid:17) α − β dt ≥ δ d Z δ (cid:16) c (1 − β )( δ − t ) (cid:17) α − β dt ≥ Cδ α − β − β − d (3.66)20here C is some constant independent of δ . Since 1 ≥ α > − d , − d − > β >
0, wehave α − β − β < d . Sending δ to 0+, the right hand side in (3.66) goes to + ∞ . Thus, for anyirrigation plan χ of µ , E W,α ( χ ) = + ∞ , thus I W,α ( µ ) = + ∞ . Acknowledgement
This research was partially supported by NSF grant DMS-1714237,”Models of controlled biological growth”. The author wants to thank his thesis advisor Pro-fessor Alberto Bressan for his many useful comments and suggestions.
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