A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces

Abstract

The present paper presents a counterexample to the sequential weak density of smooth maps between two manifolds $\mathcal {M}$ and $\mathcal {N}$ in the Sobolev space $W^{1, p} (\mathcal {M}, \mathcal {N})$, in the case p is an integer. It has been shown (see e.g. Bethuel in Acta Math 167:153–206, 1991) that, if $p<\dim \mathcal {M}$ is not an integer and the [p]-th homotopy group $\pi _{[p]}(\mathcal {N})$ of $\mathcal {N}$ is not trivial, [p] denoting the largest integer less then p, then smooth maps are not sequentially weakly dense in $W^{1, p} (\mathcal {M}, \mathcal {N})$. On the other hand, in the case $p< \dim \mathcal {M}$ is an integer, examples of specific manifolds $\mathcal {M}$ and $\mathcal {N}$ have been provided where smooth maps are actually sequentially weakly dense in $W^{1, p} (\mathcal {M}, \mathcal {N})$ with $\pi _{p}(\mathcal {N})\not = 0$, although they are not dense for the strong convergence. This is the case for instance for $\mathcal {M}=\mathbb {B}^m$, the standard ball in $\mathbb {R}^m$, and $\mathcal {N}=\mathbb {S}^p$ the standard sphere of dimension p, for which $\pi _{p}(\mathcal {N}) =\mathbb {Z}$. The main result of this paper shows however that such a property does not holds for arbitrary manifolds $\mathcal {N}$ and integers p. Our counterexample deals with the case $p=3$, $\dim \mathcal {M}\ge 4$ and $\mathcal {N}=\mathbb {S}^2$, for which the homotopy group $\pi _3(\mathbb {S}^2)=\mathbb {Z}$ is related to the Hopf fibration. We explicitly construct a map which is not weakly approximable in $W^{1,3}(\mathcal {M}, \mathbb {S}^2)$ by maps in $C^\infty (\mathcal {M}, \mathbb {S}^2)$. One of the central ingredients in our argument is related to issues in branched transportation and irrigation theory in the critical exponent case, which are possibly of independent interest. As a byproduct of our method, we also address some questions concerning the $\mathbb {S}^3$-lifting problem for $\mathbb {S}^2$-valued Sobolev maps.

Appendix: related notions on branched transportation

In this “Appendix”, we recall and recast some aspects of branched transportation, an optimization problem involved in a wide area of applications, including practical ones, for instance leafs growth, or network design. We focus on questions directly related to our main problem, keeping however this part completely self-contained.

Branched transportation appears when one seeks to optimize transportation costs when the average cost decreases with density. Consider a finite set A of points belonging to the closure of a bounded open domain $\Omega$ of $\mathbb {R}^m$: we wish to connect (or transport) them to the boundary $\partial \Omega$. The total cost be to be minimized is the sum of the length of paths joining the given points to the boundary multiplied by a density function$\varphi$, depending on the density representing the number of points using the same portion of paths. For minimizers, such paths are unions of segments, but possibly with varying densities. The intuitive idea is that, when $\varphi$ is sublinear, it is cheaper to share the same path than to travel alone, so that high densities are selected by the minimization process. This induces branching points, i.e. points where segments join to induce higher multiplicity. The density function appearing in our context, as well as in a large of part of the literature, is given by the power law $\displaystyle {\varphi (d)=d^\upalpha }$, with given parameter $0<\upalpha < 1$. Notice that $\varphi$ is sublinear, $\left( d_1+d_2\right) ^\alpha<<d_1^\alpha + d_2^\alpha$ for large numbers. Our aim is to describe the behavior of minimal branched transportation when the number of points increases and ultimately goes to$+ \infty$. Special emphasis is put on the critical case $\upalpha =\upalpha _m=1-1/m$. Our presentation closely follows [29, 30] and also [4]: we perform the necessary adaptation for connections to the boundary, which have been less considered so far. As far as we are aware of, the main result of this “Appendix”, presented in Theorem A.1, is new.

1.1 Directed graphs connecting a finite set to the boundary

1.1.1 Directed graphs and charges

The theory of oriented graphs offers an appropriate framework to describe the objects we have in mind.²⁰ Such oriented graphs involve:

• Points. These points are of two kinds: the points in A we wish to connect to the boundary, but also additional points, the branching points and points on the boundary.

• Oriented segments. These segments join points described points above. Orientation is important, as well as multiplicity, which is a positive integer.

A general directed graphs G is defined by a finite collection E(G) of oriented segments with endpoints belonging to$\overline{\Omega }$: if e is a segment in E(G), then we denote by $e^-$ and $e^+$ the endpoints of e, $e^-$ (resp $e^+$) denoting the entrance point (resp the exit point), so that $e=[e^-, e^+]$ and $\partial e=\{e^-, e^+\}$. We assume that for any segment e in E(G) the additional condition

$$\begin{aligned} \mathrm{if \ } [e^-, e^+] \in E(G), \mathrm{\ then \ } [e^+, e^-] \not \in E(G) \end{aligned}$$

(A.1)

holds, i.e. if an oriented segment belongs to the graph, the segment with opposite direction does not. Segments may be repeated with multiplicity. If $e\in E(G)$, we denote by $d(e, G)\in \mathbb {N}^\star$ its multiplicity²¹ and simply write d(e) is this is not a source of confusion. We denote by ${\mathcal {G}}(\Omega )$ the set of graphs having the previous properties, namely

$$\begin{aligned} {\mathcal {G}}(\Omega )=\left\{ \mathrm{graphs \ } G \mathrm{\ such \ that \ } \mathrm{(A.1)} \mathrm{\ holds } \right\} . \end{aligned}$$

We denote by V(G) be set of vertex of the graph G, i.e.

$$\begin{aligned} V(G)=\underset{e\in E(G)}{\cup }\partial e=\underset{e\in E(G)}{\cup }\{e^-, e^+\}\subset \overline{ \Omega }. \end{aligned}$$

Given a vertex $\upsigma \in V(G)$, we set

$$\begin{aligned} E^\pm (\upsigma , G)= \{ e\in E(G), e^\mp =\upsigma \} \mathrm{\ and \ } E(\upsigma , G)=E^+(\upsigma , G) \cup E^-(\upsigma , G), \end{aligned}$$

so that $E^+ (\upsigma , G)$ (resp $E^- (\upsigma , G)$) represents the set of segments of G having $\upsigma$ as entrance point (resp. exit point) and $E(\upsigma , G)$ the set of segments having $\upsigma$ as endpoint. We set

$$\begin{aligned} \sharp \left( E^\pm (\upsigma , G)\right) =\underset{e\in E^\pm (\upsigma , G)}{\sum }d(e) \in \mathbb {N}^\star , \end{aligned}$$

and introduce the notion of charge of a point $\upsigma \in V(G)$ as

$$\begin{aligned} \mathrm{Ch_g}(\upsigma , G)=\sharp \left( E^+(\upsigma , G)\right) -\sharp \left( E^-(\upsigma , G)\right) \in {\mathbb {Z}} . \end{aligned}$$

(A.2)

We consider the subsets $V_0(G)$, ${V_{\mathrm{chg}}}(G)$ and ${V_{\mathrm{bd}}}(G)$ of V(G) defined by

$$\begin{aligned} \left\{ \begin{aligned} V_0(G)&=\{\upsigma \in V(G), \mathrm{Ch_g}(\upsigma , G)=0\}\\ {V_{\mathrm{chg}}}(G)&=\{\upsigma \in V(G), \mathrm{Ch_g}(\upsigma ,G)\ne 0, \upsigma \in V(G){\setminus }( V_0(G)\cup \partial \Omega ) \} \\ {V_{\mathrm{bd}}}(G)&=\{\upsigma \in V(G), \upsigma \in \partial \Omega \}. \end{aligned} \right. \end{aligned}$$

(A.3)

A point $\upsigma \in V_0(G)$ will be termed a pure branching point, a point in $V_{\mathrm{chg}}(G)$ a charged point or simply a charge.²² The set of graphs with only positive charges plays a distinguished role in the later analysis. We set

$$\begin{aligned} \left\{ \begin{aligned} {\mathcal {G}}^+(\Omega )&=\left\{ G\in {\mathcal {G}}(\Omega ), \mathrm{\ s.t.\ }\mathrm{Ch_g}(\upsigma , G)\ge 0 \ \forall \upsigma \in V(G){\setminus } \partial \Omega \right\} \ \\ {\mathcal {G}}_0(\Omega )&=\left\{ G\in {\mathcal {G}}(\Omega ), \mathrm{\ s.t.\ }\mathrm{Ch_g}(\upsigma , G)= 0 \ \forall \upsigma \in V(G){\setminus } \partial \Omega \right\} . \end{aligned} \right. \end{aligned}$$

(A.4)

In several places, we will invoke the fact that, if $G\in {\mathcal {G}}^+(\Omega )$, then

$$\begin{aligned} E^+(\upsigma , G) \ne \emptyset , \mathrm{\ for \ any \ } \upsigma \in V(G). \end{aligned}$$

(A.5)

Indeed, by definition $E(\upsigma , G)$ contains at least one element, and since the charge is positive there are at least as many elements in $E^+(\upsigma , G)$ as in $E^-(\upsigma , G)$.

1.1.2 Elementary operations on directed graphs

Gluing graphs. Let $G_1$ and $G_2$ be two graphs in ${\mathcal {G}} (\Omega )$. We assume furthermore that

$$\begin{aligned}&\mathrm{if \ } e_1\in E(G_1), \ e_2 \in E(G_2)\mathrm{\ then \ either \ } e_1=e_2, \nonumber \\&\quad \mathrm{\ or \ } e_1\cap e_2 \mathrm{\ contains \ at \ most \ one \ point.} \end{aligned}$$

(A.6)

If condition (A.6) is not met, one may add new points and divide some segments in two so that the transformed graphs satisfy the condition. Given $e \in E(G)$, we denote $-e$ the segment with opposite orientation, i.e. if $e=[e^-, e^+]$, then $-e\equiv [e^+ , e^-]$. We introduce the following subsets of $E(G_1)\cup E(G_2)$

$$\begin{aligned} \left\{ \begin{aligned} E_0(G_1, G_2)&\equiv \{e \in G_1 \mathrm{\ s.t. \ }, -e\in G_2 \mathrm{\ with \ } d(e, G_1)=d(-e, G_2) \}\\ E^+(G_1, G_2)&\equiv \{e \in G_1 \mathrm{\ s.t. \ } -e \not \in G_2 \} \cup \{e \in G_2 \mathrm{\ s.t. \ } -e \not \in G_1 \} \\ E^\pm (G_1, G_2)&\equiv \{e \in G_1 \mathrm{\ s.t. \ } -e \in G_2 \mathrm{\ with \ } d(e,G_1)>d(-e,G_2)\} \\ E^\mp (G_1, G_2)&\equiv \{e \in G_2 \mathrm{\ s.t. \ } -e \in G_1 \mathrm{\ with \ } d(e,G_2) >d(-e,G_1)\} \end{aligned} \right. \end{aligned}$$

We define the glued graph

$$\begin{aligned} G=G_1 \curlyvee G_2 \in {\mathcal {G}} (\Omega ), \end{aligned}$$

(A.7)

given by the set of its directed segments

$$\begin{aligned} \begin{aligned} E(G)&\equiv E(G_1)\cup E(G_2){\setminus } E_0(G_1,G_2) \\&= E^+(G_1, G_2)\cup E^\pm (G_1, G_2)\cup E^\mp (G_1, G_2), \end{aligned} \end{aligned}$$

(A.8)

with, for $e\in E(G)$, multiplicities d(e, G) given by

$$\begin{aligned} \left\{ \begin{aligned} d(e, G)&=d(e, G_1)+d(e, G_2) \mathrm{\ if \ } e \in E^+(G_1,G_2)\\ d(e, G)&=d(e, G_1)-d(e, G_2) \mathrm{\ if \ } e \in E^\pm (G_1,G_2)\\ d(e, G)&=d(e, G_2)-d(e, G_1) \mathrm{\ if \ } e \in E^\mp (G_1,G_2), \end{aligned} \right. \end{aligned}$$

(A.9)

where we have used the convention, for $i=1, 2$, that $d(e, G_i)=0$ if $e \not \in G_i$. The vertex set $V(G_1\curlyvee G_2)$ is then provided by the endpoints of the segments in $E(G_1\curlyvee G_2)$, so that $\displaystyle {V(G) \subset V(G_1) \cup V(G_2)}$. The inclusion might be strict in the general case. We have:

Proposition A.1

Let $\upsigma \in V(G_1\curlyvee G_2)$. We have

$$\begin{aligned} \mathrm{Ch_g}(\upsigma , G_1 \curlyvee G_2)= \mathrm{Ch_g}(\upsigma , G_1)+ \mathrm{Ch_g}(\upsigma , G_2), \end{aligned}$$

(A.10)

with the convention, for $i=1, 2$, that $\mathrm{Ch_g}(\upsigma , G_i)=0$ if $\upsigma \not \in G_i$. If $G_i \in {\mathcal {G}}^+(\Omega )$ for $i=1, 2$, then we have

$$\begin{aligned} {V_{\mathrm{chg}}}(G)= {V_{\mathrm{chg}}}(G_1) \cup {V_{\mathrm{chg}}}(G_2). \end{aligned}$$

(A.11)

The result is a direct consequence of (A.9). We reader may check also that the gluing operation $\curlyvee$ enjoys classical properties as commutativity and associativity. Finally we write

$$\begin{aligned} G=G_1{\overset{\star }{\curlyvee }}G_2 \end{aligned}$$

(A.12)

in the case when, if a segment e belongs to $E(G_1)$, then the opposite segment does not belong to $G_2$, so that no cancellations for segments occur in the gluing process. The set E(G) is in that case the union $E(G_1)\cup E(G_2)$, the multiplicities being simply summed.

Subgraphs. Let $G_1$ and G be two graphs in ${\mathcal {G}}(\Omega )$. We say that $G_1$ is a subgraph of G if $E(G_1) \subset E(G)$ and if the multiplicities satisfy the conditions

$$\begin{aligned} d(e, G_1)\le d(e, G), \mathrm{\ for \ any \ segment \ } e \in E(G_1). \end{aligned}$$

(A.13)

If the two previous conditions are fullfilled, then we write $G_1\Subset G$. We introduce the complement $G_2=G{\smallsetminus } G_1$ of $G_1$ with respect to G, defining the set of oriented segments of $G_2$ as

$$\begin{aligned} E( G{\smallsetminus } G_1)=E(G_2) \equiv \left[ E(G){\setminus } E(G_1) \right] \cup E_{\mathrm{comp}}(G_1, G) \end{aligned}$$

where $E_{\mathrm{comp}}(G_1, G)$ is defined as

$$\begin{aligned} E_{\mathrm{comp}}(G_1, G)\equiv \{ e \in E(G_1), d(e, G_1)<d(e, G)\}, \end{aligned}$$

and with multiplicities given by

$$\begin{aligned} \left\{ \begin{aligned} d(e, G{\smallsetminus } G_1)&=d(e, G), \mathrm{\ if \ } e \in E(G){\setminus } E(G_1),\\ d(e, G{\smallsetminus } G_1)&=d(e, G)-d(e, G_1), \mathrm{\ if \ } e \in E_{\mathrm{comp}}(G_1, G). \end{aligned} \right. \end{aligned}$$

(A.14)

Notice that there are no segments in $G_1$ and $G_2$ with opposite orientations. It follows from these definitions that

$$\begin{aligned} G=G_1 {\overset{\star }{\curlyvee }}G_2=G_1{\overset{\star }{\curlyvee }}(G{\smallsetminus } G_1). \end{aligned}$$

In view of Proposition A.1 , if G and $G_1$ belong to ${\mathcal {G}}^+(\Omega )$ with $G_1 \Subset G$, and if furthermore

$$\begin{aligned} \mathrm{Ch_g}(\upsigma , G_1)\le \mathrm{Ch_g}(\upsigma , G), \mathrm{\ for \ any \ } \upsigma \in {V_{\mathrm{chg}}}(G), \end{aligned}$$

(A.15)

then $G_2 \in {\mathcal {G}}^+(\Omega )$. If $G_1$ and G are two graphs in ${\mathcal {G}}(\Omega )$ such that $G_1\Subset G$ and such that condition (A.15) is satifies, then we write $G_1 \overset{\star }{\Subset }G$.

Restrictions of graphs to subdomains. Let $\Omega _1 \subset \Omega$ be a subdomain of $\Omega$ and assume for the sake of simplicity (and also for further applications) that both $\Omega _1$ and $\Omega$ are polytopes. Let G be a graph in ${\mathcal {G}}(\Omega )$. We define the restriction of G to $\Omega _1$ as the graph in ${\mathcal {G}}(\Omega _1)$ such that its set of segments is given by

Its set of vertices is then given by

$$\begin{aligned} V(G_1)=\left( V(G)\cap \Omega _1\right) \cup \left( \underset{ e\in E(G)}{\cup }\partial \left( {\bar{e}} \cap \bar{\Omega } _1\right) \right) . \end{aligned}$$

One may check that $G_1\in {\mathcal {G}}(\Omega _1)$ and also $G_1\in {\mathcal {G}}(\Omega )$. If we assume moreover that $G\in {\mathcal {G}}^+(\Omega )$, then we have $G\in {\mathcal {G}}^+(\Omega _1)$, but it does not belong, in general to ${\mathcal {G}}^+(\Omega )$, since negative charges may be created on $\partial \Omega _1$.

1.1.3 The single path property

The next property, termed the single path property, has been considered in [4, 29, 30].

Definition A.1

Let $G \in {\mathcal {G}}(\Omega )$. We say that G possesses the single path property, if for any vertex $\upsigma \in V(G)\cap \Omega$ there is at most one segment e in E(G), possibly repeated with multiplicity, such that $\upsigma$ is the entrance point of e, that is $E^+(\upsigma , G)$ is either a singleton or is empty.

In other words, if G possesses the single path property, then there might be several segments ending at the same vertex, but at most one starting from it. This property possibly models some intuitive features, as for instance in river networks. We denote by ${\mathcal {G}}_{\mathrm{sp}}(\Omega )$ (resp. ${\mathcal {G}}_{\mathrm{sp}}^+(\Omega )$) the set of all graphs in ${\mathcal {G}}(\Omega )$ (resp. ${\mathcal {G}}^+ (\Omega )$) which possess the single path property. Notice that if $G \in {\mathcal {G}}^+_{\mathrm{sp}}(\Omega )$, then $E^+(\upsigma , G)$ is not empty for $\upsigma \in \Omega$, so that it is necessarily a singleton.

1.1.4 Threads, loops and bridges

A heuristic image of the notion of thread we describe next, is provided by a a curve for with one end is given by a point in $\overline{\Omega }$, reaching to the boundary $\partial \Omega$, and constructed using only segments in E(G). This image suggests the following definition.

Definition A.2

A directed graph G is said to be a polygonal curve in $\Omega$, in short a $\mathrm{P}_{\Omega }$-curve, if and only if there exists an ordered collection $B=(b_1, \ldots , b_{\mathrm{q}})$ of q not necessary distinct points in ${\bar{\Omega }}$ such that G satisfies $V(G)=B$, relation (A.1) holds, and

$$\begin{aligned} E(G)=\{[b_i, b_{i+1}],\mathrm{\ with \ multiplicity \ } 1, i=1, \ldots , \mathrm{q}\} \mathrm{\ and \ } b_\mathrm{q}\in \partial \Omega \mathrm{\ or \ } b_\mathrm{q}=b_1. \end{aligned}$$

(A.16)

Since the $\mathrm{P}_{\Omega }$-curve G is completely determined by the orderet set B, we may set

$$\begin{aligned} G= \mathrm{G}_{\mathrm{rp}} (B). \end{aligned}$$

Even if in (A.16) each segment $[b_i, b_{i+1}]$ appears with multiplicity one, the same segment may be repeated further in the sequence, so that its final multiplicity might be larger that one.

Definition A.3

Let $G= \mathrm{G}_{\mathrm{rp}} (B)$ be a $\mathrm{P}_{\Omega }$-curve. We say that $\mathrm{G}_{\mathrm{rp}} (B)$ is

• a loop if $b_1=b_{\mathrm{q}}$.

• a bridge if $b_1\in \partial \Omega$ and $b_\mathrm{q}\in \partial \Omega$.

• A thread emanating from a point $p \in \Omega$ if $p=b_1$ and $b_{\mathrm{q}} \in \partial \Omega$.

A first elementary observation is:

Lemma A.1

Let $B=(b_1, \ldots , b_{\mathrm{q}})$ of q be an ordered collected of points in ${\bar{\Omega }}$. We have $\mathrm{Ch_g}(b_i, \mathrm{G}_{\mathrm{rp}} (B))=0$ for $i=2, \ldots , q-1$ and for $i=1, q$ if $\mathrm{G}_{\mathrm{rp}} (B)$ is a loop. If $\mathrm{G}_{\mathrm{rp}} (B)$ is not a loop, then we have

$$\begin{aligned} \mathrm{Ch_g}(b_1, \mathrm{G}_{\mathrm{rp}} (B))=1 \mathrm{\ and \ } \mathrm{Ch_g}(b_q, \mathrm{G}_{\mathrm{r}} (B))=-1. \end{aligned}$$

(A.17)

Proof

Given $b_i$ in B, we denote by m(i) its multiplicity: if $i\in \{2, q-1\}$, then $b_i$ is m(i) times an entrance point as well as an exit point, so that the first assertion follows. The others follow similar arguments. $\square$

We denote by ${\mathcal {T}}_{\mathrm{hread}}(p, \Omega )$ the set of all threads emanating from p. We notice that

$$\begin{aligned} \left\{ \begin{aligned} {V_{\mathrm{chg}}}(\mathrm{G}_{\mathrm{rp}}(B))&=\emptyset \mathrm{\ when \ } \mathrm{G}_{\mathrm{rp}}(B) \mathrm{\ is \ a \ loop \ or \ a \ bridge}, \\ {V_{\mathrm{chg}}}(\mathrm{G}_{\mathrm{rp}}(B))&=\{a\} \mathrm{\ with \ } \mathrm{Ch_g}(a)=1,\mathrm{\ if \ } \mathrm{G}_{\mathrm{rp}}(B) \mathrm{\ is \ in \ } {\mathcal {T}}_{\mathrm{hread}}(p, \Omega ). \end{aligned} \right. \end{aligned}$$

(A.18)

Loops and bridges are elements in ${\mathcal {G}}_0(\Omega )$. We denote by ${\mathcal {L}}_{\mathrm{oop}} (\Omega )$ the set of loops. We say that a graph G has a loop if there exists a loop L such that $L \Subset G$. In particular a thread $G= \mathrm{G}_{\mathrm{rp}} (B)$ has a loop if there exists a subset formed of consecutive points in B yielding a loop. Given a point $p \in \Omega$ we denote by

the set of all threads without loops emanating from p. It follows rather straightforwardly from the definitions that the segments of a thread in have multiplicity one and that, if a thread has the single path property, then it has no loops. One may moreover verify:

Lemma A.2

Let $\upsigma \in \Omega$ and let $T \in {\mathcal {T}}_\mathrm{hread }(\upsigma , \Omega )$. There exists a finite family $(L_j)_{j \in J}$ of loops such that

(A.19)

Proof

We may write $T=\mathrm{G}_{\mathrm{rp}}(B)$ where B denotes an ordered set $B=\{b_1=\upsigma , b_2, \ldots , b_{\mathrm{q}}\}$, with $b_{\mathrm{q}} \in \partial \Omega$. If all points in B are distinct, then and there is nothing to prove. Otherwise there are two points, say $b_{i_1}$ and $b_{i_2}$ with $1\le i_1<i_2 <b_{\mathrm{q}}$ which are identical. Then, we set $L_1=\mathrm{G}_{\mathrm{rp}}\{b_{i_1}, \ldots , b_{i_2}=b_{i_1}\}$ and ${{\tilde{T}}}_1=\mathrm{G}_{\mathrm{rp}}\{b_1, \ldots , b_{i_1}, b_{i_2+1}, \ldots , b_{\mathrm{q}}\}$. We verify that

$$\begin{aligned} T={\tilde{T}}_1 {\overset{\star }{\curlyvee }}L_1, \mathrm{\ with \ } {\tilde{T}}_1 \in {\mathcal {T}}_{\mathrm{hread}}(p, \Omega ) \mathrm{\ and \ } L_1 \mathrm{\ is \ a \ loop}. \end{aligned}$$

If ${\tilde{T}}_1$ has no loop, then we are done. Otherwise, we start the process again with T replaced by ${\tilde{T}}_1$. It stops in a finite number of iterations, since the number of points is finite. $\square$

1.1.5 Maximal subcurves, subthreads and subloops

Consider a graph G in ${\mathcal {G}}(\Omega )$ and an ordered set $B=(b_1, \ldots , b_{\mathrm{q}})$ of elements of V(G).

Definition A.4

The $P_\Omega$-curve $\mathrm{G}_{\mathrm{rp}}(B)$ is said to be a maximal subcurve of G if $\mathrm{G}_{\mathrm{rp}}(B)\Subset G$, if $b_i \in \Omega$ for $i=1, \ldots , b_{\mathrm{q}-1}$ and either $b_\mathrm{q}\in \partial \Omega$ or $E^+\left( b_\mathrm{q}, G {\setminus } \mathrm{G}_{\mathrm{rp}}(B)\right)$ is empty.

If $\mathrm{G}_{\mathrm{rp}}(B)$ is a maximal subcurve of G and $b_{\mathrm{q}}\in \Omega$, then we have $E^+\left( b_\mathrm{q}, G {\setminus } \mathrm{G}_{\mathrm{rp}}(B)\right) =\emptyset$, which implies that $\mathrm{Ch_g}(b_\mathrm{q}, G{\setminus } \mathrm{G}_{\mathrm{rp}}(B))\le 0$, and hence

$$\begin{aligned} \mathrm{Ch_g}(b_\mathrm{q}, G)=\mathrm{Ch_g}(b_\mathrm{q}, G{\setminus } \mathrm{G}_{\mathrm{rp}}(B))+ \mathrm{Ch_g}(b_\mathrm{q}, \mathrm{G}_{\mathrm{rp}}(B))\le \mathrm{Ch_g}(b_\mathrm{q}, \mathrm{G}_{\mathrm{rp}}(B)). \end{aligned}$$

(A.20)

Our next result readily follows from the definition:

Lemma A.3

Let $G \in {\mathcal {G}}(\Omega )$ and $\upsigma \in V(G)$. There exists an ordered set $B=(b_1, \ldots , b_{\mathrm{q}})$ such that $b_1=\upsigma$ and such that $\mathrm{G}_{\mathrm{rp}}(B)$ is a maximal subcurve of G.

Proof

We construct the maximal subcurve inductively. If $E^+(\upsigma , G)=\emptyset$, we take $B=\{\upsigma \}$ and we are done. Otherwise, we choose some $b_2\in E^+(\upsigma , G)$, so that $[\upsigma , b_2] \in E^+(\upsigma , G)$ and therefore $\mathrm{G}_{\mathrm{rp}}\{\upsigma , b_2\} \Subset G$. If $b_2 \in \partial \Omega$ or if $E^+(b_2, G)=\emptyset$, then $B\equiv \{\upsigma , b_2\}$ is maximal and we are done. Otherwise, since $E^+(b_2, G)$ is not empty, there exists some point $b_3 \in V(G)$ such that $[b_2, b_3] \in E^+(b_2, G{\setminus } \mathrm{G}_{\mathrm{rp}}\{\upsigma , b_2\})$ and therefore $\mathrm{G}_{\mathrm{rp}}\{\upsigma , b_2, b_3\} \Subset G$. If $b_3 \in \partial \Omega$ or if $E^+(b_2, G{\setminus } \mathrm{G}_{\mathrm{rp}}\{\upsigma , b_2\})$, then $B\equiv \{\upsigma , b_2, b_3\}$ is maximal and we are done. Otherwise we go on, until we reach the boundary or have no more segments available to go on. $\square$

Lemma A.4

Let $G\in {\mathcal {G}}^+(\Omega )$. A maximal subcurve $\mathrm{G}_{\mathrm{rp}}(B)$ of G is either a thread emanating from a point $\upsigma \in {\bar{\Omega }}$ or a loop.

Proof

If $b_{\mathrm{q}} \in \partial \Omega$, then $\mathrm{G}_{\mathrm{rp}}(B)$ is a thread emanating from $\upsigma$ and the statement is proved. If $b_{\mathrm{q}} \in \Omega$, we argue by contradiction, and assume that $\mathrm{G}_{\mathrm{rp}}(B)$ is not a loop, so that $b_1\not = b_\mathrm{q}$. It then follows from Lemma A.1 that $\mathrm{Ch_g}(b_\mathrm{q}, \mathrm{G}_{\mathrm{rp}}(B))=-1$. Hence, invoking (A.20), we obtain $\mathrm{Ch_g}(b_\mathrm{q}, G)\le -1$. This is a contradiction with the fact that $G \in {\mathcal {G}}^+(\Omega ))$. $\square$

Lemma A.5

Let $B=(b_1, \ldots , b_{\mathrm{q}})$ be such that $\mathrm{G}_{\mathrm{rp}}(B)$ is a maximal subcurve of G. If $\mathrm{Ch_g}(b_1, G)>0$, then $\mathrm{G}_{\mathrm{rp}}(B)$ is a thread emanating from $b_1$.

Proof

We argue by contradiction: if the statement is not true, it follows from Lemma A.3 that $\mathrm{G}_{\mathrm{rp}}(B)$ is a loop, so that $b_1=b_\mathrm{q}$, and in view of lemma A.1, $\mathrm{Ch_g}(b_1, \mathrm{G}_{\mathrm{rp}}(B))=0$. Invoking (A.20), we derive that $\mathrm{Ch_g}(b_1, G)=\mathrm{Ch_g}(b_q, G)\le 0$, in contradiction with the assumption $\mathrm{Ch_g}(b_1, G)>0$. $\square$

Combining Lemmas A.2, A.3, A.4 and A.5, we deduce:

Corollary A.1

Let $\upsigma \in {V_{\mathrm{chg}}}{G}$. There exists a thread such that $T_\upsigma \overset{\star }{\Subset }G$.

1.1.6 Decomposing graphs into threads and loops and bridges

Consider a graph $G \in {\mathcal {G}}^+(\Omega )$. Since V(G) is a finite set, we may write

$$\begin{aligned} {V_{\mathrm{chg}}}(G)=\{p_1, \ldots , p_{_{\ell _{\mathrm{c}}}}\}, \end{aligned}$$

each point $p_i$ in the collection having multiplicity $\mathrm{M}_i\equiv \mathrm{Ch_g}(p_i, G)\in \mathbb {N}^*$. The following result emphasizes the importance of threads in this context:

Proposition A.2

Let $G \in {\mathcal {G}}^+(\Omega )$. We may decompose the graph G as

(A.21)

If moreover $G \in {\mathcal {G}}_{\mathrm{sp}}^+(\Omega )$, that is if G possesses the single path property, then decomposition (A.21) is unique and, for any $i\in \{1, \ldots , \ell _{\mathrm{c}}\}$, we have

$$\begin{aligned} T_{i, j}=T_{i, j'}, \mathrm{\ for \ } j \mathrm{\ and \ } j' \mathrm{\ in \ } \{1, \ldots , \mathrm{M}_i\}. \end{aligned}$$

(A.22)

Proof

We present first the construction of the subgraph $T_{1, 1}$ and then proceed recursively.

Step 1: Construction of$T_{1, 1}$. Since the point $p_1$ has positive charge $\mathrm{M}_1$ with respect to G, we may apply Corollary A.1 and choose $T_{1, 1}=T_{p_1}$, to define the graph

$$\begin{aligned} G_{1, 1}= G{\setminus } { T}_{1, 1}, \ \mathrm{and \ hence \ } G=G_{1, 1} {\overset{\star }{\curlyvee }}\, {T}_{1, 1}, \mathrm{\ with \ } G_1\in {\mathcal {G}}^+(\Omega ). \end{aligned}$$

the total charge of $G_{1,1}$ compared to the total charge of G has decreased by 1. More precisely, it follows from the rules (A.14) and (A.18) for charges that for $i=2, \ldots , \ell _{\mathrm{c}}$, we have

$$\begin{aligned}&\mathrm{Ch_g}(p_i, G_{1, 1})=\mathrm{Ch_g}(p_i, G), \mathrm{\ for \ } i=2, \ldots , \ell _{\mathrm{c}}, \\&\quad \mathrm{\ and \ } \mathrm{Ch_g}(p_1, G_{1, 1})=\mathrm{Ch_g}(p_1, G)-1, \end{aligned}$$

in case $p_1\in V(G_{1, 1})$, which occurs in particular in $p_1$ has multiplicity. In the case $\ell _{\mathrm{c}}=1$ and $\mathrm{M}_1=1$, we deduce that $G_{1, 1}\in {\mathcal {G}}_0(\Omega )$, so that setting $T_0=G_{1, 1}$, we obtain (A.21). Otherwise, we proceed recursively.

Step 2: Iterating the construction. We proceed as in step 1, but with G replaced by $G_{1,1}$. If $\mathrm{M}_1 >1$, we invoke Corollary A.1 again to assert that there exists a thread which is a subgraph of $G_{1, 1}$. We set $G_{1, 2} =G_{1, 1}{\setminus } { T}_{1, 2}$ so that we have $G_{1, 1}=G_{1, 2} {\overset{\star }{\curlyvee }}\, { T}_{1, 1}$ and

$$\begin{aligned}&\mathrm{Ch_g}(p_i, G_{1, 2})=\mathrm{Ch_g}(p_i, G), \mathrm{\ for \ } i=2, \ldots , \ell _{\mathrm{c}}, \\&\quad \mathrm{\ and \ } \mathrm{Ch_g}(p_1, G_1)=\mathrm{Ch_g}(p_1, G)-2. \end{aligned}$$

If $\mathrm{M}_1=2$ and $\ell _{\mathrm{c}}=1$, then we are done, we obtain (A.21) with $T_0=G_{1, 2}$. Otherwise, we proceed with $G_{1, 2}$ and construct iteratively the threads ${\tilde{T}}_{1, 3}, \ldots , {\tilde{T}}_{1, \mathrm{M}_1}$, and then ${\tilde{T}}_{2, 1}, \ldots , T_{2, \mathrm{M}_2}$, ... ${\tilde{T}}_{\ell _{\mathrm{c}}, 1}, \ldots , {\tilde{T}}_{\ell _{\mathrm{c}}, \mathrm{M}_{\ell _{\mathrm{c}}}}$. Setting $\ T_0=G_{\ell _{\mathrm{c}}, \mathrm{M}_{\ell _{\mathrm{c}}}}$ we obtain formula (A.21). $\square$

1.1.7 Prescribing charges and the Kirchhoff law

We are now in position to model connections of discrete sets to the boundary with possible branching points. Consider a finite set $A \subset \overline{\Omega }$, with points repeated with multiplicity ${\mathrm{M}}(a) \in \mathbb {N}^{\star }$, so that $\sharp (A)=\underset{a\in A}{\sum }\mathrm{M} (a)$. We restrict ourselves to graphs $G \in {\mathcal {G}}^+(\Omega )$ satisfying

$$\begin{aligned} {V_{\mathrm{chg}}}(G)=A \mathrm{\ and \ } \mathrm{Ch_g}(a, G)={\mathrm{M}}(a), \ \forall a \in A \cap \Omega . \end{aligned}$$

(A.23)

Condition (A.23) is equivalent to Kirchhoff’s law

$$\begin{aligned} \left\{ \begin{aligned}&\sharp \left( E^+(a , G)\right) = \sharp \left( E^- (a, G)\right) + \mathrm{M}(a), \mathrm{\, \ for \ } a\in A\cap \Omega \subset V(G) \\&\sharp \left( E^+(\upsigma , G)\right) =\sharp \left( E^- (\upsigma , G)\right) , \mathrm{\, \ for \ any \ } \upsigma \in V(G)\cap \Omega {\setminus } A. \end{aligned} \right. \end{aligned}$$

(A.24)

We introduce the class of graphs aimed to model connections of points in A to the boundary

$$\begin{aligned} {\mathcal {G}}(A, \partial \Omega )=\{ G \in {\mathcal {G}}^+(\Omega ) \ \mathrm{such \ that \ } A\subset V(G) \mathrm{\ and \ }(A.23) \mathrm{\ holds }\}. \end{aligned}$$

(A.25)

If G belongs to ${\mathcal {G}}(A, \partial \Omega )$, then the points of A are the only “source” points of the graph inside $\Omega$, with charge $\mathrm{M}(a)$, whereas all the other points have charge 0. The simplest example $G_0$ of an element in ${\mathcal {G}}(A, \partial \Omega )$ when $\Omega$ is convex is provided by the graph for which each element a in A is connected by a segment to an element of the boundary b so that in this case $\displaystyle {V(G_0)=\underset{a \in A}{\cup }\{a, b\}}$ and $\displaystyle {E(G)=\underset{a \in A}{\cup }\{[a, b]\}}$. Going back to (A.21), if $G \in {\mathcal {G}}(A, \partial \Omega )$, then we have (Fig. 23)

$$\begin{aligned} G_{\mathrm{thread}}\equiv \underset{ i \in \{1, \ldots , \ell _{\mathrm{c}}\} }{{\overset{\star }{\curlyvee }}}\left( \underset{j\in \{1, \ldots , \mathrm{M}_i\}}{{\overset{\star }{\curlyvee }}}T_{i, j}\right) \in G(A, \partial \Omega ). \end{aligned}$$

(A.26)

Fig. 23

Branched transport of the points $a_i$

Remark A.1

In definition (A.25) we allow points in A to belong to $\partial \Omega$: this, perhaps unnatural choice, is motivated by the fact that we face such a situation in Sect. A.3.1, the chosen convention simplifying somewhat the presentation. However, one may verify that

$$\begin{aligned} {\mathcal {G}}(A, \partial \Omega )={\mathcal {G}}(A{\setminus } \partial \Omega , \partial \Omega ). \end{aligned}$$

(A.27)

1.2 The functional and minimal branched connections to the boundary

Given $0\le \upalpha \le 1$, we consider the functional $\mathrm{W}_\upalpha$ defined on the set ${\mathcal {G}}( \Omega )$ by

$$\begin{aligned} \mathrm{W}_\upalpha (G)= \underset{ e\in E(G)}{\sum }(d(e))^\upalpha {\mathcal {H}}^1 (e), \, \mathrm{\ for \ } G \in {\mathcal {G}}(\Omega ), \end{aligned}$$

(A.28)

and the non-negative quantity

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )= \inf \left\{ \mathrm{W}_\upalpha (G), G \in {\mathcal {G}} (A, \partial \Omega )\right\} , \end{aligned}$$

(A.29)

which we will term the branched connection of order $\upalpha$ of the set A to the boundary $\partial \Omega$. Notice that the case $\upalpha =1$ has already been introduced in [11] as minimal connection to the boundary. Using, among other arguments, the fact that

$$\begin{aligned} \mathrm{W}_\upalpha (G) \le \mathrm{W}_\upalpha (G_{\mathrm{thread}}), \, \mathrm{\ where \ } G_{\mathrm{thread}} \mathrm{\ is \ defined \ in \ } (A.26), \end{aligned}$$

with equality if and only if $T_0$ in (A.21) is empty, it can be proved, as in [29]:

Lemma A.6

The infimum in (A.29) is achieved by some graph $G_{\mathrm{opt}} ^{^\upalpha } \in {\mathcal {G}} (A, \partial \Omega )$. The graph $G_{\mathrm{opt}}^{^\upalpha }$ has no loops and we may therefore write

(A.30)

Moreover, we have $d(e)\le \sharp (A)$ for any $e \in E(G_{\mathrm{opt}}^{^\upalpha } ).$

We notice that, as a straightforward consequence of (A.27), we have

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )=\mathrm{L}^\upalpha _{\mathrm{branch}}(A{\setminus } \partial \Omega , \partial \Omega ). \end{aligned}$$

(A.31)

We next show, similar to results in [4, 29, 30]:

Lemma A.7

Assume that $0\le \upalpha <1$. Then the graph $G_{\mathrm{opt}}^{^\upalpha }$ possesses the single path property.

Proof

We argue by contradiction and assume that there exists some vertex $\upsigma _0\in V(G_{\mathrm{opt}}^{^\upalpha } )$ and two distinct vertices $\upsigma _1$ and $\upsigma _2$ in $V(G_{\mathrm{opt}}^{^\upalpha } )$ such that $[\upsigma _0, \upsigma _i] \in E(G_{\mathrm{opt}}^{^\upalpha } )$ for $i=1, 2$. In view of the decomposition (A.30), there are two points $a_1$ and $a_2$ in A such that, for $i=1, 2,$ the segment $[\upsigma _0, \upsigma _i]$ belongs to $E(T_i)$ where $T_i$ is a thread of the form $T_{a_i, j_i}$ appearing in (A.30).

In the case the two threads have a common vertex $b_{\mathrm{com}}$ past $\upsigma _0$ we decompose $T_i$ as

$$\begin{aligned} T_i=\mathrm{G}_{\mathrm{rp}}(B_i)=\mathrm{G}_{\mathrm{rp}}(B_{0, i}) {\overset{\star }{\curlyvee }}\mathrm{G}_{\mathrm{rp}}(B_{1, i}){\overset{\star }{\curlyvee }}\mathrm{G}_{\mathrm{rp}}(B_{2, i}), \end{aligned}$$

(A.32)

where $B_{0, i}=\{a_i, \ldots , \upsigma _0\}$, $B_{1, i}=\{\upsigma _0, \upsigma _i, b_{i, 2}, \ldots , b_{i, \ell _i}=b_{\mathrm{com}}\}$ and $B_{2, i}=\{b_{\mathrm{com}}, \ldots , b_{\mathrm{q}_i}\}$, with $b_{\mathrm{q}_i} \in \partial \Omega$. In the case the two threads have no vertex in common past the vertex $\upsigma _0$, we use also decomposition (A.32), $B_{2, i}$ being void. In order to obtain a contradiction, we compare the energy of $G_{\mathrm{opt}}^{^\upalpha }$ with the energy of two comparison graphs ${\tilde{G}}_1$ and ${\tilde{G}}_2$, which correspond to an interchange of the threads $T_1$ and $T_2$, or more precisely to the parts $B_{1, 1}$ and $B_{1, 2}$. We first consider the modified threads

$$\begin{aligned}&{\tilde{T}}_1=\mathrm{G}_{\mathrm{rp}}(B_{0, 1}) {\overset{\star }{\curlyvee }}\mathrm{G}_{\mathrm{rp}}(B_{1, 2}){\overset{\star }{\curlyvee }}\mathrm{G}_{\mathrm{rp}}(B_{2, 1}) \mathrm{\ and \ } \\&\quad {\tilde{T}}_2=\mathrm{G}_{\mathrm{rp}}(B_{0, 2}) {\overset{\star }{\curlyvee }}\mathrm{G}_{\mathrm{rp}}(B_{1, 1}){\overset{\star }{\curlyvee }}\mathrm{G}_{\mathrm{rp}}(B_{2, 2}). \end{aligned}$$

We then define

$$\begin{aligned} {\tilde{G}}_1=\left( G {\setminus } T_1\right) {\overset{\star }{\curlyvee }}{\tilde{T}}_1 \mathrm{\ and \ } {\tilde{G}}_2=\left( G {\setminus } T_2\right) {\overset{\star }{\curlyvee }}{\tilde{T}}_2. \end{aligned}$$

One verifies that ${\tilde{G}}_i \in {\mathcal {G}}(A, \partial \Omega )$ for $i=1, 2$. For $i=1, 2$ and $j=0, \ldots , \ell _i-1$ we set $e_{i, j}\equiv [b_{i, j}, b_{i, j+1}]$ where $b_{i, 0}=\upsigma _0$ and $b_{i, 1}=\upsigma _i$. Setting $d_{i, j}=d(e_{i, j}, G)$, we observe that

$$\begin{aligned} \left\{ \begin{aligned}&d(e_{1, j}, {\tilde{G}}_1)= d_{1, j} -1, \mathrm{\ for \ } j=1, \ldots , \ell _1, \mathrm{\ and \ }\\&d(e_{2, j}, {\tilde{G}}_2)=d_{2, j} +1, \mathrm{\ for \ } j=1, \ldots , \ell _2, \\&d(e_{2, j}, {\tilde{G}}_1)= d_{i, j} +1, \mathrm{\ for \ } j=1, \ldots , \ell _1, \mathrm{\ and \ } \\&d(e_{1, j}, {\tilde{G}}_2)=d_{2, j} -1, \mathrm{\ for \ } j=1, \ldots , \ell _2. \end{aligned} \right. \end{aligned}$$

All other segments have the same density as for $G_{\mathrm{opt}}^{^\upalpha }$. It follows:

$$\begin{aligned} \left\{ \begin{aligned} W_\upalpha ({\tilde{G}}_1)- W_\upalpha (G_{\mathrm{opt}}^{^\upalpha } )&= \underset{j=0}{\overset{\ell _1-1}{\sum }}\left[ \left( d_{1, j}-1\right) ^\upalpha - d_{1, j}^\upalpha \right] \vert e_{1, j} \vert \\&\quad + \underset{j=0}{\overset{\ell _2-1}{\sum }} \left[ (d_{2, j}+1)^\upalpha -d_{2, j}^\upalpha \right] \vert e_{2, j} \vert \ge 0\\ W_\upalpha ({\tilde{G}}_2)- W_\upalpha (G_{\mathrm{opt}}^{^\upalpha } )&= \underset{j=0}{\overset{\ell _1-1}{\sum }}\left[ \left( d_{1, j}+1\right) ^\upalpha - d_{1, j}^\upalpha \right] \vert e_{1, j} \vert \\&\quad + \underset{j=0}{\overset{\ell _2-1}{\sum }} \left[ (d_{2, j}-1)^\upalpha -d_{2, j}^\upalpha \right] \vert e_{2, j} \vert \ge 0. \end{aligned} \right. \end{aligned}$$

Adding these inequalities we obtain

$$\begin{aligned} { \underset{i=1}{\overset{2}{\sum }}} {\underset{j=0}{\overset{\ell _i-1}{\sum }}} \left[ \left( d_{i, j}+1\right) ^\upalpha +\left( d_{i, j}-1\right) ^\upalpha -2 d_{i, j}^\upalpha \right] \ge 0. \end{aligned}$$

By concavity of the density function $\phi (d)=d^\upalpha$, we have however for $d\ge 1$, and since we assume $0\le \upalpha <1$,

$$\begin{aligned} \displaystyle {(d+1)^\upalpha +(d-1)^\upalpha -2d^\upalpha <0}, \end{aligned}$$

so that we have reached a contradiction which establishes the announced result. $\square$

Remark A.2

Using simple comparison arguments, one may easily prove that if A and B are disjoint finite subsets of $\Omega$, then

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A \cup B, \partial \Omega ) \le \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )+ \mathrm{L}^\upalpha _{\mathrm{branch}}(B, \partial \Omega ). \end{aligned}$$

(A.33)

Similarily, if $0<\upalpha ' \le \upalpha$ and $A \subset \Omega \subset \Omega '$, then we have

$$\begin{aligned} \mathrm{L}^{\upalpha '}_{\mathrm{branch}} (A, \partial \Omega ) \le \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega ) \mathrm{\ and \ } \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega ) \le \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega '). \end{aligned}$$

(A.34)

Remark A.3

We will be led to consider the case $\Omega$ is a polytope, $\overline{\Omega }=\overline{\Omega }_1 \cup \overline{\Omega }_2$, where $\Omega _1 \cap \Omega _2=\emptyset$, $\Omega _1$ and $\Omega _2$ being polytopes. Given a graph $G\in {\mathcal {G}}^+(\Omega )$, one verifies that

(A.35)

Assume next that $G\in {\mathcal {G}}(A, \Omega )$, where $A \subset \Omega$ is a finite set. We set, for $\mathfrak {p}=1, 2$, $A_\mathfrak {p}= \overline{\Omega }_\mathfrak {p}\cap A$. We have $G_\mathfrak {p}\in {\mathcal {G}}(A_\mathfrak {p}, \Omega _\mathfrak {p})$, so that

(A.36)

In the next suqbsection, we will be concerned with the asymptotic behavior of $\mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )$ as the number of elements in A tends to $+\infty$, specially in the case they are equi-distributed. Our methods rely on various decompositions.

1.3 Decomposing the domain and the graphs

We discuss here issues related to partitions of the domain $\Omega$, assuming it is a polytope. We consider the case where the set $\Omega$ is decomposed as a finite union

$$\begin{aligned}&\overline{\Omega }=\underset{{\mathfrak {p}} \in \mathfrak {P}}{\cup }\overline{ \Omega }_{{\mathfrak {p}}}, \mathrm{\ where \ the \ sets \ } \Omega _\mathfrak {p}\mathrm{\ are \ disjoint \ polytopes \ i. e. \ } \nonumber \\&\quad \Omega _\mathfrak {p}\cap \Omega _\mathfrak {p}'=\emptyset \mathrm{\ for \ } \mathfrak {p}\not = \mathfrak {p}'. \end{aligned}$$

(A.37)

Given a finite subset A of $\Omega$, we have the lower bound, in view of Remark A.2

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )\ge \underset{\mathfrak {p}\in \mathfrak {P}}{\sum }\mathrm{L}^\upalpha _{\mathrm{branch}}(A_\mathfrak {p}, \partial \Omega _\mathfrak {p}), \mathrm{\ where \ } A_\mathfrak {p}= \overline{\Omega }_\mathfrak {p}\cap A. \end{aligned}$$

(A.38)

Indeed, if G is a graph in ${\mathcal {G}}(A, \partial \Omega )$, then the restriction $G_\mathfrak {p}$ to the subset $\Omega _\mathfrak {p}$ belongs to ${\mathcal {G}}(A_\mathfrak {p}, \partial \Omega _\mathfrak {p})$. On the other hand, we have

$$\begin{aligned} \mathrm{W}_\upalpha (G)=\underset{\mathfrak {p}\in \mathfrak {P}}{\sum }\mathrm{W}_\upalpha (G_\mathfrak {p}), \end{aligned}$$

from which the conclusion (A.38) is deduced. We assume next that $\mathfrak {P}=\{1,2\}$, that is $\overline{\Omega }=\overline{\Omega }_1 \cup \overline{\Omega }_2$, where $\Omega _1 \cap \Omega _2=\emptyset$, $\Omega _1$ and $\Omega _2$ being polytopes. Our next result, is an improvement of (A.38) for this case.

Proposition A.3

Assume that (A.37) holds, with $\mathfrak {P}=\{1,2\}$, so that $\overline{\Omega }=\overline{ \Omega }_1 \cup \overline{ \Omega }_2$, with $\Omega _1\cap \Omega _2=\emptyset$. We have the lower bound

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )\ge & {} \mathrm{L}^\upalpha _{\mathrm{branch}}(A_1, \partial \Omega _1) +\mathrm{L}^\upalpha _{\mathrm{branch}}(A_2, \partial \Omega _2) \nonumber \\&+ \upkappa _\upalpha \frac{ \sharp (A_1)}{ \sharp (A)} (\mathrm{N_{\mathrm{el}}})^\upalpha \mathrm{\mathrm{dist \, }}(\Omega _1, \partial \Omega ), \end{aligned}$$

(A.39)

where $\upkappa _\upalpha >0$ depends only on $\upalpha$ and where $\mathrm{N_{\mathrm{el}}}=\sharp (A)$ denotes the number of elements in A, repeated with multiplicity.

The previous result is only of interest in the case $\mathrm{\mathrm{dist \, }}(\Omega _1, \partial \Omega )\not =0$, that is when ${\bar{\Omega }}_1 \subset \Omega$. The proof involves concavity properties, in particular the next elementary result.

Lemma A.8

Let $0<\upalpha \le 1$, $a\ge 1$ and $b\ge 1$ be two given numbers. There exists some universal constant $\upkappa _\upalpha >0$ depending only on $\upalpha$ such that

$$\begin{aligned} (a+b)^\upalpha \ge a^\upalpha + \upkappa _\upalpha b\, \inf \{b^{\upalpha -1}, \, a^{\upalpha -1} \}. \end{aligned}$$

Proof of Lemma A.8

We rely on the Taylor expansion of the expression $(1+s)^\upalpha$. It yields

$$\begin{aligned} ( 1+s)^\upalpha\ge & {} 1 + \upalpha s +\frac{1}{2} \upalpha (\upalpha -1) s^2 \ge 1 + \frac{1}{2} \upalpha s (1+ (\upalpha -1)s )\nonumber \\\ge & {} 1+ \frac{1}{2} \upalpha s (1-s) \mathrm{\ for \ } s \in [0, 1] . \end{aligned}$$

(A.40)

We distinguish three cases.

Case 1:$\displaystyle {b\le \frac{a}{2} }$. We apply (A.40) with $\displaystyle {s= \frac{b}{a}\le \frac{1}{2}}$, so that $\displaystyle {1-s \ge \frac{1}{2}}$, leading to the inequality

$$\begin{aligned} (a+b)^\upalpha \ge a^\upalpha (1+\frac{1}{4} \upalpha s)\ge a^\upalpha + \frac{1}{4} \, \upalpha b \, a^{\upalpha -1}. \end{aligned}$$

(A.41)

Case 2:$\displaystyle { 8a\ge b\ge \frac{a}{2} }$. In this case, we obtain invoking (A.40) with $\displaystyle {s=\frac{1}{2}}$

$$\begin{aligned} (a+b)^\upalpha\ge & {} (\frac{3}{2} a)^\upalpha =(1+\frac{1}{2})^\upalpha a^\upalpha \ge (1+\frac{1}{8} \upalpha ) a^\upalpha \nonumber \\\ge & {} a^\upalpha + \frac{1}{8} \upalpha \left( \frac{ b}{8}\right) ^\upalpha \ge a^\upalpha + \upalpha \left( \frac{1}{8}\right) ^{\upalpha +1}b^\upalpha . \end{aligned}$$

(A.42)

Case 3:$\displaystyle { 8a\le b}.$ In this case, we write

$$\begin{aligned} ( a+b)^\upalpha \ge b^\upalpha = \frac{1}{8^\upalpha }b^\upalpha + ( 1-\frac{1}{8^\upalpha } ) b^\upalpha \ge a^\upalpha +( 1-\frac{1}{8^\upalpha } ) b^\upalpha . \end{aligned}$$

(A.43)

We set $\displaystyle { \upkappa _\upalpha =\inf \{ \upalpha /4, \upalpha \left( 1 /8\right) ^{\upalpha +1}, \left( 1- {1}/{8^\upalpha }\right) \}.}$ Combining (A.41), (A.42) and (A.43) in the three cases, we complete the proof of the lemma. $\square$

We use Lemma A.8 in the case we have the additional assumption

$$\begin{aligned} a +b \le \mathrm{N_{\mathrm{ber}}}, \end{aligned}$$

(A.44)

where $\mathrm{N_{\mathrm{ber}}}\gg 1$ is some large number. It follows from (A.44) that $\displaystyle {b^\upalpha \ge b (\mathrm{N_{\mathrm{ber}}})^{\upalpha -1}}$ and $\displaystyle {a^{\upalpha -1} \ge (\mathrm{N_{\mathrm{ber}}})^{\upalpha -1}}$ so that in this case, (A.41) leads to the inequality

$$\begin{aligned} (a+b)^\upalpha \ge a^\upalpha + \upkappa _\upalpha \, b (\mathrm{N_{\mathrm{ber}}})^{\upalpha -1}, \end{aligned}$$

(A.45)

and hence the right hand side of (A.45) behaves linearily with respect to b. This observation will be important to establish the following:

Lemma A.9

Assume that $\Omega , \Omega _1$ and $\Omega _2$ are as in Proposition A.3 and let $G \in {\mathcal {G}}(A, \partial \Omega )$ be a graph without loop. Set, for $\mathfrak {p}=1, 2$, and $A_\mathfrak {p}=A \cap \Omega$. We have, the improved lower bound for $W_\upalpha (G_2)$,

$$\begin{aligned} W_\upalpha (G_2) \ge \mathrm{L}^\upalpha _{\mathrm{branch}}(A_2, \partial \Omega _2) + \upkappa _\upalpha { \sharp (A_1)} (\mathrm{N_{\mathrm{el}}})^{\upalpha -1} \mathrm{\mathrm{dist \, }}(\Omega _1, \partial \Omega ), \end{aligned}$$

(A.46)

where $\upkappa _\upalpha >0$ is the constant provided by Lemma A.8, and where $\mathrm{N_{\mathrm{el}}}=\sharp (A)$ denotes the number of elements in A, repeated with multiplicity.

Remark A.4

Notice that , so that we obtain the straighforward bound

$$\begin{aligned} W_\upalpha ( G_{2}) \ge \mathrm{L}^\upalpha _{\mathrm{branch}}(A_2, \partial \Omega _2). \end{aligned}$$

(A.47)

Comparing (A.47) with (A.46), we see that the later contains an additional non-negative terme on the r.h.s, corresponding to the announced improvement. Roughly speaking, this additional terms accounts for the fact that joining points in $\Omega _1$ to the boundary $\partial \Omega$, one needs to cross the region $\Omega _2$.

Proof of Lemma A.9

In view of Lemma A.6, we decompose the graph G as in (A.30), that is into threads emanating from the set A, distinguishing the threads with sources points in $A_1$ from those with source points in $A_2$ namely we write

(A.48)

We observe that, for $\mathfrak {p}=1,2$, the graph $\displaystyle {\Theta _\mathfrak {p}\equiv \underset{a \in A_\mathfrak {p}}{\curlyvee }T_a}$ belongs to ${\mathcal {G}}(A_\mathfrak {p}, \partial \Omega )$. Throughout the rest of the proof, we focus on the graph , which we decompose, using identity (A.48), as

It follows from these definitions that $G_{2,2} \in {\mathcal {G}}(A_2, \partial \Omega _2)$ whereas $G_{2,1} \in {\mathcal {G}}_0( \Omega _2)$. Notice in particular that the fact that $G_{2,2} \in {\mathcal {G}}(A_2, \partial \Omega _2)$ yields

$$\begin{aligned} W_\upalpha (G_{2,2})\ge \mathrm{L}^\upalpha _{\mathrm{branch}}(A_2, \partial \Omega _2), \end{aligned}$$

(A.49)

which provides the first term on the r.h.s of (A.46). The additional term on the r.h.s of the improved lower bound (A.46), is related to the restriction to $\Omega _2$ of the threads emanating from points in $A_1$. We will provide a lower bound of their contribution to $W_\upalpha (G_2)$, a major difficulty being that some parts might be merged with segment of $G_{2, 2}$.

Given a segment e of the graph $G_2=G_{2,1}{\overset{\star }{\curlyvee }}G_{2,2}$, we denote therefore by $d_{2,1}(e)$ (resp. $d_{2,2}(e)$) its multiplicity according to the graph $G_{2,1}$ (resp $G_{2,2}$), with the convention that $d_{2,1}(e)=0$ (resp. $d_{2,2}(e)=0$) if the segment does not belong to $E( G_{2,1})$ (resp. $E(G_{2,2}))$. It follows from the last statement in Lemma A.6 that

$$\begin{aligned} d(e, G)= d(e, G_2)=d_{2,2}(e) + d_{2,1} (e) \le \mathrm{N_{\mathrm{el}}}, \end{aligned}$$

(A.50)

and the definition of $W_\upalpha$ leads to the identity

$$\begin{aligned} W_\upalpha (G_2)=W_\upalpha ( G_{2,2} \curlyvee G_{2,1})= \underset{e \in E( G_2)}{\sum }\left( d_{2,2}(e) + d_{2,1} (e)\right) ^\upalpha {\mathcal {H}}^1(e). \end{aligned}$$

(A.51)

In order to derive a lower bound of the l.h.s of (A.51), we invoke the “linear” lower bound provided in Lemma A.8, in particular its consequence (A.45) applied with $\mathrm{N_{\mathrm{ber}}}=\mathrm{N_{\mathrm{el}}}$, $a= d_{2,2}(e)$ and $b= d_{2,1} (e)$. In view of (A.50) we have $a+b\le \mathrm{N_{\mathrm{el}}}$, so that (A.44) holds in the case considered. We have hence, for $e \in E(G_2)$,

$$\begin{aligned} \left( d_{2,2}(e) + d_{2,1} (e)\right) ^\upalpha \ge d_{2,2}(e)^\upalpha +\upkappa _\upalpha d_{2,1}(e) \left( \mathrm{N_{\mathrm{el}}}\right) ^{\upalpha -1}. \end{aligned}$$

Going back to (A.51), we are led to

$$\begin{aligned} W_\upalpha ( G_{2}) \ge \underset{e \in E(G_2)}{\sum }d_{2,2}(e)^\upalpha {\mathcal {H}}^1(e)+ \upkappa _\upalpha \, \left( \mathrm{N_{\mathrm{el}}}\right) ^{\upalpha -1} \, \underset{e \in E(G_2)}{\sum }d_{2,1}(e) {\mathcal {H}}^1(e). \end{aligned}$$

(A.52)

Since, by definition, we have $\displaystyle { W_\upalpha (G_{2,2})=\underset{e \in E( G_2)}{\sum }d_{2,2}(e)^\upalpha {\mathcal {H}}^1(e) },$ we obtain the lower bound

$$\begin{aligned} \begin{aligned} W_\upalpha (G_2)&=W_\upalpha (G_{2,2} \curlyvee G_{2,1}) \ge W_\upalpha ( G_{2,2} )\\&\quad + \upkappa _\upalpha \, \left( \mathrm{N_{\mathrm{el}}}\right) ^{\upalpha -1} \, \underset{e \in E(G_2)}{\sum }d_{2,1}(e) {\mathcal {H}}^1(e), \\&\ge \mathrm{L}^\upalpha _{\mathrm{branch}}(A_2, \partial \Omega _2) + \upkappa _\upalpha \, \left( \mathrm{N_{\mathrm{el}}}\right) ^{\upalpha -1} \, \underset{e \in E(G_2)}{\sum }d_{2,1}(e) {\mathcal {H}}^1(e). \\ \end{aligned} \end{aligned}$$

(A.53)

For the last inequality in (A.53), we have used (A.49). We next split the remainder of the proof of (A.46) into two steps.

Step 1. We have the lower bound

$$\begin{aligned} \underset{e \in E(G_{2})}{\sum }d_{2,1}(e) {\mathcal {H}}^1(e) \ge \sharp (A_1) \mathrm{dist} (\Omega _1, \partial \Omega ). \end{aligned}$$

(A.54)

Proof of (A.54). We take advantage of the linearity of the l.h.s with respect to multiplicity. Indeed, we notice that

$$\begin{aligned} \underset{ e \in E(G_{2})}{\sum }d_{2,1}(e) {\mathcal {H}}^1(e) = \underset{ a \in A_1}{\sum }{\mathcal {H}}^1({\mathcal {C}}_a \cap \Omega _2), \end{aligned}$$

where ${\mathcal {C}}_a$ denotes the polygonal curve related to the thread $T_a$. Since any thread $T_a$, joins a point in $\Omega _1$ to the boundary $\partial \Omega$, we have

$$\begin{aligned} H^1({\mathcal {C}}_a \cap \Omega _2) \ge \mathrm{dist} (\Omega _1, \partial \Omega ), \end{aligned}$$

so that the conclusion (A.54) follows combining the two previous relations.

Step 2. Proof of Lemma A.9completed. Combining the lower bound (A.53) and (A.54), we derive the lower bound (A.46), which completes the proof of Lemma A.9. $\square$

Proof of Proposition A.3

We assume that ${\bar{\Omega }}_1 \subset \Omega$, since otherwise the result (A.39) is a immediate consequence of (A.38). In view of the assumptions, we have $\Omega _2=\Omega {\setminus } {\bar{\Omega }}_1$. Let $G_{\mathrm{opt}}^{^\upalpha }$ be an optimal graph for $\mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )$, assuming for simplicity that all multiplicities in A are equal to one. We proceed first to a spatial decomposition of this graph, introducing the subgraphs . Going back to Remark A.3, we have

$$\begin{aligned} W_\upalpha (G_{\mathrm{opt}}^{^\upalpha } )=W_\upalpha (G_{\mathrm{opt},1}) + W_\upalpha (G_{\mathrm{opt},2}) \ge \mathrm{L}^\upalpha _{\mathrm{branch}}(A_1, \partial \Omega _1) + W_\upalpha (G_{\mathrm{opt},2}). \end{aligned}$$

(A.55)

In order to estimate $W_\upalpha (G_{\mathrm{opt},2})$, we invoke Lemma A.9 applied to $G_{\mathrm{opt}}^{^\upalpha }$. Combining inequality (A.46) applied to $G_{\mathrm{opt},2}$ with inequality (A.55), we derive inequality (A.39) which completes the proof of Proposition A.3. $\square$

1.3.1 Estimates for minimal branched connections

An important observation made²³ in [29] to obtain the following:

Proposition A.4

Assume that $\upalpha \in (\upalpha _m, 1]$, where $\upalpha _m=1-\frac{1}{m}$. Then we have, for some constant $C(\Omega , \upalpha )$ depending only on $\Omega$ and $\upalpha$,

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega ) \le C(\Omega , \upalpha ) \left( \sharp (A)\right) ^\upalpha . \end{aligned}$$

(A.56)

The proof is obvious for $\upalpha =1$. Indeed in this case, one obtain an upper bound for ${\mathfrak {L}}_{\mathrm{brbd}}^{1} (A, \partial \Omega )$ estimating $W_1(G_0)$, where $G_0$ is constructed as in Sect. A.1 connecting each point in A to its nearest point on the boundary. We obtain $\displaystyle { W_1(G_0)\le \mathrm{diam }(\Omega )\left( \sharp (A)\right) , }$ yielding the result in the case considered.

In the case $\upalpha _m\le \upalpha <1$, estimate (A.56) yields an improvement on the growth in terms of $\sharp A$. This is achieved in [29] replacing the elementary comparison graph $G_0$ by graphs having branching points obtained through a dyadic decomposition.

Remark A.5

The result of Proposition (A.4) is optimal: one may find simple distributions of points for which the asymptotic behavior is of order $\left( \sharp (A)\right) ^\upalpha$, for instance putting all points at the same location, far from the boundary.

1.4 The case of a uniform grid

We next focus on the behavior of $\mathrm{L}^\upalpha _{\mathrm{branch}}$ in the special case where $\Omega$ is the m-dimensional unit cube that is $\Omega =(0,1)^m$ and the points of A are located on an uniform grid (see Fig. 24). We consider therefore for an integer $\mathrm{k}$ in $\mathbb {N}^*$ the distance $h=\frac{1}{\mathrm{k}}$ and the set of points

$$\begin{aligned} \mathbf{A}^\mathrm{k}_m\equiv \boxplus ^\mathrm{k}_m(h)=\left\{ a^k_\mathrm{I}\equiv h\, I=h\, (i_1, i_2,\ldots , i_m),\mathrm{\ for \ } I \in \left\{ 1, \ldots ,\mathrm{k}\right\} ^m \right\} , \end{aligned}$$

so that $\displaystyle {\sharp (\mathbf{A}^\mathrm{k}_m )=\mathrm{k}^m}$. Notice that $\mathbf{A}^\mathrm{k}_m \cap \partial \left( (0,1)^m\right) \ne \emptyset$ (see Remark A.1). We set

$$\begin{aligned} \Uplambda _m^\upalpha (\mathrm{k})= \mathrm{L}^\upalpha _{\mathrm{branch}}(\mathbf{A}^\mathrm{k}_m, \partial (0,1)^m) \mathrm{\ and \ } \Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{k})\equiv \mathrm{k}^{-m\upalpha } {\Uplambda _m^\upalpha }(\mathrm{k}). \end{aligned}$$

We are interested in the asymptotic behavior of the quantities $\Uplambda _m^\upalpha (\mathrm{k})$ and $\Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{k})$ as $\mathrm{k}\rightarrow + \infty$. It follows from Proposition A.4 that, if $\upalpha >\upalpha _m$ then, we have the upper bound

$$\begin{aligned} {\Uplambda _m^\upalpha }(\mathrm{k}) \le C_\upalpha \mathrm{k}^{m \upalpha } \ \mathrm{\ i.e.\ } \ \Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{k})\le C_\upalpha , \end{aligned}$$

(A.57)

where the constant $C_\upalpha >0$ does not depend on $\mathrm{k}$. In the critical case $\upalpha =\upalpha _m$, the upper bound (A.57) no longer holds, as our next result shows.

Theorem A.1

There exists some constant $\mathrm{C}_m>0$ such that we have the lower bound

$$\begin{aligned}&{\Uplambda _m^{\upalpha _m}}(\mathrm{k}) \ge \mathrm{C}_m \mathrm{k}^{m \upalpha _m}\log \mathrm{k}=\mathrm{C}_m \mathrm{k}^{m-1}\log \mathrm{k}, \ \mathrm{i. e. \ } \\&\quad \Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{k})\ge \mathrm{C}_m \log \mathrm{k}, \mathrm{\ for \ } k \in \mathbb {N}^*. \end{aligned}$$

Remark A.6

The fact that the quantity $\Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{k})=\mathrm{k}^{1-m} {\Uplambda _m^{\upalpha _m}}(\mathrm{k})$ does not remain bounded as $\mathrm{k}\rightarrow + \infty$ is related to and may also presumably be deduced from the fact that the Lebesgue measure is not irrigible for the critical value $\upalpha =\upalpha _m$, a result proved in [12] (see also [4]).

Fig. 24

The case of an uniform grid

The proof of Theorem A.1 will rely on several preliminary results we present first, starting with elementary scaling laws. Let $\mathrm{q}\in \mathbb {N}^*$ be given, and consider for $\mathrm{k}\in \mathbb {N}$ the set

$$\begin{aligned} \frac{1}{\mathrm{q}}\mathbf{A}^{ k}_m =\mathbf{A}^{\mathrm{q}k}_m\cap [0, \frac{1}{\mathrm{q}}]^m=\boxplus ^\mathrm{k}_m(\frac{h}{\mathrm{q}})=\left\{ a^k_\mathrm{I}\equiv \frac{1}{\mathrm{q}\mathrm{k}}\, I, I \in \left\{ 1, \ldots ,\mathrm{k}\right\} ^m \right\} . \end{aligned}$$

The set $\frac{1}{\mathrm{q}}\mathbf{A}^{k}_m$ hence contains $\mathrm{k}^m$ distinct elements. The scaling law writes as

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}\left( \frac{1}{\mathrm{q}}\mathbf{A}^{ k}_m, \partial \left( [0, \frac{1}{\mathrm{q}}]^m\right) \right) = \mathrm{q}^{-1} \mathrm{L}^\upalpha _{\mathrm{branch}}\left( \mathbf{A}^{ k}_m, \partial \left( [0, 1]^m\right) \right) = \mathrm{q}^{-1} {\Uplambda _m^\upalpha }(\mathrm{k}). \end{aligned}$$

(A.58)

The main ingredient in the proof of Theorem A.1 is a consequence of Proposition A.3:

Lemma A.10

Let $\mathrm{q}\in \mathbb {N}^*$ be given. There exists some constant $\mathrm{C}_\mathrm{q}^\upalpha >0$ such that

$$\begin{aligned} \Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{q}\, \mathrm{k}) \ge \mathrm{q}^{m(\upalpha _m-\upalpha )} \, \Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{k})+ \mathrm{C}_\mathrm{q}^\upalpha , \ \ \mathrm{\ for \ any \ } \mathrm{k}\in \mathbb {N}^* . \end{aligned}$$

Proof

We consider the set $\mathbf{A}^{\mathrm{q}k}_m$ and decompose the domain $\overline{\Omega } =[0,1]^m$ as an union of cubes $\overline{\mathrm{Q}_\mathbf{J}}$, with $\mathbf{J}\equiv (j_1, j_2, \dots , j_m)\in {\mathfrak {J}}\equiv \{0,\ldots , \mathrm{q}-1\}^m$, and

$$\begin{aligned} \mathrm{Q}_\mathbf{J}= \frac{1}{\mathrm{q}}\mathbf{J}+ (0, \frac{1}{\mathrm{q}})^m =\frac{1}{\mathrm{q}}(j_1, j_2, \dots , j_m) + (0, \frac{1}{\mathrm{q}})^m. \end{aligned}$$

We have therefore $\mathrm{Q}_\mathbf{J}\cap \mathrm{Q}_{\mathbf{J}'} \not = \emptyset$ if $\mathbf{J}\not = \mathbf{J}'$ and $\displaystyle {[0,1]^m=\underset{ \mathbf{J}\in {\mathfrak {J}}}{\cup }\bar{\mathrm{Q}}_\mathbf{J}}.$ We set

$$\begin{aligned} A_{\mathbf{J}} \equiv A^{\mathrm{q}\mathrm{k}}_m \cap \mathrm{Q}_\mathbf{J}, \mathrm{\ so \ that \ } A_{\mathbf{J}}= \frac{1}{\mathrm{q}}\mathbf{J}+ \frac{1}{\mathrm{q}} \mathbf{A}_m^k. \end{aligned}$$

It follows from the scaling law (A.58) and translation invariance that

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A_\mathbf{J}, \partial \mathrm{Q}_{\mathbf{J}})=\mathrm{q}^{-1} {\Uplambda _m^\upalpha }(\mathrm{k}), \mathrm{\ for \ } \mathbf{J}\in {\mathfrak {J}}. \end{aligned}$$

(A.59)

We next single out a cube $\mathrm{Q}_{\mathbf{J}_0}$ which is far from the boundary. For that purpose, we consider the integer $\displaystyle { \mathrm{q}_0 \equiv \left[ \frac{\mathrm{q}}{2}\right] }$, the multi-index $\mathbf{J}_0=(\mathrm{q}_0, \mathrm{q}_0, \ldots , \mathrm{q}_0)$ and the sets

$$\begin{aligned} \Omega _1 =\mathrm{Q}_{\mathbf{J}_0} \mathrm{\ and \ } \Omega _2= \underset{ \mathbf{J}\in {\mathfrak {J}} {\setminus } \{\mathbf{J}_0\}}{\cup }\mathrm{Q}_\mathbf{J}, \mathrm{\ so \ that \ } \mathrm{dist } (\Omega _1, \partial \Omega ) \ge \frac{1}{4} \mathrm{\ for \ } \mathrm{q}\ge 3. \end{aligned}$$

Applying inequality (A.39) of Proposition A.56, we are led to

$$\begin{aligned} \begin{aligned} \Uplambda _m^\upalpha (\mathrm{q}\mathrm{k})= \mathrm{L}^\upalpha _{\mathrm{branch}}(\mathbf{A}_m^{\mathrm{q}\mathrm{k}}, \partial (0,1)^m)&\ge \mathrm{L}^\upalpha _{\mathrm{branch}}(A_{\mathbf{J}_0}, \partial \mathrm{Q}_{J_0})\\&\quad + \mathrm{L}^\upalpha _{\mathrm{branch}}(\Omega _2 \cap \mathbf{A}^{\mathrm{q}\mathrm{k}}_m, \partial (0,1)^m) \\&\quad +\frac{1}{4}\upkappa _\upalpha \mathrm{k}^{m \upalpha } \mathrm{q}^{m(\upalpha -1)}. \end{aligned} \end{aligned}$$

(A.60)

We deduce from inequality (A.38) and (A.58) that

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(\Omega _2 \cap A^{\mathrm{q}\mathrm{k}}_m, \partial (0,1)^m)\ge & {} \underset{ \mathbf{J}\in {\mathfrak {J}} {\setminus } \{\mathbf{J}_0\}}{\sum }\mathrm{L}^\upalpha _{\mathrm{branch}}(A_\mathbf{J}, \partial \mathrm{Q}_\mathbf{J})\nonumber \\= & {} \left[ \mathrm{q}^{m}-1\right] \mathrm{q}^{-1} {\Uplambda _m^\upalpha }(\mathrm{k}). \end{aligned}$$

(A.61)

Combining (A.60), (A.61) with (A.59) for $\mathbf{J}=\mathbf{J}_0$, we are led to the lower bound

$$\begin{aligned} \Uplambda _m^\upalpha (\mathrm{q}\mathrm{k}) \ge \mathrm{q}^{m-1} {\Uplambda _m^\upalpha }(\mathrm{k}) + \frac{1}{4}\upkappa _\upalpha \mathrm{k}^{m \upalpha } \mathrm{q}^{m(\upalpha -1)}. \end{aligned}$$

Multiplying both sides by $(\mathrm{q}\mathrm{k})^{-m\upalpha }$, we obtain the desired result with $\mathrm{C}^\upalpha _\mathrm{q}=\frac{1}{4} \upkappa _\upalpha \mathrm{q}^{-1}.$$\square$

Lemma A.11

We have, for any integers $1 \le \mathrm{k}' \le \mathrm{k}$,

$$\begin{aligned} {\Uplambda _m^\upalpha }(\mathrm{k}') \le \frac{\mathrm{k}'}{\mathrm{k}} {\Uplambda _m^\upalpha }(\mathrm{k}) \ \mathrm{\ and \ hence \ \ } \Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{k}') \le \left( \frac{\mathrm{k}'}{\mathrm{k}} \right) ^{m\upalpha +1} \Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{k}). \end{aligned}$$

Proof

consider the cube ${\mathcal {Q}}_\mathrm{k}'=(0, \frac{\mathrm{k}'}{ \mathrm{k}} )^m \subset (0,1)^m$ and the set $A'=\mathbf{A}^\mathrm{k}_m \cap {\mathcal {Q}}_\mathrm{k}'$. It follows from inequality (A.38) that

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}(A', \partial {\mathcal {Q}}_\mathrm{k}') \le \mathrm{L}^\upalpha _{\mathrm{branch}}(A^\mathrm{k}_m, \partial (0,1)^m)={\Uplambda _m^\upalpha }(\mathrm{k}), \end{aligned}$$

whereas the scaling property yields $\displaystyle {\mathrm{L}^\upalpha _{\mathrm{branch}}(A', \partial {\mathcal {Q}}_\mathrm{k}')=\frac{\mathrm{k}'}{\mathrm{k}} {\Uplambda _m^\upalpha }(\mathrm{k}'). }$ The conclusion follows combining the previous inequalities. $\square$

Proof of Theorem A.1 completed

In the special case $\upalpha =\upalpha _m$, the exponent of $\mathrm{q}$ in the r.h.s of the inequality of Lemma A.10 vanishes, so that we obtain

$$\begin{aligned} \Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{q}\, \mathrm{k}) \ge \Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{k})+ \mathrm{C}_\mathrm{q}^{\upalpha _m}, \mathrm{\ for \ any \ integer \ } \mathrm{q}\ge 3. \end{aligned}$$

Iterating this lower bound, we obtain, for any any integer $\ell >0$, to the lower bound

$$\begin{aligned} \Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{q}^\ell )\ge \mathrm{C}^{\upalpha _m}_\mathrm{q}\ell . \end{aligned}$$

(A.62)

On the other hand, it follows from Lemma A.11 that for any $\mathrm{q}^\ell \le \mathrm{k}\le \mathrm{q}^{\ell +1}$ we have

$$\begin{aligned} \Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{k}) \ge \mathrm{q}^{-m} \ \Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{q}^\ell ). \end{aligned}$$

(A.63)

Combining (A.63) with (A.62) we deduce that, for any $\mathrm{k}\in \mathbb {N}^*$, we obtain the inequality

$$\begin{aligned} \Uplambda ^{m, \upalpha _m}_{\mathrm{norm}}(\mathrm{k}) \ge \mathrm{q}^{-m} \mathrm{C}^{\upalpha _m}_\mathfrak {q}\left[ \frac{\log \mathrm{k}}{\log \mathrm{q}}\right] , \end{aligned}$$

which leads immediately to the conclusion, fixing the value of $\mathrm{q}$ for instance as $\mathrm{q}=5$. $\square$

Remark A.7

For $\upalpha <\upalpha _m$ the same type of argument show that $\displaystyle \Uplambda ^{m,\upalpha }_{\mathrm{norm}}(\mathrm{k}) \rightarrow + \infty \mathrm{\ as \ } \mathrm{k}\rightarrow + \infty .$

1.5 Charges with opposite signs

We consider here, as in the introduction, the possibility of having also points with negative charges. We consider therefore a collection of points $\mathbf{P}=\{P_i\}_{i \in I}$ in $\Omega$ with positive charge $+1$, a collection of points $\mathbf{Q}=\{Q_j\}_{j \in J}$ in $\Omega$, with negative charge $-1$, and set $A=\mathbf{P}\cup \mathbf{Q}$. We define the set $\mathcal {G} (\mathbf{P}, \mathbf{Q}, \Omega )$ of graph satisfying $A\subset V(G)\subset {\bar{\Omega }},$ and the Kirchhoff law (A.23), setting $\mathrm{M}(P_i)=+1$ and $\mathrm{M}(Q_j)=-1$. For $0\le \upalpha \le 1$, we set

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}( \mathbf{P}, \mathbf{Q}, \partial \Omega )=\mathrm{L}^\upalpha _{\mathrm{branch}}(A, \partial \Omega )=\inf \left\{ \mathrm{W}_\upalpha (G), G \in {\mathcal {G}}\left( \mathbf{P}, \mathbf{Q}, \partial \Omega \right) \right\} . \end{aligned}$$

(A.64)

Next assume that we are given a collections of disjoint subdomains $(\Omega _\mathfrak {p})_{\mathfrak {p}\in \mathfrak {P}}$ of $\Omega$ such that

$$\begin{aligned} \mathbf{Q}\cap \Omega _\mathfrak {p}=\emptyset \mathrm{\ and \ set \ } \ A_\mathfrak {p}= \mathbf{P}\cap \Omega _\mathfrak {p}, \end{aligned}$$

(A.65)

so that the subdomains $\Omega _\mathfrak {p}$ contain only possibly positive charges.

Lemma A.12

If (A.65) is satisfied, then we have the inequality

$$\begin{aligned} \mathrm{L}^\upalpha _{\mathrm{branch}}\left( \mathbf{P}, \mathbf{Q}, \partial \Omega \right) \ge \underset{\mathfrak {p}\in \mathfrak {P}}{\sum }\mathrm{L}^\upalpha _{\mathrm{branch}}( A_\mathfrak {p}, \partial \Omega _\mathfrak {p}). \end{aligned}$$

Proof

Let G be a graph in ${\mathcal {G}} \left( A, \partial \Omega \right)$ and set $\displaystyle {G_\mathfrak {p}=G \cap \Omega _\mathfrak {p}}$, for $\mathfrak {p}\in \mathfrak {P}$. Since there are no negative charges in $\Omega _\mathfrak {p}$, it turns out that $G_\mathfrak {p}\in {\mathcal {G}}({\mathfrak {A}}_\mathfrak {p}, \partial \Omega _\mathfrak {p})$, so that $\displaystyle { {\mathrm{W}}_\upalpha (G_\mathfrak {p}) \ge \mathrm{L}^\upalpha _{\mathrm{branch}}({\mathfrak {A}}_\mathfrak {p}, \partial \Omega _\mathfrak {p}). }$ On the other hand, we have $\displaystyle { \mathrm{W}_\upalpha (G) \ge \underset{\mathfrak {p}\in \mathfrak {P}}{\sum }{\mathrm{W}}_\upalpha (G_\mathfrak {p})}$ so that the conclusion follows. $\square$