An Averaging Principle for Stochastic Flows and Convergence of Non-Symmetric Dirichlet Forms

Abstract

We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include:

• a proof of an original averaging principle for stochastic flows of kernels;

• the definition and study of a convergence of sequences of non-symmetric bilinear forms defined on different spaces;

• the study of weighted Sobolev spaces on metric graphs or “books”.

Appendix:

Notation: In a topological space $A\Subset B$ means that $\overline {A}\subset \mathring {B}$ and $\overline {A}$ is compact.

1.1 Weighted Sobolev Spaces

For k ≥ 1, the class of Muckenhoupt weights A₂ consists all mappings $\omega :\mathbb {R}^{k}\to [0,\infty ]$, for which there is a constant C such that for all ball B in $\mathbb {R}^{k}$, we have

$$\frac{1}{|B|}\left( {\int}_{B} \omega(x) dx\right)\times \frac{1}{|B|}\left( {\int}_{B} \frac{1}{\omega(x)} dx\right) \le C.$$

For m ≥ 2, set $B_{m,1}:=\left (\cup _{i=1}^{m} ]0,\infty [\times \{i\}\right )\cup \{0\}$, equipped with the distance d₁ defined by d₁((x,i), (y,i)) = |x − y| and if i≠j, d₁((x,i), (y,j)) = x + y, and d₁((x,i), 0) = x. Let $i_{1}:B_{m,1}\to \{0,\dots ,m\}$ be defined i₁(x,i) = i and i₁(0) = 0. Then B_m,1 is a metric graph constituted of m half lines joined at 0. Set also $B_{m,2}:=\mathbb {R}\times B_{m,1}$, equipped with the distance d₂ defined by d₂((x,i), (y,i)) = ∥x − y∥, $d_{2}((x,i),(y,j))=\|x-y^{\prime }\|$ if i≠j and where $y^{\prime }=(y_{1},-y_{2})$, and d₂((x,i),y) = ∥x − y∥ if y = (y₁, 0), where ∥⋅∥ is the Euclidean norm on $\mathbb {R}^{2}$. Then B_m,2 is a metric space constituted of m half planes joined along a line. Let also $i_{2}: B_{m,2}\to \{0,\dots ,m\}$ be defined by i₂(x₁,x₂,i) = i and i₂(x₁, 0) = 0. To simplify the notation we will simply denote d₁ and d₂ by d. For 1 ≤ i ≤ m, we will use the notation (0,i) = 0 ∈ B_m,1 and for $x_{1}\in \mathbb {R}$, (x₁, 0,i) = (x₁, 0) ∈ B_m,2.

For m = 1 and k ∈{1, 2}, set $B_{1,k}=\mathbb {R}^{k}$.

Let $\omega : B_{m,k}\to [0,\infty ]$ be such that $\omega \in L^{1}_{loc}(B_{m,k})$, i.e. such that for all $A\Subset {\Omega }$, $\mu (A):= {\int \limits }_{A} \omega (x) dx < \infty$, with dx the measure on B_m,k that coincides with the Lebesgue measure on E_i := {x ∈ B_m,k : i_k(x) = i} for each i. For i≠j, set E_i,j := E_i ∪ E_j ∪ E₀ (which is isometric to $\mathbb {R}^{k}$, and will be thus identified to $\mathbb {R}^{k}$).

In the following, we fix k ∈{1, 2} and m ≥ 1 and we let Ω be an open subset of B_m,k. For i ∈{1,⋯ ,m}, set Ω_i = Ω ∩ E_i and for 1 ≤ i≠j ≤ m, set Ω_i,j = Ω ∩ E_i,j. Then Ω_i and Ω_i,j are open subsets of $\mathbb {R}^{k}$. Denote by L²(Ω,ω) the space of all measurable functions f on B_m,k such that ${\int \limits }_{\Omega } f^{2}(x) \omega (x) dx < \infty$. For a function f ∈ L²(Ω,ω), weakly differentiable on Ω_i,j for all 1 ≤ i≠j ≤ m, define the norm of f by

$$ \|f\|^{2}_{W^{1}({\Omega},\omega)}={\int}_{\Omega} \big(f^{2} + \|\nabla f\|^{2}\big)(x) \omega(x) dx. $$

(A.1)

Let W¹(Ω,ω) be the completion with respect to the norm $\|\cdot \|_{W^{1}({\Omega },\omega )}$ of the vector space of the functions f ∈ L²(Ω,ω), that are weakly differentiable on Ω_i,j for all 1 ≤ i≠j ≤ m and with $\|f\|_{W^{1}({\Omega },\omega )}<\infty$. Suppose also that $\frac {1}{\omega }\in L^{1}_{loc}({\Omega })$. Then, for all $i\in \{1,\dots ,k\}$ (resp. all 1 ≤ i≠j ≤ k), the restriction of f ∈ W¹(Ω,ω) to Ω_i (resp. to Ω_i,j) belongs to $W^{1,1}_{loc}({\Omega }_{i})$ (resp. to $W^{1,1}_{loc}({\Omega }_{i,j})$.

We also define H¹(Ω,ω) (resp. ${H^{1}_{0}}({\Omega },\omega )$) to be the completion of C(Ω) ∩ W¹(Ω,ω) (resp. of C_c(Ω) ∩ W¹(Ω,ω)), with respect to $\|\cdot \|_{W^{1}({\Omega },\omega )}$. Define also ${W^{1}_{0}}({\Omega },\omega )$ to be the set of all f ∈ W¹(Ω,ω) such that the function F = f1_Ω ∈ W¹(B_m,k,ω). Equipped with the inner product

$$ \langle f,g\rangle_{W^{1}({\Omega},\omega)}={\int}_{\Omega} (fg+\nabla f\cdot\nabla g)(x)\omega(x)dx, $$

(A.2)

W¹(Ω,ω), ${W^{1}_{0}}({\Omega },\omega )$, H¹(Ω,ω) and ${H^{1}_{0}}({\Omega },\omega )$ are Hilbert spaces.

Lemma A.1

Let m ≥ 1 and $O\Subset {\Omega }$ be open subsets of B_m,k. Then there is δ > 0 such that K_δ := {x ∈ B_m,k : d(x,O) ≤ δ} is a compact set with O ⊂ K_δ ⊂Ω. Suppose that there is $\bar {\omega }:B_{m,k}\to [0,\infty ]$ such that $\bar {\omega }=\omega$ on Ω and such that

• when m ≥ 2, for all i≠j, the restriction $\bar {\omega }_{i,j}$ of $\bar {\omega }$ to E_i,j belongs to the class A₂.

• when m = 1, $\bar {\omega }$ belongs to the class A₂.

Then if $f\in {W^{1}_{0}}(O,\omega )$, there is g ∈ L²(O,ω) such that g = 0 on O ∖ K_δ and such that for all (x,y) ∈ O²,

$$|f(x)-f(y)|\le d(x,y)\big(g(x)+g(y) \big).$$

Moreover, there is a sequence of lipschitzian functions $f_{n}\in {W^{1}_{0}}(O,\omega )$ such that $\lim _{n\to \infty }\|f-f_{n}\|_{W^{1}(O,\omega )}=0$.

Proof

We only consider the case m ≥ 2, the case m = 1 being simpler.

The existence of δ and K_δ is a standard exercise.

Let $f\in {W^{1}_{0}}(O,\omega )$ and denote by f_i,j the restriction f to E_i,j, and set O_i,j := O ∩ E_i,j. Then $f_{i,j}\in {W^{1}_{0}}(O_{i,j},\bar {\omega }_{i,j})$. Set F := f1_O and F_i,j the restriction of F on E_i,j. Then $F_{i,j}\in W^{1}(E_{i,j},\bar {\omega }_{i,j})$. Recall that E_i,j is isometric to $\mathbb {R}^{k}$. Note that for x ∈ O_i = O ∩ E_i, we have for all j≠i, f(x) = f_i,j(x) = F_i,j(x).

Since $\bar {\omega }\in A_{2}$, we have (see [11]) $G_{i,j}\in L^{2}(E_{i,j},\bar {\omega }_{i,j})$ such that for all $(x,y)\in E_{i,j}^{2}$,

$$|F_{i,j}(x)-F_{i,j}(y)|\le d(x,y)\big(G_{i,j}(x)+G_{i,j}(y)\big).$$

For x ∈Ω, define $g(x):={\sum }_{j\ne i} G_{i,j}(x)$ if x ∈Ω_i. Then g ∈ L²(Ω,ω), and we have for all (x,y) ∈Ω²,

$$|f(x)-f(y)|\le d(x,y)\big(g(x)+g(y)\big).$$

Set now $g_{\delta }:=g 1_{K_{\delta }}+\delta ^{-1}|f|$. Then, we have that for all (x,y) ∈Ω,

$$|f(x)-f(y)|\le d(x,y)\big(g_{\delta}(x)+g_{\delta}(y)\big).$$

Indeed, this inequality is straightforward to check if $(x,y)\in K_{\delta }^{2}$ or if (x,y) ∈ (Ω ∖ O)². If (x,y) ∈ O × (Ω ∖ K_δ), we have d(x,y)(g_δ(x) + g_δ(y)) ≥ d(x,y)δ^− 1|f(x)|≥|f(x) − f(y)|. This shows the first part of the lemma.

For the second part, it suffices to follow the proof of Theorem 5 of [11] (with the function g_δ, and since f = f_λ on E_λ and that $E_{\lambda }\supset {\Omega }\setminus K_{\delta }$, we have f_λ = 0 sur Ω ∖ K_δ). □

For an open set U ⊂Ω, define the capacity of U by

$$\text{Cap}(U):= \inf\{\|h\|_{W^{1}({\Omega},\omega)}^{2}: h\ge 1 \text{ on} U\text{ and } h\in W^{1}({\Omega},\omega)\}.$$

Lemma A.2

Let m ≥ 1, Ω an open subset of B_m,k and $\omega :{\Omega }\to [0,\infty ]$ a measurable mapping. For R > 0, set Ω_R := {x ∈Ω : ∥x∥ < R}. Suppose that, for all R > 0, there exists a non decreasing sequence of open subsets (Ω_R,n)_n≥ 1 such that

• (i) ∪_n≥ 1Ω_R,n = Ω_R,

• (ii) $\lim _{n\to \infty } \text {Cap}({\Omega }_{R}\setminus \overline {\Omega }_{R,n})=0$,

• (iii) for all n ≥ 1, ${W^{1}_{0}}({\Omega }_{R,n},\omega )={H^{1}_{0}}({\Omega }_{R,n},\omega )$.

Then we have $W^{1}({\Omega },\omega )={H^{1}_{0}}({\Omega },\omega )$.

Proof

Let f ∈ W¹(Ω,ω) and 𝜖 > 0. For R > 1, define $f_{R}:{\Omega }\to \mathbb {R}$ defined by f_R(x) = f(x) if ∥x∥≤ R − 1, f_R(x) = (R −∥x∥)f(x) if ∥x∥∈ [R − 1,R] and f_R(x) = 0 if ∥x∥ > R. Then we have that $f_{R}\in {W^{1}_{0}}({\Omega }_{R},\omega )$ and it is easy to check that there is an R₀ > 0 such that for all R > R₀, $\|f-f_{R}\|_{W^{1}({\Omega },\omega )}<\epsilon$.

From now on, we fix R > R₀. Applying Theorem III.2.11 in [19], conditions (i) and (ii) ensures that there is a sequence of functions $f_{R,n}\in {W^{1}_{0}}({\Omega }_{R,n},\omega )$ such that $\lim _{n\to \infty }\|f_{R}-f_{R,n}\|_{W^{1}({\Omega }_{R},\omega )}=0$.

And we conclude using (iii). □

Let now G be a connected metric space, and fix k ∈{1, 2}. Suppose that there is a locally finite covering $({\Omega }_{\ell })_{\ell \in {\mathscr{L}}}$ of G, with open sets and and with ${\mathscr{L}}$ a countable set. Suppose also that this covering is such that for each ℓ, Ω_ℓ is isometric to an open subset of B_m,k for some m ≥ 1. Suppose that there is a sequence of functions $(\varphi _{\ell })_{\ell \in {\mathscr{L}}}$ such that φ_ℓ : G → [0, 1], φ_ℓ = 0 on G ∖Ω_ℓ, φ_ℓ restricted to ${\Omega }_{\ell }^{i}$ is C¹, with bounded derivatives, for each $i\in \{1,\dots , m_{\ell }\}$ and such that ${\sum }_{\ell \in {\mathscr{L}}}\varphi _{\ell }=1$.

Let $\omega :G\to [0,\infty ]$ be a measurable mapping. For each ℓ, then there is m ≥ 1 such that Ω_ℓ ⊂ B_m,k, we let $\omega _{\ell }:B_{m,k}\to [0,\infty ]$ be such that ω_ℓ = ω on Ω_ℓ. Suppose that $\omega _{\ell }\in L^{1}_{loc}(B_{m,k})$ and $\frac {1}{\omega _{\ell }}\in L^{1}_{loc}({\Omega }_{\ell })$.

Define W¹(G,ω) to be the set of all measurable functions $f:G\to \mathbb {R}$ such that for each ℓ there is $f_{\ell }\in W^{1}({\Omega }_{\ell },\omega _{\ell })$ with f = f_ℓ on Ω_ℓ with $\|f\|_{W^{1}(G,\omega )}<\infty$ where

$$\|f\|^{2}_{W^{1}(G,\omega)}={\int}_{G} \big(f^{2} + \|\nabla f\|^{2}\big)(x) \omega(x) dx.$$

Then W¹(G,ω) equipped with the innner product

$$\langle f,g\rangle^{2}_{W^{1}(G,\omega)}={\int}_{G} \big(fg + \nabla f\cdot \nabla g \big)(x) \omega(x) dx$$

is a Hilbert space. Define also ${H^{1}_{0}}(G,\omega )$ as the completion of C_c(G) ∩ W¹(G,ω). Note that if f ∈ W¹(G,ω), we have that for each ℓ, $f\varphi _{\ell }\in W^{1}({\Omega }_{\ell },\omega _{\ell })$.

Lemma A.3

Suppose that for all ℓ and all R > 0 there exists a non decreasing sequence of open subsets (Ω_ℓ,R,n)_n≥ 1 such that

• (i) $\bigcup _{n\ge 1}{\Omega }_{\ell ,R,n}={\Omega }_{\ell ,R}$,

• (ii) $\lim _{n\to \infty } \text {Cap}({\Omega }_{\ell ,R}\setminus \overline {\Omega }_{\ell ,R,n})=0$,

• (iii) for all n ≥ 1, ${W^{1}_{0}}({\Omega }_{\ell ,R,n},\omega _{\ell })={H^{1}_{0}}({\Omega }_{\ell ,R,n},\omega _{\ell })$.

Then we have $W^{1}(G,\omega )={H^{1}_{0}}(G,\omega )$.

Proof

The proof of this lemma is almost identical to the one of Lemma A.2. We write $f={\sum }_{\ell } f_{\ell }$, where f_ℓ := fφ_ℓ. For each 𝜖 > 0 there is a finite ${\mathscr{L}}_{0}\subset \mathcal L$ such that $\|f-{\sum }_{\ell \in {\mathscr{L}}_{0}}f_{\ell }\|_{W^{1}(G,\omega )}<\epsilon$. And then we can follow the proof of Lemma A.2, for Ω = Ω_ℓ and f = f_ℓ and to find, for all $\ell \in {\mathscr{L}}_{0}$, an integer n ≥ 1 and a function $g_{\ell }\in C_{c}({\Omega }_{\ell ,R,n})\cap W^{1}({\Omega }_{\ell },\omega _{\ell })$ such that $\|f_{\ell } -g_{\ell }\|_{W^{1}(G,\omega )}<\frac {\epsilon }{|{\mathscr{L}}_{0}|}$. Then $\|f -{\sum }_{\ell \in {\mathscr{L}}_{0}} g_{\ell }\|_{W^{1}(G,\omega )}<2 \epsilon$. □

1.2 Regularity for Section ??

In this paragraph, we complete the proofs of Proposition 7.5 and of Proposition 7.6.

Let us first recall some notation of Section ??. The space $\widetilde M=V\cup \bigcup _{i\in I} E_{i}$ is a metric graph, with E_i =]l_i,r_i[×{i}. There is a continuous map $\pi :\mathbb {R}^{2}\to \widetilde M$ such that if x ∈Ω_i, π(x) = (H(x),i) ∈ E_i. For v ∈ V, $I^{+}_{v}=\{i\in I, (l_{i},i)=v\}$, $I^{-}_{v}=\{i\in I, (r_{i},i)= v\}$ and $I_{v}=I^{+}_{v}\cup I^{-}_{v}$. Let d(v) be the cardinal of the set I_v, which is the degree of v in the graph $\widetilde M$. Choose i ∈ I_v and set h_v = l_i if v = (l_i,i) or h_v = r_i if v = (r_i,i). Remark that h_v does not depend on the particular choice i ∈ I_v. For v ∈ V, we define $\gamma (v):=\bigcup _{(h,i)\sim v}\gamma _{i}(h)$, the connected level set associated to the vertex v. Then for all x ∈ γ(v), H(x) = h_v.

The space $\widetilde { {\mathscr{H}}}=\{f: f\circ \pi \in H^{1}(\mathbb {R}^{2})\}$ equipped with the inner product $\langle f,g\rangle _{\widetilde H}:=\langle f\circ \pi ,g\circ \pi \rangle _{H^{1}(\mathbb {R}^{2})}$ is a Hilbert space. Let now $f\in \widetilde {{\mathscr{H}}}$. For i ∈ I, let $f_{i}:]l_{i},r_{i}[\to \mathbb {R}$ be defined by f_i(h) = f(h,i). Then (see Lemma 7.4), for all i ∈ I, f_i is weakly differentiable and we have

$$\|f\|^{2}_{\widetilde H} ={\sum}_{i\in I} {\int}_{l_{i}}^{r_{i}} (f^{\prime}_{i}(h))^{2} a_{i}(h) dh + {\sum}_{i\in I} {\int}_{l_{i}}^{r_{i}} (f_{i}(h))^{2} T_{i}(h) dh$$

where $\displaystyle a_{i}(h)=\oint _{\gamma _{h,i}} |\nabla H| d\ell$ and $\displaystyle T_{i}(h)=\oint _{\gamma _{h,i}} \frac {d\ell }{|\nabla H|}$. For $\widetilde x=(h,i)\in E_{i}$, set $a(\widetilde x)=a_{i}(h)$ and $T(\widetilde x)=T_{i}(h)$.

In the following lemma we recall asymptotics given in chapter 8 in [9].

Lemma A.4

Let v ∈ V and i ∈ I(v). Then v = (h_v,i) and as h → h_v, with (h,i) ∈ E_i,

• (1) If d(v) = 1,

  $$a_{i}(h)\sim a_{i,v} |h-h_{v}| \quad\text{ and }\quad T_{i}(h)\sim t_{i,v}$$

• (2) If d(v) ≥ 2,

  $$a_{i}(h)\sim a_{i,v} \quad\text{ and }\quad T_{i}(h)\sim t_{i,v} |\log|h-h_{v}||$$

where a_i,v and t_i,v are two finite positive constants.

Proof

By definition of v, either γ(v) contains exactly one extremum of H and d(v) = 1 or γ(v) contains a saddle point of H and d(v) ≥ 3 (d(v)≠ 2 since the stationary points of H are non-degenerate).

Suppose first that d(v) = 1. Then γ(v) = {x^∗}, with x^∗ a local extremum of H. If H is quadratic at x^∗ then, the computation of $a(\tilde x)$ and $T(\tilde x)$ can easily be done explicitly and gives the stated asymptotics. In the general case, one can use an asymptotic expansion or the Morse Lemma to conclude.

Suppose now that d(v) ≥ 2. Let $\tilde x=(h,i)\to v$. Since |∇H| is continuous and that γ(v) is compact with finite length, $a(\tilde x)$ converges to a constant $a_{i,v}=\oint _{\gamma _{h_{v},i}}|\nabla H|d \ell$ as $\tilde x\to v$. The main contribution for T(h) comes when the orbit γ_h,i is near a stationary point x^∗∈ γ(v) because otherwise |∇H| is bounded away from 0. Note that there could be several such stationary points, each is a saddle. If H is quadratic in a small neighborhood $\mathcal {V}$ of x^∗ then, the computation of an asymptotic can easily be done explicitly and gives $\oint _{\gamma _{h,i}\cap \mathcal {V}}\frac 1{|\nabla H|}d \ell \sim C^{*}|\log |h-h_{v}||$ for some C^∗ > 0 which only depends on the number of times the orbit comes near x^∗ (one or two times since x^∗ is not degenerated) and on the Hessian matrix of H at x^∗. In the general case, one can use the Morse Lemma to obtain the same asymptotics. Since each saddle point of γ(v) gives an asymptotic term of the same order, we obtain the stated result. □

Proof Proof of Proposition 7.5

Let $f\in \widetilde {\mathcal H}$. Let $v\in \mathcal {V}$ with d(v) ≥ 2, and i ∈ I_v. Let A_i be an open set of ]l_i,r_i[ such that $\bar {A}_{i}\subset ]l_{i},r_{i}[\cup \{h_{v}\}$. Note that there is c > 0 such that T_i ≥ a_i ≥ c on A_i. Thus f_i restricted to A_i belongs to H²(A_i) and as a consequence f_i can be extended to a continuous function on ]l_i,r_i[∪{h_v}. Set Γ_i,v = ∂Ω_i ∩ H^− 1(h_v). Since f ∘ π(x) = f_i ∘ H(x), we have $f_{|{\Gamma }_{i,v}}=f_{i}(h_{v})$.

Let now j ∈ I_v with j≠i. Suppose first that Γ_i,v ∩Γ_j,v≠∅, then f_i(h_v) = f_j(h_v). If Γ_i,v ∩Γ_j,v = ∅, then there is $(i_{0},i_{1},\dots ,i_{\ell })\in I_{v}^{\ell +1}$ such that i₀ = i, i_ℓ = j and for all 1 ≤ k ≤ ℓ, ${\Gamma }_{i_{k-1},v}\cap {\Gamma }_{i_{k},v}\ne \emptyset$ and so $f_{i}(h_{v})=f_{i_{1}}(h_{v})={\dots } =f_{i_{\ell -1}}(h_{v})=f_{j}(h_{v})$. This proves that f is continuous on $\widetilde M\setminus \mathcal {V}_{1}$, and completes the proof of the first implication of the Proposition 7.5.

Let now $f:M\to \mathbb {R}$ be a measurable map such that f is continuous on $\widetilde M\setminus \mathcal {V}_{1}$, f_i is weakly differentiable for all i ∈ I and $\|f\|_{\widetilde {\mathcal H}}<\infty$. Then, for all i, $f\circ \pi \in H^{1}({\Omega }_{i})$ and f ∘ π is continuous on M ∖Γ₁, where Γ₁ is the set of local extrema of H. Since Γ₁ is a finite set, $f\circ \pi \in H^{1}(\mathbb {R}^{2})$. This proves that $f\in \widetilde {\mathcal H}$. □

Proof Proof of Proposition 7.6

To complete this proof, it remains to check that Assumption 6.9-(ii) is satisfied, i.e. that $C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}$ is dense in $\widetilde { {\mathscr{H}}}$.

Let i and j in I be such that d(E_i,E_j) = 0. Denote by v_i,j the unique vertex in V such that v_i,j ∈ ∂E_i ∩ ∂E_j. Set E_i,j := E_i ∪ E_j ∪{v_i,j} and $f_{i,j}=f_{|E_{i,j}}$. Let A_i,j be an open subset of E_i,j such that $v_{i,j}\in A_{i,j}\Subset E_{i,j}$. Then (since there is c > 0 such that $T(\widetilde x)\ge a(\widetilde x)\ge c$ in a neighborhood of v), f_i,j ∈ H¹(A_i,j ∩ E_i) and u_i,j ∈ H¹(A_i,j ∩ E_j). Since f_i,j is continuous, we have that f_i,j is weakly differentiable.

For each i, let k_i :]l_i,r_i[→]0,L_i[ be defined such that k_i(l_i) = 0 and $dk_{i}(h)=\sqrt {\frac {T_{i}(h)}{a_{i}(h)}}dh$. Then k_i is properly defined (in particular $\displaystyle {\int \limits }_{l_{i}+} \sqrt {\frac {T_{i}(h)}{a_{i}(h)}}dh <\infty$ and $\displaystyle L_{i}={\int \limits }_{l_{i}}^{r_{i}} \sqrt {\frac {T_{i}(h)}{a_{i}(h)}}dh <\infty$ only for the unique i for which Ω_i is unbounded). Define a new metric graph with edges ${E^{G}_{i}}:=]0,L_{i}[\times \{i\}$ with the same adjacency rules as for $\widetilde M$. Denote this new metric graph by G. By abuse of notation, The set of vertices of G will also be denoted by V.

For v ∈ V, let I(v) be the set of all i such that ${E^{G}_{i}}$ is an edge adjacent to v. For i ∈ I(v), set ${O^{i}_{v}}:= \{x \in {E^{G}_{i}}: d(x,v)<\frac {2L_{i}}{3}\}$ and $O_{v}:=\{v\}\cup \cup _{i\in I(v)}{O^{i}_{v}}$. When d(v) = 1 set ${O^{0}_{v}}=O_{v}\setminus \{v\}$ and when d(v) ≥ 2, set ${O^{0}_{v}}=O_{v}$. Then ${O^{0}_{v}}$ is on open subset of B_m,1 with m = d(v) = |I(v)|.

Let G⁰ be the metric graph obtained out of G by taking out of G the vertices of degree 1, and denote by V⁰ the set of vertices of G⁰. Then it is easy to check that $({O^{0}_{v}})_{v\in V^{0}}$ is a covering of G⁰ and that there are functions φ_v, v ∈ V⁰, such that φ_v : G⁰ → [0, 1], φ_v = 0 on $G\setminus {O^{0}_{v}}$, φ_v restricted to ${O^{0}_{v}}$ is C¹, with bounded derivatives, for each $i\in \{1,\dots , d(v)\}$ and such that ${\sum }_{v\in V^{0}}\varphi _{v}=1$.

For $f\in \widetilde {{\mathscr{H}}}$, let $g:G^{0}\to \mathbb {R}$ be defined by g(v) = f(v) if v ∈ V⁰ and such that g(k_i(h),i) = f(h,i) if (h,i) ∈ E_i.

Let i≠j such that $d({E^{G}_{i}},{E^{G}_{j}})=0$ and set $E^{G}_{i,j}:={E^{G}_{i}}\cup {E^{G}_{j}}\cup \{x_{i,j}\}$, we then have that $g_{i,j}:=g_{|E^{G}_{i,j}}$ is weakly differentiable. This holds because f_i,j is weakly differentiable and g_i,j = f_i,j ∘ h_i,j, where $h_{i,j}:E^{G}_{i,j}\to E_{i,j}$ is the continuous function, differentiable on ${E^{G}_{i}}$ and on ${E^{G}_{j}}$, defined by $h_{i,j}(k,i)=(k_{i}^{-1}(k),i)$ if $(k,i)\in {E^{G}_{i}}$, $h_{i,j}(k,j)=(k_{j}^{-1}(k),j)$ if $(k,j)\in {E^{G}_{j}}$ and h_i,j(x_i,j) = x_i,j.

Then we have that $f\in \widetilde {{\mathscr{H}}}$ if and only if g defined above belongs to W(G⁰,ω) with ω a measurable function on G⁰ such that if $(k,i)\in {E^{G}_{i}}$, $\omega (k,i)=\omega _{i}(k)=\sqrt {a_{i}T_{i}}\circ k^{-1}_{i}(k)$.

Lemma A.5

For $\widetilde x\in G^{0}$ and v ∈ V. We have as $\widetilde x\to v$,

• (1) If d(v) = 1, $\omega (\widetilde x)\sim c_{v} d(\widetilde x,v)$

• (2) If d(v) ≥ 2, $\omega (\widetilde x)\sim c_{v}\sqrt {|\log (d(\widetilde x,v))|} \to \infty$

where c_v > 0.

Proof

The case d(v) = 1 is straightforward. For the case d(v) ≥ 2, we let $\widetilde x=(k,i)\in G^{0}$ and assume v = (0,i), then $k=d(\widetilde x,v)$ and $h:=k^{-1}_{i}(k)$ with $k_{i}^{\prime }(h)=\sqrt {\frac {T_{i}(h)}{a_{i}(h)}}\sim C_{v}\sqrt {|\log h|}$ as h → 0. An integration by part yields that $k(h)\sim C h\sqrt {|\log h|}$ and thus $|\log (k)|\sim |\log h|$. Then, since $\omega (\widetilde x)=\sqrt {a_{i}T_{i}}(h)\sim C\sqrt {|\log h|}$, we get the result. The same result holds if v = (L_i,i). □

To prove that $C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}$ is dense in $\widetilde {{\mathscr{H}}}$, we will now use Lemma A.3.

Fix R > 0. For v ∈ V and n ≥ 1, set O_v,R,n = O_v,R if d(v) ≥ 2 and set O_v,R,n := {x ∈ O_v,R : d(x,v) > n^− 1} if d(v) = 1. Then (i) of Lemma A.3 is satisfied.

Let us now check that (ii) is satisfied. This has only to be checked for v ∈ V with d(v) = 1. Let ${E^{G}_{i}}$ be the unique edge adjacent to v. Without loss of generality we will suppose that v = (0,i). We then have that O_v,R ∖ O_v,R,n =]0,n^− 1[×{i}. To prove that Cap(O_v,R ∖ O_v,R,n) → 0. In this case, fix A > 0 such that A < R and $A<\frac {2L_{i}}{3}$ and take (for n > A^− 1), g_n(k) = 1 if k ≤ n^− 1 and g_n(k) = 0 if k > A and $g_{n}(k)=\frac {\log (A)-\log (k)}{\log (A)-\log (n^{-1})}$ if n^− 1 ≤ k ≤ A. Then $Cap(O_{v,R}\setminus O_{v,R,n})\le {\int \limits }_{n^{-1}}^{A} \frac {\omega _{i}(k)}{k^{2}(\log (A)+\log (n))^{2}} dk + {\int \limits }_{n^{-1}}^{A} \frac {(\log (A)-\log (k))^{2}\omega _{i}(k)}{(\log (A)+\log (n))^{2}} dk$ which converges to 0 as $n\to \infty$. This proves (ii)

To check (iii), we shall use Lemma A.1. for the open sets O_v,R,n ⊂ B_m,1, with m = d(v). For each v,R,n, we define an extension $\bar {\omega }$ of $\omega _{|O_{v,R,n}}$, defined on B_m,1. By a slight abuse of notation, we set identify v as 0 in the definition of B_m,1 and keep the labels of I(v) to identify the branches of B_m,1.

If m = d(v) = 1 then recall that $B_{1,1}=\mathbb {R}$. O_v,R,n is an open interval ]l,r[ and we let $\bar {\omega }(k)=\omega (l)\in ]0,\infty [$ if k ≤ l and $\bar {\omega }(k)=\omega (r)\in ]0,\infty [$ if k > r. The $\bar {\omega }$ is continuous and positive, and it thus belongs to the class A₂.

If m = d(v) ≥ 2, for i ∈ I⁺(v), $O_{v,R,n}\cap {E^{G}_{i}}$ is an open interval ]0,r[ (with r = (2L_i/3) ∧ R) and for k > r, we set $\bar {\omega }(k,i)=\omega (r,i)\in ]0,\infty [$ and $\bar {\omega }(v)=\infty$. For i ∈ I⁻(v), $O_{v,R,n}\cap {E^{G}_{i}}$ is an open interval ]l,L_i[ (with l = L_i − (2L_i/3) ∧ R) and we set

$$ \bar{\omega}(k,i)= \begin{cases} \omega(L_{i}-k,i)\in ]0,\infty[&\text{ for }0<k<L_{i}-l \\ \omega(g,i)\in ]0,\infty[&\text{ for }k>L_{i}-l \end{cases} $$

and $\bar {\omega }(v)=\infty$. This defines $\bar {\omega }$ on B_m,1. It remains to check that for i≠j in I(v), we have that $\bar {\omega }_{i,j}$ belongs to the class A₂. Note that $\lim _{x\to v}\bar {\omega }_{i,j}(x)=\infty$. Since $\omega _{i,j}(x)\sim c \sqrt {\log (k)}$ as k := d(x,v) → 0, it is a simple exercise to show that $\bar {\omega }_{i,j}$ belongs to the class A₂.

Then Lemma A.3 can be applied. This proves that $W(G^{0},\omega )={H^{1}_{0}}(G^{0},\omega )$. Let $f\in \widetilde {{\mathscr{H}}}$ and g ∈ W(G⁰,ω) defined as above by $g(k,i)=f(k_{i}^{-1}(k),i)$ if $(k,i)\in {E^{G}_{i}}$ and g(v) = f(v) if v ∈ G⁰. Since $W(G^{0},\omega )={H^{1}_{0}}(G^{0},\omega )$, for all 𝜖 > 0 there is g_𝜖 ∈ C_c(G⁰) ∩ W(G⁰,ω) such that $\|g-g_{\epsilon }\|_{W(G^{0},\omega )}<\epsilon$. Set f_𝜖(h,i) = g_𝜖(k_i(h),i) if (h,i) ∈ E_i, f_𝜖(v) = g_𝜖(v) if v ∈ V⁰ and f_𝜖(v) = 0 if v ∈ V ∖ V₀. Then $f_{\epsilon }\in C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}$ and $\|f-f_{\epsilon }\|_{\widetilde {{\mathscr{H}}}}=\|g-g_{\epsilon }\|_{W(G^{0},\omega )}<\epsilon$. □

1.3 Regularity for Section ??

To prove the regularity we can apply directly Lemma A.2 since our state space will already be an open subset of B_4,2. Set I := {1, 2, 3, 4}. Recall that $\widetilde {M}=\cup _{i\in I} C_{i}$ is constituted of four cones glued along one edge. Set $\widetilde M_{0}:=\{y\in \widetilde M: y_{1}y_{2} > 0\}$.

1.3.1 The Space $\widetilde {\mathcal H}$

Recall that for t ∈]0, 1[

$$ \begin{array}{@{}rcl@{}} K(t)&=&{\int}_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-t^{2}\sin^{2}\theta}}, E(t)={\int}_{0}^{\pi/2} \sqrt{1-t^{2}\sin^{2}\theta} d\theta \end{array} $$

(A.3)

$$ \begin{array}{@{}rcl@{}} \alpha(t)&=&1-\frac{E(t)}{K(t)} \qquad\quad\text{ and } \lambda(t)=1-\frac{\alpha(t)(1+t^{2})}{2t^{2}}. \end{array} $$

(A.4)

For y ∈∪_i∈IC̈_i, let $t(y)=\frac {|y_{1}|\wedge |y_{2}|}{|y_{1}|\vee |y_{2}|}$. Then $t(y)=\frac {y_{2}}{y_{1}}$ for y ∈C̈₁ ∪C̈₃ and $t(y)=\frac {y_{1}}{y_{2}}$ for y ∈C̈₂ ∪C̈₄.

For i ∈ I, set $\widetilde {m}_{i}$ the measure on C_i with density h_i(y)e^{−W(∥y∥)} with respect to Lebesgue measure on C_i, where

$$ \begin{array}{@{}rcl@{}} h_{1}(y)=h_{3}(y)&=4\sqrt{2} |y_{2}|K\left( t(y)\right); h_{2}(y)=h_{4}(y)=4\sqrt{2} |y_{1}|K\left( t(y)\right). \end{array} $$

(A.5)

For y ∈∪_i∈IC̈_i, set a(y) := ν_y(∇π ⊗∇π). We have that $\nu _{y}({x_{3}^{2}})=2{y_{1}^{2}}\alpha (t)$ for y ∈C̈₁, then,

$$ \begin{array}{@{}rcl@{}} a(y)&=&\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} +\frac{\alpha(t)}{2t^{2}}\begin{pmatrix} -t^{2}&t\\ t&-1 \end{pmatrix} \text{ if } y\in \mathring{C}_{1}\cup \mathring{C}_{3} \text{ and } t=\frac{y_{2}}{y_{1}};\\ a(y)&=&\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} +\frac{\alpha(t)}{2t^{2}}\begin{pmatrix} -1&t\\ t&-t^{2} \end{pmatrix} \text{ if } y\in \mathring{C}_{2}\cup \mathring{C}_{4} \text{ and } t=\frac{y_{1}}{y_{2}}. \end{array} $$

(A.6)

The eigenvalues of a(y) are 1 and λ(t), with corresponding eigenvectors (y₁,y₂) and (−y₂,y₁). If i ∈ I and y ∈C̈_i, set a_i(y) = a(y).

Let us now collect some asymptotics.

Lemma A.6

As t → 0⁺,

$$ \begin{array}{@{}rcl@{}} E(t)&=&\frac{\pi}{2}\left( 1-\frac{t^{2}}{4}+O(t^{4})\right); K(t)=\frac{\pi}{2}\left( 1+\frac{t^{2}}{4}+O(t^{4})\right); \end{array} $$

(A.7)

$$ \begin{array}{@{}rcl@{}} \alpha(t)&=&\frac{t^{2}}{2}\left( 1+O(t^{2})\right); \lambda(t)=\frac{3}{4}+O(t^{2}). \end{array} $$

(A.8)

As t → 1⁻,

$$ \begin{array}{@{}rcl@{}} E(t)&=&1+o(1); K(t)=-\frac{1}{2}\log(1-t) + O(1); \end{array} $$

(A.9)

$$ \begin{array}{@{}rcl@{}} \alpha(t)&=&1+\frac{2}{\log(1-t)}(1+o(1)); \lambda(t)=\frac{-2}{\log(1-t)}(1+o(1)). \end{array} $$

(A.10)

Proof

Asymptotics for E and K are standard, the others are simple consequences. □

Let $f\in \widetilde {{\mathscr{H}}}$. For i ∈ I, the map $f_{i}:=f_{|C_{i}}$ is weakly differentiable on C_i and, using the change of variable formula Eq. ??, we have that

$$ \|f\|^{2}_{\widetilde{\mathcal{H}}}={\sum}_{i=1}^{4}{\int}_{C_{i}}f^{2} d\widetilde{m}_{i}+{\sum}_{i=1}^{4}{\int}_{C_{i}}a_{i}^{k\ell}\partial_{k} f_{i}\partial_{\ell} f_{i}d\widetilde{m}_{i}. $$

(A.11)

Set ${\Omega }:=]0,+\infty [\times B_{4,1}\subset B_{4,2}$, Ω_i := Ω ∩ E_i for i ∈ I and Ω_i,j := Ω ∩ E_i,j for i≠j ∈ I. Let us remark from Lemma A.6 that $\displaystyle \mathcal {I}:={{\int \limits }_{0}^{1}} \frac {du}{(1+u^{2})\sqrt {\lambda (u)}} <+\infty$. Set $c=\frac {\pi }{2\mathcal {I}}$ and define $\varphi :[0,1]\to [0,\frac {\pi }2]$ by

$$ \varphi(t)=c {{\int}_{t}^{1}} \frac{du}{(1+u^{2})\sqrt{\lambda(u)}}. $$

(A.12)

Define $z:\cup _{i=1}^{4} \mathring {C}_{i} \to ]0,+\infty [^{2}$ by

$$ \begin{array}{@{}rcl@{}} z(y)&=|y|\left( \cos\varphi (t(y)),\sin \varphi\left( t(y)\right)\right). \end{array} $$

(A.13)

Note that for all i ∈ I, $z_{i}= z_{|\mathring {C}_{i}}$ is one to one.

For $g:{\Omega }\to \mathbb {R}$ define $\mathcal F(g):\widetilde {M}_{0}\to \mathbb {R}$ by $\mathcal F(g)(y)=g(z(y),i)$ if y ∈C̈_i and $\mathcal {F}(g)(y)=g(y,0)$ if y ∈ D^∗. For $f:\widetilde {M}\to \mathbb {R}$ (or for $f:\widetilde {M}_{0}\to \mathbb {R}$), define $\mathcal G(f):{\Omega }\to \mathbb {R}$ such that if (z,i) ∈Ω_i, then $\mathcal {G}(f)(z,i)=f_{i}\circ z_{i}^{-1}(z)$ and if z > 0, then $\mathcal G(f)(z,0)=f(z)$. Then, $\mathcal G\circ \mathcal F(g)=g$ and we have that g ∈ C_c(Ω) if and only if $\mathcal {F}(g)\in C_{c}(\widetilde {M}_{0})$.

Define $k:[0,\frac {\pi }{2}[\to \mathbb {R}^{+}\cup \{\infty \}$ by $k(0)=\infty$ and, for $\phi \in ]0,\frac {\pi }{2}[$ and t = φ^− 1(ϕ),

$$ k(\phi):=\frac{4\sqrt{2}tK(t)\sqrt{\lambda(t)}}{c\sqrt{1+t^{2}}}. $$

(A.14)

For $z=r({\cos \limits } \phi ,{\sin \limits } \phi )$ with $(r,\phi )\in ]0,\infty [\times [0,\frac {\pi }2[$, set ω(z,i) := re^−W(r)k(ϕ) and

$$ b(z,i):=\begin{pmatrix} \cos^{2}\phi+c^{2}\sin^{2}\phi & (1-c^{2})\sin\phi\cos\phi\\ (1-c^{2})\sin\phi\cos\phi & \sin^{2}\phi+c^{2}\cos^{2}\phi \end{pmatrix} =R_{\phi} \begin{pmatrix} 1 & 0\\ 0 & c^{2} \end{pmatrix} R_{-\phi}$$

where R_ϕ is the matrix associated to the rotation of angle ϕ,

$$R_{\phi}= \begin{pmatrix} \cos\phi&-\sin\phi\\ \sin\phi&\cos\phi \end{pmatrix}.$$

Let $f\in \widetilde {\mathcal H}$ and $g=\mathcal G(f)$. The change of variable z = z_i(y) on each C_i yields

$$ \begin{array}{@{}rcl@{}} \|f\|^{2}_{\widetilde{\mathcal{H}}}={\sum}_{i=1}^{4}{\int}_{{\Omega}_{i}}g^{2}(z,i)\omega(z,i)dz+{\sum}_{i=1}^{4}{\int}_{{\Omega}_{i}}b^{k\ell}(z,i)\partial_{k} g(z,i)\partial_{\ell} g(z,i)\omega(z,i)dz. \end{array} $$

(A.15)

Recall that the norm on W¹(Ω,ω) is given by

$$ \|g\|^{2}_{W^{1}({\Omega},\omega)}={\sum}_{i=1}^{4}{\int}_{{\Omega}_{i}}(g^{2}(z,i)+|\nabla g(z,i)|^{2})\omega(z,i)dz. $$

(A.16)

We thus get that

$$ (1\wedge c^{2})\|g\|^{2}_{W^{1}({\Omega},\omega)} \leq \|f\|^{2}_{\widetilde{\mathcal{H}}} \leq(1\vee c^{2})\|g\|^{2}_{W^{1}({\Omega},\omega)}. $$

(A.17)

Lemma A.7

$f\in \widetilde {{\mathscr{H}}}$ if and only if $\mathcal G(f)\in W^{1}({\Omega },\omega )$.

Proof

In this proof, we fix $f:\widetilde M\to \mathbb {R}$ and we set $g:=\mathcal G(f)$. If $f \in \widetilde H$, then for each i, f_i is weakly differentiable on C̈_i and $g_{|{\Omega }_{i}}$ is weakly differentiable on Ω_i. And Eq. A.17 show that $g_{i}:=g_{|{\Omega }_{i}}\in W^{1}({\Omega }_{i},\omega _{i})$, where $\omega _{i}=\omega _{|{\Omega }_{i}}$. Moreover, if $\mathcal {O}$ be a bounded open set in Ω_i,j, then $g_{|\mathcal {O}_{i}}\in H^{1}(\mathcal {O}_{i})$, $g_{|\mathcal {O}_{j}}\in H^{1}(\mathcal {O}_{j})$ and coincide on $\mathcal {O}\cap E_{0}$. Since ω is bounded from below on $\mathcal {O}$, this implies that $g_{|\mathcal {O}}\in H^{1}(\mathcal {O})\supset W^{1}(\mathcal {O},\omega _{i,j})$. We thus have that $g_{|{\Omega }_{i,j}}\in W^{1}({\Omega }_{i,j},\omega _{i,j})$, which proves that g ∈ W¹(Ω,ω).

On the converse, if g ∈ W¹(Ω,ω), then using that $f\circ \pi =\mathcal {F}(g)\circ \pi$ and Eq. A.17, we have f ∘ π ∈ H¹(m) and so $f\in \widetilde H$. □

Lemma A.8

${H^{1}_{0}}({\Omega },\omega )=W^{1}({\Omega },\omega )$

We use Lemma A.2 to prove this lemma.

This lemma shows the density of $C_{c}(\widetilde M)\cap \widetilde {{\mathscr{H}}}$ in $\widetilde {{\mathscr{H}}}$. Indeed, $\mathcal {F}:\widetilde H\to W^{1}({\Omega },\omega )$ and $\mathcal {G}: W^{1}({\Omega },\omega )$ are continuous mappings and if g ∈ C_c(Ω) ∩ W¹(Ω,ω) then $\mathcal {F}(g)\in C_{c}(\widetilde M)\cap \widetilde H$. Therefore, if $f\in \widetilde H$ and $g=\mathcal {G}(f)$, there is a sequence g_n ∈ C_c(Ω) ∩ W¹(Ω,ω) approximating g in W¹(Ω,ω) and so $f_{n}:=\mathcal {F}(g_{n})$ is a sequence in $C_{c}(\widetilde M)\cap \widetilde H$ approximating f in $\widetilde H$.

Proof Proof of Lemma A.8

For R > 0, recall that Ω_R = {z ∈Ω : ∥z∥ < R} and define, for n ≥ 1, ${\Omega }_{R,n}:=\{(r{\cos \limits } \phi , r{\sin \limits } \phi ,i) : (r,\phi ,i)\in ]0,R[\times [0,\frac \pi 2-\frac 1n[\times I\}$.

In order to apply Lemma A.2, we have to check that (i), (ii) and (iii) are satisfied. Item (i) is clearly satisfied.

To prove (ii), we construct a sequence of non negative functions (g_n) on Ω, such that g_n(z) = 1 in Ω_R,n ∖Ω_R and $\lim _{n\to +\infty }\|g_{n}\|^{2}_{W^{1}({\Omega },\omega )}=0$. Let us first obtain asymptotics for ω.

Lemma A.9

We have, as $\phi \to \frac {\pi }{2}^{-}$, $k(\phi )\sim \frac {3\pi }{8c^{2}}(\frac {\pi }{2}-\phi ).$

Proof

Using Lemma A.6 and Eq. A.12, we have that $\frac {\pi }{2}-\varphi (t)\sim \frac {2c}{\sqrt {3}}t$ as t → 0⁺. Thus, since ϕ = φ(t), we have $t\sim \frac {\sqrt {3}}{2c}(\frac {\pi }{2}-\phi )$ as $\phi \to \frac {\pi }{2}^{-}$. From Lemma A.6 and Eq. A.14, we obtain $k(\phi )\sim \frac {\pi \sqrt {3}}{4c}t\sim \frac {3\pi }{8c^{2}}(\frac {\pi }{2}-\phi )$ as $\phi \to \frac {\pi }{2}^{-}$. □

Using this Lemma, we can fix $0<\phi _{0}<\frac \pi 2$ and c₀ > 0 such that $0\leq k(\phi )\leq c_{0}(\frac {\pi }{2}-\phi )$ for all $\phi \in [\phi _{0},\frac {\pi }{2}]$. Then we define $f_{1}:]0,+\infty [\to \mathbb {R}$, $f_{2,n}:[0,\frac {\pi }2[\to \mathbb {R}$ by

$$ \begin{array}{@{}rcl@{}} f_{1}(r)&=& \begin{cases} 1&\text{ for }r<R \\ 2-\frac{r}{R}&\text{ for }R\leq r<2R \\ 0&\text{ for }2R\leq r \end{cases}, \end{array} $$

(A.18)

$$ \begin{array}{@{}rcl@{}} f_{2,n}(\phi)&=& \begin{cases} 0&\text{ for }0\leq \phi\leq \phi_{0} \\ \frac{\ln(\frac{\pi}{2}-\phi)-\ln(\frac{\pi}{2}-\phi_{0})}{\ln(\frac1n)-\ln(\frac{\pi}{2}-\phi_{0})}&\text{ for }\phi_{0}\leq\phi<\frac{\pi}{2}-\frac1n \\ 1&\text{ for }\frac{\pi}{2}-\frac1n\le \phi \end{cases} \end{array} $$

(A.19)

For i ∈ I and $z=r(\cos \limits (\phi ),\sin \limits (\phi ))\in ]0,+\infty [^{2}$, set g_n(z,i) = f₁(r)f_2,n(ϕ). We then have

$$ \|g_{n}\|^{2}_{W^{1}({\Omega},\omega)}=4{\int}_{{\Omega}_{1}}({g_{n}^{2}}(z)+|\nabla g_{n}(z)|^{2})\omega(z)dz. $$

(A.20)

Since $|\nabla g_{n}(z)|^{2}=(f_{1}^{\prime })^{2}(r)f^{2}_{2,n}(\phi )+\frac 1{r^{2}}{f^{2}_{1}}(r)(f^{\prime }_{2,n})^{2}(\phi )$, we have

$$ \begin{array}{@{}rcl@{}} {\int}_{{\Omega}_{1}}{g_{n}^{2}}(z)\omega(z)dz &=&{\int}_{0}^{2R}{f^{2}_{1}}(r)r^{2}e^{-W(r)}dr{\int}_{\phi_{0}}^{\frac{\pi}{2}}f^{2}_{2,n}(\phi)k(\phi)d\phi, \end{array} $$

(A.21)

$$ \begin{array}{@{}rcl@{}} {\int}_{{\Omega}_{1}}|\nabla g_{n}(z)|^{2}\omega(z)dz &=&{\int}_{0}^{2R}(f^{\prime}_{1})^{2}(r)r^{2}e^{-W(r)}dr{\int}_{\phi_{0}}^{\frac{\pi}{2}}f^{2}_{2,n}(\phi)k(\phi)d\phi\\ &&+{\int}_{0}^{2R}{f^{2}_{1}}(r)e^{-W(r)}dr{\int}_{\phi_{0}}^{\frac{\pi}{2}}(f^{\prime}_{2,n}(\phi))^{2}k(\phi)d\phi. \end{array} $$

(A.22)

The integrals involving f₁ are finite and does not depend on n. For f_2,n, we have

$$ \begin{array}{@{}rcl@{}} {\int}_{\phi_{0}}^{\frac{\pi}{2}}f^{2}_{2,n}(\phi)k(\phi)d\phi &\leq& c_{0}{\int}_{\phi_{0}}^{\frac{\pi}{2}}f^{2}_{2,n}(\phi)\left( \frac{\pi}{2}-\phi\right) d\phi \end{array} $$

(A.23)

$$ \begin{array}{@{}rcl@{}} &=& c_{0}\left( {\int}_{\phi_{0}}^{\frac{\pi}{2}-\frac{1}{n}}f^{2}_{2,n}(\phi)\left( \frac{\pi}{2}-\phi\right) d\phi + {\int}_{\frac{\pi}{2}-\frac{1}{n}}^{\frac{\pi}{2}}\left( \frac{\pi}{2}-\phi\right) d\phi\right) \end{array} $$

(A.24)

$$ \begin{array}{@{}rcl@{}} &\leq& c_{0}\left( {\int}_{\phi_{0}}^{\frac{\pi}{2}}\left( \frac{\ln(\frac{\pi}{2}-\phi)-\ln(\frac{\pi}{2}-\phi_{0})}{\ln(\frac{1}{n})-\ln(\frac{\pi}{2}-\phi_{0})}\right)^{2} \left( \frac{\pi}{2}-\phi\right) d\phi + \frac1{2n^{2}}\right).\\ \end{array} $$

(A.25)

Thus, $\lim _{n\to +\infty }{\int \limits }_{\phi _{0}}^{\frac {\pi }{2}}f^{2}_{2,n}(\phi )k(\phi )d\phi =0$. We also have

$$ \begin{array}{@{}rcl@{}} {\int}_{\phi_{0}}^{\frac{\pi}{2}}(f^{\prime}_{2,n}(\phi))^{2}k(\phi)d\phi &\leq& c_{0}{\int}_{\phi_{0}}^{\frac{\pi}{2}-\frac{1}{n}}(f^{\prime}_{2,n})^{2}(\phi)\left( \frac{\pi}{2}-\phi\right) d\phi \end{array} $$

(A.26)

$$ \begin{array}{@{}rcl@{}} &=&\frac{c_{0}}{(\ln(n^{-1})-\ln(\frac{\pi}{2}-\phi_{0}))^{2}}{\int}_{\phi_{0}}^{\frac{\pi}{2}-\frac{1}{n}}\frac{d\phi}{\frac{\pi}{2}-\phi} \end{array} $$

(A.27)

$$ \begin{array}{@{}rcl@{}} &\leq&\frac{c_{0}}{|\ln(n^{-1})-\ln(\frac{\pi}{2}-\phi_{0})|}. \end{array} $$

(A.28)

Thus, $\lim _{n\to +\infty } {\int \limits }_{\phi _{0}}^{\frac {\pi }{2}}(f^{\prime }_{2,n}(\phi ))^{2}k(\phi )d\phi =0$ and then $\lim _{n\to +\infty }\|g_{n}\|^{2}_{W^{1}({\Omega },\omega )}=0$. Item (ii) is proven.

To prove (iii), we apply Lemma A.1. Let us first define an extension $\bar {\omega }_{n,R}$ to B_4,2 of ω which coincide with ω on Ω_R,n. We define $k_{n}:[-\pi ,\pi ]\to \mathbb {R}^{+}\cup \{\infty \}$ by

$$ k_{n}(\phi)= \begin{cases} k(|\phi|)&\text{ for } \phi\in [-\frac\pi2+\frac1n,\frac\pi2-\frac1n]\\ k(\frac\pi2-\frac1n)&\text{ elsewhere,} \end{cases} $$

(A.29)

and define the weight $\bar {\omega }_{n,R}:B_{4,2}\to \mathbb {R}^{+}$ by $\bar {\omega }_{n,R}(z,i):=re^{-W(r\wedge R)}k_{n}(\phi )$ where $z=(r\cos \limits \phi , r\sin \limits \phi )$ with $(r,\phi ,i)\in \mathbb {R}^{+}\times [-\pi ,\pi ]\times I$. Then, $\bar {\omega }_{n,R}$ coincide with ω on Ω_R,n. Note that for i≠j, the restriction of $\bar {\omega }_{n,R}$ to E_i,j (identified with $\mathbb {R}^{2}$) does not depend on i and j. Denote this common measure ω_n,R and set $\omega _{n}:\mathbb {R}^{2}\to \mathbb {R}^{+}\cup \{+\infty \}$, defined by ω_n(z) = rk_n(ϕ) where $z=r(\cos \limits \phi ,\sin \limits \phi )\in \mathbb {R}^{2}$. Since, $0<\inf _{r>0} e^{-W(r\wedge R)} \le \sup _{r>0}e^{W(r\wedge R)}$, ω_n,R belongs to the class A₂ if and only if ω_n belongs to the class A₂.

Define for ρ > 0, $z\in \mathbb {R}^{2}$,

$$C(z,\rho)=\frac1{\pi^{2} \rho^{4}}{\int}_{B(z,\rho)}\omega_{n}{\int}_{B(z,\rho)}\frac1{\omega_{n}}.$$

where B(z,ρ) is the ball at center z and radius ρ.

Let us first obtain asymptotics for ω_n.

Lemma A.10

We have, as ϕ → 0 +, $k(\phi )\sim \frac 1{2c}\left |\log \left (\phi \right )\right |^{1/2}.$

Proof

For ϕ > 0 close to 0 and t = φ^− 1(ϕ), set δ := 1 − t. We have

$$ \phi=c{\int}_{1-\delta}^{1}\frac{du}{(1+u^{2})\sqrt{\lambda(u)}}=c{\int}_{0}^{\delta}\frac{du}{(1+(1-u)^{2})\sqrt{\lambda(1-u)}} $$

(A.30)

As u → 0⁺, using Lemma A.6, we get $\sqrt {\lambda (1-u)}\sim \frac {\sqrt {2}}{\sqrt {|\ln (u)|}}$, and thus as ϕ → 0⁺, by integration by parts

$$ \begin{array}{@{}rcl@{}} \phi\sim \frac{c}{2\sqrt{2}}{\int}_{0}^{\delta}\sqrt{|\log(u)|}du\sim\frac{c}{2\sqrt{2}}\delta\sqrt{|\log(\delta)|}. \end{array} $$

(A.31)

This entails that $\log (\delta )\sim \log (\phi ).$ It is straightforward, using Lemma A.6 again, that $K(t)\sqrt {\lambda (t)}\sim \frac 1{\sqrt {2}}\sqrt {|\log (\delta )|}$. Then we obtain that $k(\phi )\sim \frac 1{2c}\sqrt {|\log (\delta )|} \sim \frac 1{2c}\sqrt {|\log (\phi )|}$. □

Let us now prove the condition A2 for ω_n.

Lemma A.11

ω_n satisfies the A₂-Muckenhoupt condition: $\sup _{z,\rho }C(z,\rho )<+\infty$.

Proof Proof of Lemma A.11

Note that k_n is a continuous positive function on [−π,π] ∖{0}. Lemma A.10 shows that $k_{n}(\phi )\sim \frac 1{2c}\sqrt {\left |\log |\phi |\right |}(1+o(1))$ as ϕ → 0. Thus the integrals ${\int \limits }_{-\pi }^{\pi }k_{n}(\phi )d\phi$ and ${\int \limits }_{-\pi }^{\pi }k_{n}(\phi )^{-1}d\phi$ are finite and ω_n and $\omega _{n}^{-1}$ are locally integrable on $\mathbb {R}^{2}$. Therefore (z,ρ)↦C(z,ρ) is a continuous function. Note also that for λ > 0 and $z\in \mathbb {R}^{2}$, ω_n(λz) = λω_n(z), then C(λz,λρ) = C(z,ρ). This leaves two cases to investigate: z = 0, ρ = 1 or |z| = 1, ρ > 0.

For z = 0, ρ = 1, we have

$$ \begin{array}{@{}rcl@{}} C(0,1)=\frac1{\pi^{2}}{\int}_{B(0,1)}\omega_{n}{\int}_{B(0,1)}\frac1{\omega_{n}} =\frac1{3\pi^{2}}\displaystyle{\int}_{-\pi}^{\pi}k_{n}{\int}_{-\pi}^{\pi}\frac1{k_{n}}<+\infty. \end{array} $$

(A.32)

Obviously, $C(0,\rho )=C(0,1)=C_{n}<+\infty$ does not depend on ρ.

For the second case, fix ρ > 0 and $z=(\cos \limits \phi ,\sin \limits \phi )$, with ϕ ∈ [−π,π]. If $\rho \geq \frac {1}{2}$, since B(z,ρ) ⊂ B(0,ρ + 1), we get $C(z,\rho )\leq \frac {(\rho +1)^{4}}{\rho ^{4}}C_{n}\le 3^{4} C_{n}$. If $\rho <\frac 12$, we set $\theta =\arcsin (\rho )\in ]0,\frac {\pi }{6}[$. We have $B(z,\rho )\subset T(z,\rho ):=\{z=r(\cos \limits \psi ,\sin \limits \psi ); (r,\psi )\in [1-\rho ,1+\rho ]\times [\phi -\theta ,\phi +\theta ]\}$. We obtain

$$ \begin{array}{@{}rcl@{}} {\int}_{B(z,\rho)}\omega_{n} \leq{\int}_{T(z,\rho)}\omega_{n} &=&{\int}_{1-\rho}^{1+\rho} r^{2}dr{\int}_{\phi-\theta}^{\phi+\theta}k_{n} =\frac23(3\rho+\rho^{3}){\int}_{\phi-\theta}^{\phi+\theta}k_{n}, \end{array} $$

(A.33)

$$ \begin{array}{@{}rcl@{}} {\int}_{B(z,\rho)}\frac1{\omega_{n}} \leq{\int}_{T(z,\rho)}\frac1{\omega_{n}} &=&2\rho{\int}_{\phi-\theta}^{\phi+\theta}\frac1{k_{n}}. \end{array} $$

(A.34)

Then we get (setting $c_{0}=\frac {13}{3\pi ^{2}}$)

$$ C(z,\rho)\leq \frac{4(3+\rho^{2})}{3\pi^{2}\rho^{2}}{\int}_{\phi-\theta}^{\phi+\theta}k_{n}{\int}_{\phi-\theta}^{\phi+\theta}\frac1{k_{n}} \leq D(\phi,\theta):=\frac{c_{0}}{\sin(\theta)^{2}}{\int}_{\phi-\theta}^{\phi+\theta}k_{n}{\int}_{\phi-\theta}^{\phi+\theta}\frac1{k_{n}}. $$

(A.35)

The function (ϕ,𝜃)↦D(ϕ,𝜃) is continuous on $[-\pi ,\pi ]\times ]0,\frac \pi 6]$. Let us now show that D is bounded in a neighborhood of 𝜃 = 0 +. Note that

$$ \begin{array}{@{}rcl@{}} D(\phi,\theta) \leq\frac{4c_{0}\theta^{2}}{\sin(\theta)^{2}} \frac{\sup_{[\phi-\theta,\phi+\theta]}k_{n}}{\inf_{[\phi-\theta,\phi+\theta]}k_{n}} \leq 8c_{0}\frac{\sup_{[\phi-\theta,\phi+\theta]}k_{n}}{\inf_{[\phi-\theta,\phi+\theta]}k_{n}} \end{array} $$

(A.36)

since $\frac {\theta ^{2}}{\sin \limits (\theta )^{2}}<2$ for all $\theta \in [0,\frac {\pi }6]$.

Fix $\theta _{0}\in ]0,\frac {\pi }{6}[$. We have $\sup _{(\phi ,\theta )\in [-\pi ,\pi ]\times [\theta _{0},\frac \pi 6]} D(\phi ,\theta )<\infty$. From Lemma A.10, we get that there are 0 < c₁ < C₁ and 0 < c₂ < C₂ such that

$$ \begin{array}{@{}rcl@{}} c_{1}&\leq &k_{n}(\phi)\leq C_{1}, \text{ for }\phi\in[-\pi,\pi]\setminus [-\theta_{0},\theta_{0}] \end{array} $$

(A.37)

$$ \begin{array}{@{}rcl@{}} c_{2}\sqrt{|\log(|\phi[)|}&\leq &k_{n}(\phi)=k(\phi)\leq C_{2}\sqrt{|\log(|\phi|)|}, \text{ for }\phi\in[-3\theta_{0},3\theta_{0}]. \end{array} $$

(A.38)

Let 𝜃 < 𝜃₀. Without loss of generality, we assume that ϕ ∈ [0,π]. If ϕ − 𝜃 ≥ 𝜃₀, we get [ϕ − 𝜃,ϕ + 𝜃] ⊂ [𝜃₀,π + 𝜃₀] and we get, using Eq. A.37, $D(\phi ,\theta )\leq \frac {4C_{1}}{c_{1}}$.

If ϕ − 𝜃 < 𝜃₀, then [ϕ − 𝜃,ϕ + 𝜃] ⊂ [−𝜃₀, 3𝜃₀], and we consider again two subcases. First subcase: ϕ ≥ 2𝜃, let u := ϕ + 𝜃 ≥ 3𝜃,

$$ \begin{array}{@{}rcl@{}} D(\phi,\theta) &\leq& \frac{8c_{0}C_{2}}{c_{2}} \left( \frac{\log(u-2\theta)}{\log(u)}\right)^{\frac12} \leq \frac{8c_{0}C_{2}}{c_{2}} \left( 1+ \frac{\log(1-\frac{2\theta}{u})}{\log(u)}\right)^{\frac12} \end{array} $$

(A.39)

$$ \begin{array}{@{}rcl@{}} &\leq& \frac{8c_{0}C_{2}}{c_{2}} \left( 1+ \frac{\log(1-\frac{2}{3})}{\log(u)}\right)^{\frac12} \leq \frac{8c_{0}C_{2}}{c_{2}} \left( 1+ \frac{\log(\frac1{3})}{\log(3\theta_{0})}\right)^{\frac12}. \end{array} $$

(A.40)

In the second subcase we have ϕ < 2𝜃. We start back from Eq. A.35:

$$ \begin{array}{@{}rcl@{}} D(\phi,\theta)\leq\frac{c_{0}}{\sin(\theta)^{2}}{\int}_{0}^{\phi+\theta}k_{n}{\int}_{0}^{\phi+\theta}\frac1{k_{n}} \leq C_{2}(\theta):=\frac{c_{0}}{\sin(\theta)^{2}}{\int}_{0}^{3\theta}k_{n}{\int}_{0}^{3\theta}\frac1{k_{n}}. \end{array} $$

(A.41)

C₂ is a continuous function on ]0,𝜃₀] and as 𝜃 → 0⁺, we get (using Lemma A.10

$$ \begin{array}{@{}rcl@{}} {\int}_{0}^{3\theta}k_{n}(\phi)d\phi \sim\frac{3\theta}{2c}\sqrt{-\log(3\theta)} \end{array} $$

(A.42)

$$ \begin{array}{@{}rcl@{}} {\int}_{0}^{3\theta}k_{n}(\phi)^{-1}d\phi \sim 6c\theta\frac1{\sqrt{-\log(3\theta)}}. \end{array} $$

(A.43)

Thus $\lim _{\theta \to 0^{+}} C_{2}(\theta )=18.$ Finally, this gives us the result. □

With Lemma A.11, conditions of Lemma A.1 are satisfied and condition (iii) of Lemma A.2 holds. Therefore, $W^{1}({\Omega },\omega )={H^{1}_{0}}({\Omega },\omega )$. □