From Statistical Polymer Physics to Nonlinear Elasticity

Abstract

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the $\Gamma$-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the $\Gamma$-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).

Appendix A

In this appendix we collect and prove some of the results we used in the paper. We start with the technical proof of the interpolation inequality.

Proof of Proposition 1

We set $\beta =1$ to reduce the notation. Given $\delta >0$ and $N\in {\mathbb {N}}$, for $i\in \{1,\dots ,N+1\}$ we introduce the open sets

$$\begin{aligned} O_i=\left\{ x\in O:\;{\mathrm{dist}}(x,\partial O)>(i+1)\frac{\delta }{2N}\right\} . \end{aligned}$$

Then the stripes $S_i:=O_{i-1}\backslash \overline{O_{i+2}}$ fulfill $S_i\cap S_j=\emptyset$ whenever $|i-j|>2$. Thus for every $u:O_{\varepsilon }^{\mathcal {L}}\rightarrow {\mathbb {R}}^n$ we obtain by averaging

$$\begin{aligned} \frac{1}{N}\sum _{i=1}^{N}H_{\varepsilon }(S_i,u)\leqq \frac{3}{N}H_{\varepsilon }(O,u), \end{aligned}$$

so that we can decompose the set ${\mathcal {N}}_p(v,O,\varepsilon ,\kappa )=\bigcup _{i=1}^N{\mathcal {P}}_{i,\varepsilon }$ (we omit the dependence on O and $\kappa$), where

$$\begin{aligned} {\mathcal {P}}_{i,\varepsilon }=\left\{ u\in {\mathcal {N}}_p(v,O,\varepsilon ,\kappa ):\;H_{\varepsilon }(S_i,u)\leqq \frac{3}{N}H_{\varepsilon }(O,u)\right\} . \end{aligned}$$

Let $\theta _i:O\rightarrow [0,1]$ be the Lipschitz-continuous cut-off function defined by

$$\begin{aligned} \theta _i(z)=\min \left\{ \max \left\{ \frac{2N}{\delta }{\mathrm{dist}}(z,\partial O)-(i+1),0\right\} ,1\right\} , \end{aligned}$$

so that $\theta _i\equiv 1$ on $\overline{O_{i+1}}$, $\theta _i\equiv 0$ on $O\backslash O_{i}$ and its Lipschitz constant can be bounded by $\mathrm {Lip}(\theta _i)\leqq \frac{2N}{\delta }$. We then define an interpolation between functions $u,\psi :O_{\varepsilon }^{\mathcal {L}}\rightarrow {\mathbb {R}}^n$ as

$$\begin{aligned} T_{i,\varepsilon }(u,\psi )(x)=\theta _i(\varepsilon x)u(x)+(1-\theta _i(\varepsilon x))\psi (x). \end{aligned}$$

Observe that if $u\in {\mathcal {P}}_{i,\varepsilon }$ as well as $\varphi \in {\mathcal {N}}_p(v,O,\varepsilon ,\kappa )$ and $\psi \in {\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )$, by the Minkowski inequality we have

$$\begin{aligned} \varepsilon ^{\frac{d}{p}}\Vert v_{\varepsilon }-\varepsilon T_{i,\varepsilon }(u,\psi )\Vert _{\ell ^p_{\varepsilon }(O)}\leqq & {} 2\kappa |O|^{\frac{1}{p}+\frac{1}{d}}+\varepsilon ^{\frac{d}{p}}\Vert \varepsilon \psi -\varepsilon \varphi \Vert _{\ell ^p_{\varepsilon }(O\backslash \overline{O_{i+1}})}\\\leqq & {} 2\kappa |O|^{\frac{1}{p}+\frac{1}{d}}+C\varepsilon |O|^{\frac{1}{p}}, \end{aligned}$$

so that $T_{i,\varepsilon }(u,\psi )\in {\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )$ for $\varepsilon$ small enough.

For technical reasons the interpolations will not suffice to prove the estimates. For every i let us choose $t_i\in [\frac{1}{4},\frac{3}{4}]$ such that, setting $S_i^t=\{x\in O:\;\theta _i(x)=t\}$, the coarea formula implies

$$\begin{aligned} \frac{1}{2}{\mathcal {H}}^{d-1}( S_i^{t_i})\leqq \int _{\frac{1}{4}}^{\frac{3}{4}}{\mathcal {H}}^{d-1}(S_i^t)\,\mathrm {d}t\leqq \int _0^1{\mathcal {H}}^{d-1}( S_i^t)\,\mathrm {d}t=\int _{O}|\nabla \theta _i|\leqq \frac{2N}{\delta }|O_i\backslash {O_{i+1}}|.\nonumber \\ \end{aligned}$$

(A.1)

We set $S_i^*=\{x\in O:\theta _i(x)<t_i\}$. Note that for $\delta$ small enough (depending only on O), we have $S_i^*\in {\mathcal {A}}^R(D)$ (see for instance [32, Lemma 2.2]). Let us introduce the product set

$$\begin{aligned} {\mathcal {U}}_{\varepsilon }^i(M):=\big ({\mathcal {P}}_{i,\varepsilon }\cap {\mathcal {S}}_M(O,\varepsilon )\big )\times {\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon ), \end{aligned}$$

as well as the integral

$$\begin{aligned} e^i_{\varepsilon }(M):=\bigg (\int _{{\mathcal {U}}_{\varepsilon }^i(M)}\exp \Big (- H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))-c_0\Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\Big )\,\mathrm {d}u\,\mathrm {d}\psi \bigg ), \end{aligned}$$

where $c_0>0$ is a small constant such that $c_0|\xi |^p\leqq f(\cdot ,\xi )+c_0^{-1}$ (cf. Hypothesis 1). This integral quantity will be the main ingredient to prove the interpolation inequality. We split the remaining argument into several steps.

Step 1. Energy bounds for the interpolation.

To bound the energy of $T_{i,\varepsilon }(u,\psi )$, we use the pointwise inequality

$$\begin{aligned} |\psi (x)-\psi (y)|^p\leqq & {} C|(\psi -\varphi )(x)-(\psi -\varphi )(y)|^p+C|\varphi (x)-\varphi (y)|^p\\\leqq & {} C+C|\varphi (x)-\varphi (y)|^p, \end{aligned}$$

which is valid for all $x,y\in (O\backslash \overline{O_{i+1}})_{\varepsilon }$. Combined with the two-sided growth condition in Hypothesis 1 we infer that

$$\begin{aligned} H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))&\leqq H_{\varepsilon }(O_{i+1},u)+H_{\varepsilon }(O\backslash \overline{O_i},\psi )+H_{\varepsilon }(S_i,T_{i,\varepsilon }(u,\psi ))\nonumber \\&\leqq H_{\varepsilon }(O_{i+1},u)+CH_{\varepsilon }(O^{\delta },\varphi )+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+H_{\varepsilon }(S_i,T_{i,\varepsilon }(u,\psi )), \end{aligned}$$

(A.2)

where $O^{\delta }$ is defined in the statement of Proposition 1. In order to estimate the last term on the right hand side we use the formula

$$\begin{aligned} T_{i,\varepsilon }(u,\psi )(x)-T_{i,\varepsilon }(u,\psi )(y)&=\big (\theta _i(\varepsilon x)-\theta _i(\varepsilon y)\big )\big (u(x)-\psi (x)\big ) \\&\quad +\theta _i(\varepsilon y)\big (u(x)-u(y)\big )\\&\quad +(1-\theta _i(\varepsilon y))\big (\psi (x)-\psi (y)\big ) \end{aligned}$$

and the bound on the Lipschitz constant of $\theta _i$ to estimate the energy on the interpolation stripe via

$$\begin{aligned}&H_{\varepsilon }(S_i,T_{i,\varepsilon }(u,\psi )) \nonumber \\&\quad \leqq C\Vert \nabla _{{\mathbb {B}}}T_{i,\varepsilon }(u,\psi )\Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+C|(S_i)^{{\mathcal {L}}}_{\varepsilon }|\nonumber \\&\quad \leqq C\left( \Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+\Vert \nabla _{{\mathbb {B}}}\psi \Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+\frac{(N\varepsilon )^p}{\delta ^p}\Vert u-\psi \Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+|(S_i)^{{\mathcal {L}}}_{\varepsilon }|\right) \nonumber \\&\quad \leqq \frac{C}{N}H_{\varepsilon }(O,u)+CH_{\varepsilon }(O^{\delta },\varphi )+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{CN^p}{\delta ^p}\kappa ^p|O|^{1+\frac{p}{d}}\varepsilon ^{-d}, \end{aligned}$$

(A.3)

where we have used again that the degree of each vertex is equibounded and that, after suitable extension, $\psi \in {\mathcal {N}}_p(v,O,\varepsilon ,2\kappa )$ for $\varepsilon$ small enough. Combining (A.2) and (A.3) we infer that

$$\begin{aligned} H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))\leqq & {} H_{\varepsilon }(O_{i+1},u)+\frac{C}{N}H_{\varepsilon }(O,u)+CH_{\varepsilon }(O^{\delta },\varphi )+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }| \nonumber \\&+\frac{CN^p}{\delta ^p}\kappa ^p|O|^{\frac{p}{d}}|O_{\varepsilon }^{\mathcal {L}}|. \end{aligned}$$

(A.4)

Step 2. Lower bound for $e^i_{\varepsilon }(M)$.

In order to prove a lower bound for the integral, first note that due to Hypothesis 1 and the definition of $S_i^*$

$$\begin{aligned} c_0\Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\leqq H_{\varepsilon }(O\setminus O_{i+1},u)+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|, \end{aligned}$$

so that, up to increasing C, we can add this inequality to (A.4) and obtain the estimate

$$\begin{aligned} H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))+c_0\Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\leqq & {} \left( 1+\frac{C}{N}\right) H_{\varepsilon }(O,u)+CH_{\varepsilon }(O^{\delta },\varphi )\\&+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{CN^p}{\delta ^p}\kappa ^p|O|^{\frac{p}{d}}|O_{\varepsilon }^{\mathcal {L}}| \end{aligned}$$

Rearranging the terms we obtain by Fubini’s Theorem that

$$\begin{aligned} e_{\varepsilon }^i(M)&\geqq \exp \bigg (-C \Big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}| \nonumber \\&\quad + H_{\varepsilon }(O^{\delta },\varphi )\Big )\bigg )\int _{{\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )}\mathrm {d}\psi \nonumber \\&\quad \times \int _{{\mathcal {P}}_{i,\varepsilon }\cap {\mathcal {S}}_M(O,\varepsilon )}\exp (- H_{\varepsilon }(O,u))\,\mathrm {d}u\nonumber \\ \geqq&\exp \bigg (-C \Big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\Big )\bigg )\nonumber \\&\quad \times Z_{\varepsilon ,O}({\mathcal {P}}_{i,\varepsilon }\cap {\mathcal {S}}_M(O,\varepsilon )) \end{aligned}$$

(A.5)

where we used that the measure of ${\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )$ can be bounded from below by $\exp (-C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)$.

Step 3. Upper bound for $e^i_{\varepsilon }(M)$ and conclusion.

To estimate $e^i_{\varepsilon }(M)$ from above, similar to [34] we perform a suitable change of variables. Define $\Phi _{i,\varepsilon }:{\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\times {\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )\rightarrow {\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\times {\mathcal {N}}_p(u,O\backslash \overline{O_{i+1}},\varepsilon ,3\kappa )$ by

$$\begin{aligned} \Phi _{i,\varepsilon }(u,\psi )(x)= {\left\{ \begin{array}{ll} (T_{i,\varepsilon }(u,\psi )(x),\psi (x)) &{}\text{ if } \theta _i(\varepsilon x)\geqq t_i,\\ (T_{i,\varepsilon }(u,\psi )(x),u(x)) &{}\text{ if } \theta _i(\varepsilon x)<t_i. \end{array}\right. } \end{aligned}$$

Note that for $\varepsilon$ small enough $\Phi _{i,\varepsilon }$ is well-defined and bijective onto its range ${\mathcal {R}}(\Phi _{i,\varepsilon })$. For the idea how to calculate the Jacobian, we refer to the proof of Proposition 3. As $t_i\in [\frac{1}{4},\frac{3}{4}]$, it holds that

$$\begin{aligned} |\det (D\Phi _{i,\varepsilon }(u,\psi ))|^{-1}= & {} \left( \prod _{x:\theta _i(\varepsilon x)\geqq t_i}|\theta _i(\varepsilon x)|^n\prod _{x:\theta _i(\varepsilon x)<t_i}|1-\theta _i(\varepsilon x)|^n\right) ^{-1}\\\leqq & {} \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|). \end{aligned}$$

Setting $(g,h)=\Phi _{i,\varepsilon }(u,\psi )$, by construction of the interpolation we have

$$\begin{aligned} \begin{aligned} g&\in {\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ), \\ h&=(h_1,h_2)\in {\mathcal {N}}_{\infty }(\varphi ,O\backslash (\overline{O_{i+1}}\cup S_i^*),\varepsilon )\\&\quad \times \underbrace{\{h:(S_i^*)_{\varepsilon }\rightarrow {\mathbb {R}}^n:\;\Vert h-\varepsilon ^{-1}v_{\varepsilon }\Vert _{\infty }\leqq C\kappa |O_{\varepsilon }|^{\frac{1}{p}+\frac{1}{d}}\}}_{=:R_{i,\varepsilon }}. \end{aligned} \end{aligned}$$

As the measure of the set ${\mathcal {N}}_{\infty }(\varphi ,O\backslash (\overline{O_{i+1}}\cup S_i^*),\varepsilon )$ can be bounded by $\exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)$, the above change of variables and Fubini’s Theorem imply

$$\begin{aligned} e_{\varepsilon }^i(M)&\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\int _{{\mathcal {R}}(\Phi _{i,\varepsilon })}\exp \Big (- H_{\varepsilon }(g,O)-c_0\Vert \nabla _{{\mathbb {B}}}h\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\Big )\,\mathrm {d}g\,\mathrm {d}h\nonumber \\&\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\int _{{\mathcal {N}}_{\infty }(\varphi ,O\backslash (\overline{O_{i+1}}\cup S_i^*),\varepsilon )}\mathrm {d}h_1\int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)})\,\mathrm {d}h_2\nonumber \\&\quad \times Z({\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ))\nonumber \\&\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^t)})\,\mathrm {d}h_2\; \nonumber \\&\quad \times Z_{\varepsilon ,O}({\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi )). \end{aligned}$$

(A.6)

In order to bound the integral on the right hand side, we apply Lemma 3.3 to the graph $G_{S_i^*,\varepsilon }$ with $\alpha =c_0$ and $\gamma =C\kappa |O_{\varepsilon }|^{\frac{1}{p}+\frac{1}{d}}$ and infer

$$\begin{aligned} \int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)})\,\mathrm {d}h_2&\leqq \Big (C\kappa ^n |O_{\varepsilon }|^{\frac{n}{p}+\frac{n}{d}}\Big )^{N_{i,\varepsilon }}C^{|(S_i^*)^{{\mathcal {L}}}_{\varepsilon }|-N_{i,\varepsilon }}, \end{aligned}$$

where we denoted by $N_{i,\varepsilon }$ the number of connected components of the graph $G_{S_i^*,\varepsilon }$. For $\varepsilon$ small enough (possibly depending on $N,\delta$), by Remark 7, (3.3) and the fact that $S_i^*\in {\mathcal {A}}^R(D)$ we can bound the number of components via

$$\begin{aligned} N_{i,\varepsilon }\leqq \#\{x\in O_{\varepsilon }^{\mathcal {L}}:\;{\mathrm{dist}}(x,\partial (S_i^*)_{\varepsilon })\leqq C_0\}\leqq C\varepsilon ^{1-d}({\mathcal {H}}^{d-1}(S_i^{t_i})+{\mathcal {H}}^{d-1}(\partial O)). \end{aligned}$$

In particular, for $N,\delta$ and $\kappa >0$ fixed, due to (A.1) there exists $\varepsilon _0$ such that for all $\varepsilon <\varepsilon _0$

$$\begin{aligned} \int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)})\,\mathrm {d}h_2\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|). \end{aligned}$$

Plugging this bound into (A.6) and comparing with (A.5) yields

$$\begin{aligned}&Z_{\varepsilon ,O}({\mathcal {P}}_{\varepsilon }^i\cap {\mathcal {S}}_M(O,\varepsilon )) \leqq Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi )) \\&\quad \times \exp \Big (C\big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\big )\Big ) \end{aligned}$$

Summing this inequality over i, by the definition of the sets ${\mathcal {P}}_{i,\varepsilon }$ we infer that

$$\begin{aligned}&Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\cap {\mathcal {S}}_M(O,\varepsilon )) \leqq \sum _{i=1}^N Z_{\varepsilon ,O}({\mathcal {P}}_{\varepsilon }^i\cap {\mathcal {S}}_M(O,\varepsilon ))\nonumber \\&\quad \leqq Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ))\nonumber \\&\qquad \times N\exp \Big (C\big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\big )\Big ) \end{aligned}$$

(A.7)

Now, choosing

$$\begin{aligned} M=2\left( \frac{1}{|O_{\varepsilon }^{\mathcal {L}}|}\log \Big (Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa ))\Big )+{\overline{C}}+\frac{\log (2)}{|O_{\varepsilon }^{\mathcal {L}}|}\right) , \end{aligned}$$

where ${\overline{C}}$ is the constant of Lemma 3.4, we obtain by the same Lemma and Remark 8 that, for any $\kappa >0$ fixed and all $\varepsilon$ small enough,

$$\begin{aligned} Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\backslash {\mathcal {S}}_M(O,\varepsilon ))\leqq \frac{1}{2}Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )). \end{aligned}$$

Thus (A.7) and the definition of M yield the final estimate

$$\begin{aligned}&Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa ))\\&\quad \leqq 2Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\cap {\mathcal {S}}_M(O,\varepsilon )) \\&\quad \leqq Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ))\,Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa ))^{\frac{C}{N}}\nonumber \\&\qquad \times 2N\exp \Big (C\big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\big (\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}+\frac{C}{N}\big )|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\big )\Big ). \end{aligned}$$

$\square$

Remark 14

Note that the restriction on $\delta$ in the interpolation inequality comes only from the requirement that tubular neighbourhoods of the boundary have again Lipschitz boundary. In particular, if $\delta$ satisfies the condition for a set $O\subset {\mathbb {R}}^d$, then $\delta ^{\prime }=\delta \rho$ satisfies the condition for all sets of the form $O^{\prime }=z+\rho O$. Applying this fact to the family of cubes $Q(z,\rho )$ with $z\in D$ and $\rho >0$, we obtain that there exists $\delta _0>0$ such that for all $\delta <\delta _0$, all $N\in {\mathbb {N}}$ and all $\kappa >0$ it holds that

$$\begin{aligned} \frac{N-C}{N}{\mathcal {F}}_{\kappa }^-(Q(z,\rho ), {\overline{\varphi }}_\Lambda )&\geqq {{\overline{W}}}(\Lambda )-C(1+|\Lambda |^p)\frac{|Q(z,\rho )^{\delta \rho }|}{|Q(z,\rho )|}\\&\quad -C\left( \frac{(N\kappa |O(z,\rho )|^{\frac{1}{d}})^p}{(\delta \rho )^p}+\frac{1}{N}\right) \\&\geqq {{\overline{W}}}(\Lambda )-C\left( (1+|\Lambda |^p)\delta +\frac{(N\kappa )^p}{\delta ^p}+\frac{1}{N}\right) \end{aligned}$$

${\mathbb {P}}$-almost surely. Here we used Lemma 4.1, Proposition 3 and (3.1) in order to pass to the limit as $\varepsilon \rightarrow 0$ in the interpolation inequality almost surely. Note that the last bound is independent of $\rho$ and z.

Lemma A.1

Let $p\in (1,+\infty )$. For all $u,v\in {\mathbb {R}}^n$ it holds that

$$\begin{aligned} |u-v|^p+|u+v|^p\leqq \max \{2^{p-1},2\}(|u|^p+|v|^p). \end{aligned}$$

Proof of Lemma A.1

For $p\geqq 2$ the claimed estimate follows from Clarkson’s inequality. If $p<2$, then $(x_1^p+x_2^p)^{\frac{1}{p}}\geqq (x_1^2+x_2^2)^{\frac{1}{2}}$ for all $x_1,x_2\geqq 0$. Moreover, with elementary analysis one can show that $(x_1^p+x_2^p)^{\frac{1}{p}}\leqq 2^{\frac{1}{p}-\frac{1}{2}}(x_1^2+x_2^2)^{\frac{1}{2}}$. Applying these two inequalities first with $x_1=|u-v|$ and $x_2=|u+v|$ and then with $x_1=|u|$ and $x_2=|v|$ we obtain

$$\begin{aligned} (|u-v|^p+|u+v|^p)^{\frac{1}{p}}\leqq & {} 2^{\frac{1}{p}-\frac{1}{2}}(|u-v|^2+|u+v|^2)^{\frac{1}{2}}=2^{\frac{1}{p}}(|u|^2+|v|^2)^{\frac{1}{2}}\\\leqq & {} 2^{\frac{1}{p}}(|u|^p+|v|^p)^{\frac{1}{p}}. \end{aligned}$$

$\square$

Lemma A.2

Let $p\in (1,+\infty )$. Then there exists a constant $c_{p}$ such that the Hausdorff measure of the sphere $S^{n-1}_{p}=\{y\in {\mathbb {R}}^n:\;|y|_p=1\}$ fulfills

$$\begin{aligned} {\mathcal {H}}^{n-1}(S^{n-1}_{p})\geqq \left( \frac{c_{p}}{n}\right) ^{\frac{n}{p}}. \end{aligned}$$

Proof of Lemma A.2

Note that $S^{n-1}_{p}$ is a compact smooth $(n-1)$-dimensional manifold. Hence we can characterize its Hausdorff measure by its Minkowski content. To be more precise, it holds that

$$\begin{aligned} {\mathcal {H}}^{n-1}(S^{n-1}_{p})=\lim _{\varepsilon \rightarrow 0}\frac{{\mathcal {H}}^{n}(S^{n-1}_{p}+B_{\varepsilon }(0))}{2\varepsilon }, \end{aligned}$$

(A.8)

where the factor 2 comes from the Lebesgue measure of the 1D unit ball $[-1,1]$. Note however that $B_{\varepsilon }(0)$ is a ball with respect to the Euclidean metric on ${\mathbb {R}}^n$. We now give a lower bound for the nominator on the right hand side of (A.8). To this end, set $c_{n,p}=\max \{1,n^{\frac{1}{2}-\frac{1}{p}}\}$. Then, for $y\ne 0$, we have

$$\begin{aligned} \left| y-\frac{y}{|y|_p}\right| _2\leqq ||y|_p-1|\frac{|y|_2}{|y|_p}\leqq ||y|_p-1|c_{n,p}, \end{aligned}$$

where we used that by definition $|y|_2\leqq c_{n,p}|y|_p$ for all $y\in {\mathbb {R}}^n$. We conclude that

$$\begin{aligned} \{y\in {\mathbb {R}}^n:\;1-c^{-1}_{n,p}\varepsilon<|y|_p<1+c^{-1}_{n,p}\varepsilon \}\subset S^{n-1}_p+B_{\varepsilon }(0). \end{aligned}$$

Hence we deduce from (A.8) and the well-know formula for the volume of p-norm balls that

$$\begin{aligned} {\mathcal {H}}^{n-1}(S^{n-1}_p)&\geqq \liminf _{\varepsilon \rightarrow 0}\frac{{\mathcal {H}}^n(\{|y|_p<1+c_{n,p}^{-1}\varepsilon \})-{\mathcal {H}}^n(\{|y|_p<1-c^{-1}_{n,p}\varepsilon \})}{2\varepsilon }\\&=\frac{(2\Gamma (\frac{1}{p}+1))^n}{\Gamma (\frac{n}{p}+1)}\lim _{\varepsilon \rightarrow 0}\frac{(1+c^{-1}_{n,p}\varepsilon )^n-(1-c^{-1}_{n,p}\varepsilon )^n}{2\varepsilon }\\&=\frac{(2\Gamma (\frac{1}{p}+1))^n}{\Gamma (\frac{n}{p}+1)}nc^{-1}_{n,p}\geqq \frac{(2\Gamma (\frac{1}{p}+1))^n}{\Gamma (\frac{n}{p}+1)}n^{\frac{1}{2}}. \end{aligned}$$

We conclude the proof using Stirling’s formula in the form of the upper bound

$$\begin{aligned} \Gamma \left( \frac{n}{p}+1\right) \leqq \left( \frac{2\pi n}{p}\right) ^{\frac{1}{2}}\left( \frac{n}{pe}\right) ^{\frac{n}{p}}\exp (p/12). \end{aligned}$$

$\square$