Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates

TL;DR

This paper investigates non-convex functionals that penalize simultaneous oscillations in functions of two variables, establishing conditions under which zero functional value implies the function is separable, with quantitative estimates depending on smoothness.

Contribution

It provides new rigidity estimates for non-convex functionals, linking zero energy to function structure and smoothness, with quantitative control of the distance to separable functions.

Findings

Zero functional value implies separability under certain smoothness conditions.

Quantitative bounds relate the functional value to the distance from the set of separable functions.

Smoothness of the function critically influences the rigidity estimates.

Abstract

We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: \Omega_1\times\Omega_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset $S(\Omega_1\times\Omega_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication "$\mathcal{E}(u) = 0 \Rightarrow u \in S(\Omega_1\times\Omega_2)$" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) $\mathcal{E}(u)$ controls in a strong sense the distance of $u$ to $S(\Omega_1\times\Omega_2)$.

Equations

$$|{\nabla }_{1}u||{\nabla }_{2}u|\equiv 0\text{ a.e. in }\Omega$$

$$|u(x_1+z_1,x_2)-u(x)||u(x_1,x_2+z_2)-u(x)|$$

$${\mathbf{R}}^n=X_1\oplus X_2,\text{with}{n}_l:=\dim X_l\ge 1\text{ for }l=1,2.$$

$$S(\Omega ):=\{u\in L(\Omega ):u(x)=u_l(x_l)\text{ in }\Omega \text{ with }l=1\text{ or }l=2\text{ and }{u}_l\in L({\Omega }_l)\}.$$

$$\rho \in L^1({\mathbf{R}}^n,{\mathbf{R}}_{+})\text{ with }\int \rho =1,\text{ and }\mathrm{supp}\rho \subset B_1.$$

$${\mathcal{E}}_{\epsilon ,p}^{{\theta }_1,{\theta }_2}(u;\Omega )={\mathcal{E}}_{\epsilon ,p}^{{\theta }_1,{\theta }_2}(u):={\int }_{{\mathbf{R}}^n}{\rho }_{\epsilon }(z){\int }_{{\Omega }^{\epsilon }}\frac{|u(x+z_1)-u(x){|}^{{\theta }_1}|u(x+z_2)-u(x){|}^{{\theta }_2}}{|z{|}^p}dxdz,$$

$${\Omega }^{\epsilon }:={\Omega }_1^{\epsilon }+{\Omega }_2^{\epsilon },\text{and }{\Omega }_j^{\epsilon }:=\bigcap _{z_l\in X_l,|z_l|<\epsilon }({\Omega }_l-z_l)\text{ for }l=1,2.$$

$${\mathcal{E}}_p^{{\theta }_1,{\theta }_2}(u;\Omega )={\mathcal{E}}_p^{{\theta }_1,{\theta }_2}(u):=\underset{\epsilon \downarrow 0}{\mathrm{liminf}}{\mathcal{E}}_{\epsilon ,p}^{{\theta }_1,{\theta }_2}(u).$$

$${\mathcal{E}}_{\epsilon ,p}(u):={\mathcal{E}}_{\epsilon ,p}^{{\theta }_1,{\theta }_2}(u),{\mathcal{E}}_p(u):={\mathcal{E}}_p^{{\theta }_1,{\theta }_2}(u).$$

$${\mathcal{E}}_{\epsilon ,p}(u)\le {\epsilon }^{q-p}{\mathcal{E}}_{\epsilon ,q}(u).$$

$${\mathcal{E}}_p(u)=\{\begin{array}{ll}0& \text{ for p<p∗(u),}\\ \infty & \text{ for p>p∗(u).}\end{array}$$

$${\mathcal{E}}_p(u)=0$$

$$\nabla u(x)\in X_1\cup X_2\text{almost everywhere in }\Omega .$$

$$\forall u\in L(\Omega ),{\mathcal{E}}_p(u)=0⟹u\in S(\Omega )$$

$$P(\theta_{1},\theta_{2})\,:=\,\left\{\begin{array}[]{clr}2&\text{ if $\theta\leq 1$,}&(a)\vskip 3.0pt plus 1.0pt minus 1.0pt\\ 1+\theta&\text{ if $\theta\geq 1$ and $\min(\theta_{1},\theta_{2})\leq 1$,}&(b)\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \min(\theta_{1},\theta_{2})+\theta&\text{ if $\theta_{1},\theta_{2}>1$.}&(c)\end{array}\right.$$

$$\mathcal{E}(u):={\mathcal{E}}_{P({\theta }_1,{\theta }_2)}(u)={\mathcal{E}}_{P({\theta }_1,{\theta }_2)}^{{\theta }_1,{\theta }_2}(u).$$

$${\int }_{B_r}{\int }_{{\Omega }^r}\frac{|u(x+z_1)-u(x){|}^{{\theta }_1}|u(x+z_2)-u(x){|}^{{\theta }_2}}{|z{|}^{n+p}}dxdz<\infty ,$$

$$\underset{\epsilon \downarrow 0}{\mathrm{liminf}}{\int }_{{\mathbf{R}}^n}{\rho }_{\epsilon }(z){\int }_{{\Omega }^{\epsilon }}\frac{|u(x+z_1+z_2)-u(x+z_1)-u(x+z_2)+u(x){|}^{\theta }}{|z{|}^{P({\theta }_1,{\theta }_2)}}dxdz\lesssim \mathcal{E}(u).$$

$$\mu [u]:={\nabla }_1{\nabla }_{2}u.$$

$$⟨\mu [u],\phi ⟩\lesssim \{\begin{array}{ll}\mathcal{E}(u)\Vert \phi {\Vert }_{\infty }& \text{ if (a) holds with }\{\begin{array}{l}\theta =1\text{ or }\\ \theta <1\text{ and }\Vert u{\Vert }_{\infty }\le 1,\end{array}\text{vskip}6.0ptplus2.0ptminus2.0pt\\ \mathcal{E}(u{)}^{1/\theta }\Vert {\nabla }_1\phi {\Vert }_1^{1-1/\theta }\Vert \phi {\Vert }_{\infty }^{1/\theta }& \text{ if (b) holds and }{\theta }_1\le 1,\text{vskip}6.0ptplus2.0ptminus2.0pt\\ \mathcal{E}(u{)}^{1/\theta }\Vert {\nabla }_1\phi {\Vert }_1^{{\theta }_2/\theta }\Vert \phi {\Vert }_{\infty }^{{\theta }_1/\theta }& \text{ if (c) holds and }{\theta }_1\le {\theta }_2.\end{array}$$

$$L_{loc}^1(\Omega )\cap \mathrm{span}S(\Omega )=\left\{x\in \Omega \mapsto u_1(x_1)+u_2(x_2):u_l\in L_{loc}^1({\Omega }_l)\text{ for }l=1,2\right\}.$$

$${\int }_{{\Omega }_l^{\epsilon }}{\int }_{{\Omega }_l^{\epsilon }}{\tilde{\rho }}_{\epsilon }(x_l-y_l)\frac{|u_l(x_l)-u_l(y_l){|}^{{\theta }_l}}{|x_l-y_l{|}^{{\theta }_l}}d{x}_{l}d{y}_l\stackrel{\epsilon \downarrow 0}{⟶}0.$$

$$\Vert u-\bar{u}{\Vert }_{\infty }+|\nabla [u-\bar{u}]|(\Omega )\lesssim \mathcal{E}(u)+\mathcal{E}(u{)}^{\frac{1}{2}}.$$

$$\Vert u-\bar{u}{\Vert }_{L^{\frac{\hat{n}}{\hat{n}-1}}(\Omega )}\lesssim \mathcal{E}(u)+\mathcal{E}(u{)}^{\frac{1}{2}}.$$

$$Du(x,z):=u(x+z)-u(x).$$

$${\mathcal{E}}_{\epsilon ,p}^{{\theta }_1,{\theta }_2}(u)={\int }_{{\mathbf{R}}^n}{\rho }_{\epsilon }(z){\int }_{{\Omega }^{\epsilon }}\frac{|Du(x,z_1){|}^{{\theta }_1}|Du(x,z_2){|}^{{\theta }_2}}{|z{|}^p}dxdz.$$

$${\mathcal{E}}_{\epsilon ,p}(v;U),\text{ for }\epsilon >0,\text{and}{\mathcal{E}}_p(v;U)$$

$${\mathcal{E}}_{\epsilon ,\theta }(u)\le {\lambda }^{\theta }{\int }_{{\Omega }^{\epsilon }}\left[{\int }_{{\mathbf{R}}^n}\frac{|z_1{|}^{{\theta }_1}|z_2{|}^{{\theta }_2}}{|z{|}^{\theta }}{\rho }_{\epsilon }(z)dz\right]dx.$$

$${\mathcal{E}}_{\theta }(u)\le {\lambda }^{\theta }|\Omega |\left({\int }_{{\mathbf{R}}^n}\frac{|z_1{|}^{{\theta }_1}|z_2{|}^{{\theta }_2}}{|z{|}^{\theta }}\rho (z)dz\right),$$

$${\mathcal{E}}_{\epsilon ,\theta }^{{\theta }_1,{\theta }_2}(u)={\int }_{{\Omega }^{\epsilon }}\left[{\int }_{{\mathbf{R}}^n}\left(\frac{|{\nabla }_{1}u(x)\cdot z_1{|}^{{\theta }_1}|{\nabla }_{2}u(x)\cdot z_2{|}^{{\theta }_2}}{|z{|}^{\theta }}+o(|z|)\right){\rho }_{\epsilon }(z)dz\right]dx.$$

Full text

1 Introduction

Given two bounded non-empty connected and open sets $\Omega_{1}\subset\mathbf{R}^{n_{1}}$, $\Omega_{2}\subset\mathbf{R}^{n_{2}}$, we consider the non-convex set $S(\Omega)$ of measurable functions $u$ defined on $\Omega=\Omega_{1}\times\Omega_{2}\subset\mathbf{R}^{n_{1}+n_{2}}$ which only depend on the first $n_{1}$ coordinates or on the last $n_{2}$ coordinates, that is $u(x)=u_{1}(x_{1})$ or $u(x)=u_{2}(x_{2})$. We present and study a family of non-convex functionals that detect whether a measurable function $u$ defined on $\Omega$ belongs to $S(\Omega)$. As a first guess, we could expect that for $u$ with Sobolev regularity the relation

[TABLE]

would imply $u\in S(\Omega)$. As we will see later on, this is not the case even for $u\in Lip(\Omega)$ and therefore (1.1) cannot be the starting point for defining our functionals. However, our construction relies on a discrete version of (1.1). If $u\in S(\Omega)$ then at any point $x=(x_{1},x_{2})\in\Omega$, and any $z=(z_{1},z_{2})\in\mathbf{R}^{n_{1}+n_{2}}$ with $|z|$ small enough, the product

[TABLE]

is well defined and vanishes. The functionals that we consider are based on the integration of (1.2) in $x$ and $z$ with a weight depending on $z$ and a limiting process that localizes the integration around $z=0$.

1.1 Definitions and context

Let us introduce some notation. For convenience, we write the Cartesian product as a sum: we decompose the $n$-dimensional Euclidean space $X=\mathbf{R}^{n}$ as,

[TABLE]

We will assume for simplicity that $X_{1}\perp X_{2}$. The domain $\Omega\subset\mathbf{R}^{n}$ is $\Omega:=\Omega_{1}+\Omega_{2}$ where, for $l=1,2$, $\Omega_{l}$ is a non-empty bounded domain (i.e. open and connected) of the subspace $X_{l}$. We note $L(U)$ the space of Lebesgue-measurable functions defined on a measurable set $U$ and for $x\in\mathbf{R}^{n}$, we note $x=x_{1}+x_{2}$ its decomposition in $X_{1}+X_{2}$. With this notation we define the set

[TABLE]

(This set depends on $\Omega_{1}$, $\Omega_{2}$ but as no ambiguity arises we choose this short notation.)

We also fix a radial non-negative kernel

[TABLE]

As usual, for $\varepsilon>0$ we introduce the rescaled kernel $\rho_{\varepsilon}:=\varepsilon^{-n}\rho(\varepsilon^{-1}\cdot)$ so that $\{\rho_{\varepsilon}\}_{\varepsilon>0}$ forms a family of radial mollifiers. We introduce three real parameters $p$, $\theta_{1}$, $\theta_{2}>0$ and define for any measurable function $u:\Omega\to\mathbf{R}$ and any $\varepsilon>0$, the quantity

[TABLE]

with

[TABLE]

Eventually, we send $\varepsilon$ to [math] and define the functional

[TABLE]

Most of the time, we omit the dependency on the parameters $\theta_{1},\theta_{2}$ and note

[TABLE]

We are interested in the qualitative and quantitative properties of functions with finite energy $\mathcal{E}_{p}(u)$. We first observe that $u$, $\theta_{1}$ and $\theta_{2}$ being fixed, there exists at most one value of $p$ for which $0<\mathcal{E}_{p}(u)<\infty$.

Remark A**.**

From the properties of the kernel $\rho$, we have for $p<q$,

[TABLE]

Sending $\varepsilon$ to [math], we see that $[\mathcal{E}_{p}(u)>0]\Rightarrow[\mathcal{E}_{q}(u)=\infty]$ and we deduce that there exists $p^{*}(u)\in[0,\infty]$ such that

[TABLE]

By construction, $\mathcal{E}_{\varepsilon,p}(u)=\mathcal{E}_{p}(u)=0$ for every $u\in S(\Omega)$. A first natural question is:

[TABLE]

We will see that the answer is “yes” for $p$ large enough depending on $\theta_{1},\theta_{2}$. This question initially appeared (with $u$ being a characteristic function and $n_{2}=1$) in the study of pattern formation in some variational models involving competition between a local attractive term and a non-local repulsive one. Indeed, in [GR16, DR18]), energies related to $\mathcal{E}_{p}$ are used to show that some sets $S\subset\mathbf{R}^{n}$ are union of stripes. The functionals $\mathcal{E}_{p}$ extend this setting to general functions and to general dimensions $(n_{1},n_{2})$.

Our second main result may be seen as an answer to a quantitative version of Question (1.4). Indeed, we prove (at least for some values of $\theta_{1}$ and $\theta_{2}$), that the non-convex energy $\mathcal{E}_{p}(u)$ controls the distance from $u$ to the non-convex set $S(\Omega)$ in a strong norm. Of course, as seen from Remark A, this is an interesting question only for the borderline exponent $p$ for Question (1.4).

Remark B**.**

Let us point out that when investigating Question (1.4), there is no loss of generality in assuming that $u\in L^{\infty}$. Indeed, $\arctan u\in S(\Omega)$ if and only if $u\in S(\Omega)$ and $\mathcal{E}_{p}(\arctan u)\leq\mathcal{E}_{p}(u)$.

Remark C**.**

Notice that the functionals $\mathcal{E}_{p}^{\theta_{1},\theta_{2}}$ can be seen as variants of the non-local functionals used by Brezis et al to characterize Sobolev spaces [BBM01, Bre02, DMMS08]. It turns out that the present non-convex setting is rather different but we do use their results in our analyses at some point: when $u(x)$ splits as $u_{1}(x_{1})+u_{2}(x_{2})$.

1.2 The vanishing energy case: Question (1.4)

To get some insight into the behavior of the functional $\mathcal{E}_{p}$, let us first consider the simple situation $u\in C^{1}(\overline{\Omega})$. Within this setting, the problem is rigid as soon as $p\geq\theta_{1}+\theta_{2}$.

Proposition D** (Proposition 2.1).**

Let us note $\theta:=\theta_{1}+\theta_{2}$.

• (i)

For every $u\in\operatorname{Lip}(\Omega)$, $\mathcal{E}_{\theta}(u)<\infty$ (and therefore $\mathcal{E}_{p}(u)=0$ for $p<\theta$);

• (ii)

If $u\in C^{1}(\overline{\Omega})$, then  $[\mathcal{E}_{\theta}(u)=0]\Longrightarrow u\in S(\Omega)$.

To obtain the point (i) we simply plug the inequality $|u(x+z_{l})-u(x)|\leq\|\nabla u\|_{\infty}|z_{l}|$ in the definition of $\mathcal{E}_{\varepsilon,\theta}$. This first point shows that the parameter $p=\theta$ is sharp in (ii). The proof of (ii) runs as follows. Using the relation $u(x+z_{l})-u(x)=\nabla u(x)\cdot z_{l}+o(|z|)$ in the definition of $\mathcal{E}_{\varepsilon,\theta}(u)$, we obtain that $\mathcal{E}_{\theta}(u)=0$ implies that $\nabla u$ satisfies the differential inclusion (equivalent to (1.1))

[TABLE]

This differential inclusion is rigid for $u\in C^{1}(\Omega)$: it yields $u\in S(\Omega)$. On the contrary, since the convex hull of $X_{1}\cup X_{2}$ is $\mathbf{R}^{n}$, the differential inclusion (1.5) is not rigid in the class of Lipschitz continuous functions. Indeed, the set of Lipschitz functions satisfying (1.5) is dense in $W^{1,1}(\Omega)$ (see [Dac08, Theorem 10.18]). As a consequence we cannot substitute Lipschitz continuity to the $C^{1}$-regularity assumption in Proposition D (ii). More precisely, at least in the range $\min(\theta_{1},\theta_{2})\leq 1$, when we pass from $C^{1}$-regularity to Lipschitz continuity, the threshold jumps from $p=\theta$ to $p=1+\theta$ as shown by the two following propositions. The first one follows from Theorem I (with $\theta_{1},\theta_{2}$ in the range (b) of (1.6) below).

Proposition E** (Proposition 3.9).**

Assume $\min(\theta_{1},\theta_{2})\leq 1$ then for $u\in Lip(\Omega)$, $[\mathcal{E}_{1+\theta}(u)=0]\Longrightarrow u\in S(\Omega)$.

Proposition F** (Proposition 2.2 (i)).**

There exists $u\in Lip(\Omega)\backslash S(\Omega)$ with $0<\mathcal{E}_{1+\theta}(u)<\infty$.

The typical example of a function $u\in Lip(\Omega)\backslash S(\Omega)$ with $\mathcal{E}_{1+\theta}(u)<\infty$ is the “roof” function $u(x):=\min(x_{1},x_{2})$ defined on $\Omega=(0,1)\times(0,1)$ with $n_{1}=n_{2}=1$. This function is locally independent of $x_{1}$ or of $x_{2}$ away from the diagonal $\{x_{1}=x_{2}\}$, so that the integrand in (1.3) vanishes outside $\{(x,z):|x_{1}-x_{2}|\leq|z|<\varepsilon\}$. With this remark, it is not difficult to guess that $\mathcal{E}_{1+\theta}(u)$ is finite and positive.

In this work we consider lower regularities than Lipschitz continuity, like merely measurable functions or $L^{1}_{loc}$ or $L^{\infty}$ functions for the finest results (recall however Remark B). In this setting, a second important example of a function which “almost” belongs to $S(\Omega)$ is given by the characteristic function of a “corner”, $u:=\boldsymbol{1}_{(0,1)^{2}}$ defined in $\Omega=(-1,1)^{2}$. Here, the integrand of (1.3) vanishes outside $\{(x,z):|x|<\varepsilon,|z|<\varepsilon\}$ and it is easy to check that $\mathcal{E}_{2}(u)<\infty$.

Proposition G** (Proposition 2.2 (ii)).**

There exists $u\in L^{\infty}(\Omega)\backslash S(\Omega)$ with $0<\mathcal{E}_{2}(u)<\infty$.

Propositions F and  G show that the implication

[TABLE]

could only hold under the condition $p\geq\max(1+\theta,2)$. Our first main result shows that in many cases, this bound is sharp.

Definition H**.**

For $\theta_{1},\theta_{2}>0$, using the notation $\theta:=\theta_{1}+\theta_{2}$, we define

[TABLE]

To lighten notation, from now on, $\theta_{1},\theta_{2}>0$ being given, we note

[TABLE]

Theorem I** (Theorem 3.7).**

For every $u\in L(\Omega)$, there holds $\mathcal{E}(u)=0\ \Longrightarrow u\in S(\Omega)$.

Remark J**.**

The counter-examples of Propositions F, G show that the exponent $P(\theta_{1},\theta_{2})$ cannot be improved in the cases (a) and (b) of (1.6). On the contrary, in case $(c)$ we believe that the sharp exponent should still be $1+\theta$ although we only succeed to prove that the optimal exponent was not larger than $P(\theta_{1},\theta_{2})=\min(\theta_{1},\theta_{2})+\theta>1+\theta$.

Since $\mathcal{E}(u)<\infty$ implies $\mathcal{E}_{p}(u)=0$ for every $p<P(\theta_{1},\theta_{2})$ (see Remark A), we obtain as direct corollary of Theorem I:

Corollary K**.**

If $u\in L(\Omega)$ satisfies $\mathcal{E}_{p}(u)<\infty$ for some $p>P(\theta_{1},\theta_{2})$, then $u\in S(\Omega)$.

Another consequence of the theorem is the following generalization of [GR16, Proposition 4.3].

Corollary L** (Corollary 3.8).**

For every $p\geq P(\theta_{1},\theta_{2})$, $r>0$, and every $u\in L(\Omega)$, if

[TABLE]

then $u\in S(\Omega^{r})$.

This is indeed a far reaching generalization of [GR16, Proposition 4.3] since the same conclusion was obtained there under the assumptions that $n_{2}=1$, $u=\boldsymbol{1}_{E}$ for some set $E$ of finite perimeter satisfying an extra technical assumption and $p>n$ (see also [DR18] where the condition $p>n$ was independently relaxed to $p\geq 2$). As opposed to [GR16] and [DR18] where the proofs are somewhat geometrical and based on slicing, our proof is purely analytical.

The main insight in the proof of Theorem I is that if $\mathcal{E}(u)$ does not control first order differential quotients it does control second order ones. Very roughly speaking (at least for $\theta\leq 1$), the main observation is that333By convention $a\lesssim b$ means that there exists a non-negative constant $C$ which may only depend on $\theta_{1},\theta_{2}$, $n$, $\Omega$ or on the kernel $\rho$ such that $a\leq Cb$.

[TABLE]

This yields a quantitative control on the distribution444For $u\in L^{1}_{loc}(\Omega)$ we note $\nabla_{l}$ the distributional gradient with respect to the variables in $X_{l}$.

[TABLE]

Proposition M** (Proposition 3.3).**

Let $u\in L^{1}_{loc}(\Omega)$. For every $\varphi\in C^{\infty}_{c}(\Omega,\mathbf{R}^{n_{1}\times n_{2}})$,

[TABLE]

Remark N**.**

For some estimates (as the first one above), $u$ is required to be bounded. In these situations, we avoid complex formulas involving $\|u\|_{\infty}$ by assuming $\|u\|_{\infty}\leq 1$. The general case can be recovered by scaling.

The proof of Theorem I continues as follows. Proposition M implies that if $\mathcal{E}(u)=0$ then $\mu[u]=0$. Obviously, the space of functions $u\in L^{1}_{loc}(\Omega)$ with $\mu[u]=0$ is

[TABLE]

Plugging the decomposition $u(x)=u_{1}(x_{1})+u_{2}(x_{2})$ into the definition of $\mathcal{E}_{\varepsilon,p}(u)$, the integrations over $\Omega_{1}$ and $\Omega_{2}$ decouple. Using again $\mathcal{E}(u)=0$, we deduce that for $l=1$ or $l=2$, we have a control of the form

[TABLE]

We then use ideas from [BBM01] in either $\Omega_{1}$ or $\Omega_{2}$ to obtain Theorem I.

Before developing further the consequences of Proposition M let us comment about its optimality. In the cases (a) and (b) of Proposition M, the estimates are optimal in the sense that for $p<P(\theta_{1},\theta_{2})$, (1.7) does not hold in general. Indeed, we can precise Propositions F, G as follows.

Proposition O** (Proposition 2.2).**

• (i)

There exists $u\in Lip(\Omega)\backslash S(\Omega)$ with $0<\mathcal{E}_{1+\theta}(u)<\infty$ and $\mu[u]\neq 0$ (typically, $u$ is a “roof” function);

• (ii)

There exists $u\in L^{\infty}(\Omega)\backslash S(\Omega)$ with $0<\mathcal{E}_{2}(u)<\infty$ and $\mu[u]\neq 0$ (typically, $u$ is a“corner” function).

Let us make the important observation that Proposition M gives more information than $\mathcal{E}(u)=0$ implies $u\in S(\Omega)$. Indeed, it shows that if $0<\mathcal{E}(u)<\infty$, then $\mu[u]$ lies in some Sobolev space with null or negative regularity exponent. In particular, in case $(a)$ ($\theta=1$ or $\theta<1$ and $u\in L^{\infty}(\Omega)$), if $\mathcal{E}(u)=\mathcal{E}_{2}(u)$ is finite then $\mu[u]$ is a finite Radon measure. By construction, we prove that it is not necessarily true in the other cases.

Proposition P** (Proposition 2.3).**

For every $\theta>1$, there exists $u\in Lip(\mathbf{R}^{2})$, compactly supported, with $\mathcal{E}_{1+\theta}(u)<\infty$ and for which $\mu[u]=\partial_{1}\partial_{2}u$ is not a finite Radon measure.

1.3 Control of the distance of $u$ to $S(\Omega)$

We then focus on case $(a)$ of (1.6) with the assumption $u\in L^{\infty}(\Omega)$. We prove that the energy gives a quantitative control on the distance of $u$ to $S(\Omega)$. We obtain the strongest result in this direction for $n=2$.

Theorem Q** (Theorem 4.1).**

Assume that $n_{1}=n_{2}=1$ and $\theta\leq 1$. Then, for every $u\in L^{\infty}(\Omega)$ with $\|u\|_{\infty}\leq 1$ and $\mathcal{E}(u)<\infty$, there exists $\bar{u}\in S(\Omega)$ such that $u-\bar{u}\in BV(\Omega)\cap L^{\infty}(\Omega)$ with the estimate

[TABLE]

The idea of the proof is to first decompose $u$ as $u(x)=u_{1}(x_{1})+u_{2}(x_{2})+w(x)$ where $w$ satisfies $\partial_{1}\partial_{2}w=\mu[u]=\partial_{1}\partial_{2}u$ and $\|w\|_{\infty}+|\nabla w|(\Omega)\lesssim\mathcal{E}(u)$. Using this (and in particular the $L^{\infty}$ bound on $w$), we can quantify how much the integration with respect to $x_{1}$ and $x_{2}$ in the definition (1.3) of $\mathcal{E}_{\varepsilon,2}(u)$ decouples. In higher dimension, the failure of the Sobolev embedding $BV(\Omega_{l})\subset L^{\infty}(\Omega_{l})$ makes the situation more complex (and in particular the energy does not control the corresponding $w$ in $L^{\infty}$) and we were not able to obtain a $BV$ estimate. Nevertheless, we have,

Theorem R** (Theorem 4.5).**

Assume that $\hat{n}:=\max(n_{1},n_{2})\geq 2$, that $\Omega_{1}$ and $\Omega_{2}$ are bounded extension domains and that $\theta\leq 1$. Let $u\in L^{\infty}(\Omega)$ with $\|u\|_{\infty}\leq 1$ and $\mathcal{E}(u)<\infty$. Then, there exists $\bar{u}\in L^{\infty}(\Omega)\cap S(\Omega)$ such that

[TABLE]

Notice that the $L^{\frac{\hat{n}}{\hat{n}-1}}$ norm (which comes from the embedding of $BV(\Omega_{l})$ into $L^{\frac{n_{l}}{n_{l}-1}}(\Omega_{l})$) is stronger than the $L^{\frac{n}{n-1}}$ norm which would come from the embedding of $BV(\Omega)$.

1.4 Further results

In a second paper [GM19], we will focus on the case $\theta\leq 1$ with $u\in L^{\infty}(\Omega)$ and $\mathcal{E}(u)<\infty$ and study the structure of the defect measure $\mu=\mu[u]$ (which is then a Radon measure).

In dimension 2 (i.e. $n_{1}=n_{2}=1$) we show that if $\theta=1$, then $\mu$ concentrates on a set with Hausdorff dimension at most 1. Moreover, if $u$ is Lipschitz continuous, then $\mu$ is absolutely continuous with respect to the Hausdorff measure $\mathcal{H}^{1}$ (and satisfies the differential inclusion (1.5)). On the contrary, if $\theta<1$, we show that $\mu$ concentrates on a countable set: there exist sequences $(x_{k})_{k\geq 1}\subset\Omega$ and $(m_{k})_{k\geq 1}\subset\mathbf{R}$ such that $\sum|m_{k}|^{\theta}\lesssim\mathcal{E}_{2}(u)$ and $\mu=\sum m_{k}\delta_{x_{k}}$. Moreover, if $u$ is a characteristic function, then $m_{k}\in\pm\{1,2\}$ for every $k\geq 1$. As a consequence of the estimate $\sum|m_{k}|^{\theta}\lesssim\mathcal{E}_{2}(u)$ there exists some $\eta>0$ such that $\mathcal{E}_{2}(u)<\eta$ implies $\mu=0$, which in turn leads to $u\in S(\Omega)$. This improves Theorem I and Theorem Q in this particular case.

In higher dimensions, we assume $\theta<1$. Using tools from Geometric Measure Theory (mainly the rectifiability criterion for flat chains of White [Whi99]), we prove that $\mu$ is a $(n-2)$-rectifiable measure with a tensor structure: for $l=1,2$, there exist $\Sigma_{l}\subset X_{l}$ $(n_{l}-1)-$rectifiable and a Borel function $m:\Sigma_{1}+\Sigma_{2}\to\mathbf{R}$ such that $\mu=m\nu_{1}\otimes\nu_{2}\mathcal{H}^{n-2}\hskip 2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule height=0.5pt,width=7.0pt,depth=0.0pt}\,(\Sigma_{1}+\Sigma_{2})$, where for $l=1,2$, $\nu_{l}\in X_{l}$ is a normal to $\Sigma_{l}$. This gives a relatively good understanding of the case where the typical function $u$ with $\mathcal{E}(u)<\infty$ is a “corner” (recall Proposition O).

In the case $\theta=1$, in order to distinguish between “corners” and “roofs” (see Proposition O), one needs to impose more regularity on $u$. To understand this better, we plan to investigate in a future work, the set of Lipschitz continuous functions which satisfy the differential inclusion (1.5) and are such that $\mu[u]=\nabla_{1}\nabla_{2}u$ is a Radon measure.

1.5 Conventions and notation

In all the paper, we consider $\theta_{1},\theta_{2}>0$ and note $\theta=\theta_{1}+\theta_{2}$ their sum. For $x,z\in\mathbf{R}^{n}$, and $u:\Omega\to\mathbf{R}$ we note

[TABLE]

In particular, the expression (1.3) of $\mathcal{E}_{\varepsilon,p}^{\theta_{1},\theta_{2}}(u)$ simplifies to

[TABLE]

As already said, except at two points (case (c) in the proofs of Proposition M and Theorem I), we omit the superscript $\theta_{1},\theta_{2}$ by writing $\mathcal{E}_{\varepsilon,p}$ for $\mathcal{E}_{\varepsilon,p}^{\theta_{1},\theta_{2}}$ and $\mathcal{E}_{p}$ for $\mathcal{E}_{p}^{\theta_{1},\theta_{2}}$. When $p=P(\theta_{1},\theta_{2})$ as defined in (1.6), we simply write $\mathcal{E}$ for $\mathcal{E}_{p}$.

If $v\in L(\Omega)$ and $U=U_{1}+U_{2}$ with $U_{l}\subset\Omega_{l}$ a subdomain of $\Omega_{l}$ for $l=1,2$, we note,

[TABLE]

the energies of the restriction $v_{|U}$.

In the sequel $(e_{1},\cdots,e_{n_{1}})$ (respectively $(f_{1},\cdots,f_{n_{2}})$) denotes an orthonormal basis of $X_{1}$ (respectively of $X_{2}$).

Given $x\in\mathbf{R}^{n}$, we note $x_{l}$ its orthogonal projection on $X_{l}$, for $l=1,2$, so that $x=x_{1}+x_{2}$. Similarly, given a function $\varphi\in C^{\infty}_{c}(\Omega)$ and $x\in\Omega$, we note $\nabla_{1}\varphi(x)+\nabla_{2}\varphi(x)$ the decomposition of $\nabla\varphi(x)$ in $X_{1}+X_{2}$. By duality, the distributional derivative of a function $u\in L^{1}_{loc}(\Omega)$ decomposes as $\nabla u=\nabla_{1}u+\nabla_{2}u$, where $\nabla_{l}u$ is a distribution on $\Omega$ with values into $X_{l}$.

For $r>0$ and $x\in\mathbf{R}^{n}$, $B_{r}(x)$ denotes the open ball in $\mathbf{R}^{n}$ with center $x$ and radius $r$. If $x=0$, we simply write $B_{r}$. For $l=1,2$, we note $B^{X_{l}}_{r}(x_{l})$ the open ball in $X_{l}$ with center $x_{l}$ and radius $r$.

We use standard notation for the function spaces ($L^{1}_{loc}(\Omega)$, $L^{q}(\Omega)$, $L^{\infty}(\Omega)$, $Lip(\Omega)$, $BV(\Omega)$).

For $a\in\mathbf{R}$, we note $a_{+}:=\max(a,0)$ its non-negative part.

If $A\subset\mathbf{R}^{n}$ is a measurable subset, we note $|A|$ its volume and $\mathcal{H}^{k}(A)$ its $k-$dimensional Hausdorff measure.

We note \mbox{{\scriptsize\diagup}}\int_{\Omega}u(x)\,dx=|\Omega|^{-1}\int_{\Omega}u the mean value of a function $u$ over $\Omega$.

Eventually, by convention, $a\lesssim b$ means that there exists a non-negative constant $C$ which may only depend on $\theta_{1},\theta_{2}$, $n$, $\Omega$ or on the kernel $\rho$ such that $a\leq Cb$.

1.6 Outline of the paper

In the first section, we prove Proposition D and we build all the examples and counter-examples introduced above. In Section 3, we consider the zero-energy case and prove Theorem I (and all the results from Proposition E to P). Section 4 is dedicated to the quantitative control of the distance of $u$ to $S(\Omega)$ in terms of $\mathcal{E}(u)$.

2 The case of $C^{1}$-functions and counter-examples

We first settle Question (1.4) in the setting of $C^{1}$-functions.

Proposition 2.1** (Proposition D).**

• (i)

For every $u\in\operatorname{Lip}(\Omega)$, $\mathcal{E}_{\theta}(u)<\infty$ (and therefore $\mathcal{E}_{p}(u)=0$ for $p<\theta$);

• (ii)

For $u\in C^{1}(\overline{\Omega})$, we have $\mathcal{E}_{\theta}(u)=0\Longrightarrow u\in S(\Omega)$.

Proof.

(i) Let $u\in Lip(\Omega)$ with Lipschitz constant $\lambda$. We have

[TABLE]

Using the change of variable $z=\varepsilon\tilde{z}$ and sending $\varepsilon$ to 0, we get

[TABLE]

which proves (i).

(ii) Let $u\in C^{1}(\overline{\Omega})$. Plugging the relation $u(x+z_{l})-u(x)=\nabla_{l}u(x)\cdot z_{l}+o(|z_{l}|)$ into the definition of $\mathcal{E}_{\varepsilon,\theta}^{\theta_{1},\theta_{2}}(u)$, we observe that

[TABLE]

Sending $\varepsilon$ to 0, we have

[TABLE]

where $v_{j}\in X_{l}$ are (arbitrary) unit vectors.

Hence, if $\mathcal{E}_{\theta}^{\theta_{1},\theta_{2}}(u)=0$, (2.1) implies that $|\nabla_{1}u(x)||\nabla_{2}u(x)|=0$ in $\Omega$.

Next, suppose that at some point $x^{0}\in\Omega$, $w_{1}:=\nabla_{1}u(x^{0})\neq 0$. Let $C_{2}$ be the connected component of the set

[TABLE]

which contains $x_{2}^{0}$. By continuity of $\nabla_{1}u$, $C_{2}$ is closed in $\Omega_{2}$. Let us show that it is also open. Let $y_{2}\in C_{2}$, by continuity of $\nabla_{1}u$ we have $\nabla_{1}u\not\equiv 0$ in some neighborhood $N_{\varepsilon}(x^{0}_{1}+y_{2}):=B^{X_{1}}_{\varepsilon}(x^{0}_{1})+B^{X_{2}}_{\varepsilon}(y_{2})$. Since $|\nabla_{1}u||\nabla_{2}u|$ vanishes, we have $\nabla_{2}u\equiv 0$ in $N_{\varepsilon}(x^{0}_{1}+y_{2})$. In particular, $\nabla_{1}u(x_{1}^{0}+y^{\prime}_{2})=\nabla_{1}u(x_{1}^{0}+y_{2})=w_{1}$ for every $y^{\prime}_{2}\in B^{X_{2}}_{\varepsilon}(y_{2})$. We conclude that $B^{X_{2}}_{\varepsilon}(y_{2})\subset C_{2}$ which proves that $C_{2}$ is relatively open in $\Omega_{2}$. Finally, the open set $\Omega_{2}$ being connected, we have $C_{2}=\Omega_{2}$ and we conclude that $\nabla_{1}u(x^{0}_{1}+y_{2})\neq 0$ for every $y_{2}\in\Omega_{2}$.

Assuming by contradiction that there exists also some point $\tilde{x}^{0}\in\Omega$ with $\nabla_{2}u(\tilde{x}^{0})\neq 0$ we obtain similarly that $\nabla_{2}u(y_{1}+\tilde{x}^{0}_{2})\neq 0$ for every $y_{1}\in\Omega_{1}$. Choosing $y_{1}=x^{0}_{1}$ and $y_{2}=\tilde{x}^{0}_{2}$ we obtain at the point $x=x^{0}_{1}+\tilde{x}^{0}_{2}$, the contradiction

[TABLE]

We conclude that $\nabla_{l}u\equiv 0$ for $l=1$ or $l=2$: in short, $u\in S(\Omega)$. This proves (ii). ∎

We now, give lower-bounds for $p$ in Question (1.4) in the setting of Lipschitz continuous and of bounded functions.

Proposition 2.2** (Proposition O).**

• (i)

There exists $u\in Lip(\Omega)\backslash S(\Omega)$ with $0<\mathcal{E}_{1+\theta}(u)<\infty$ and $\mu[u]\neq 0$;

• (ii)

There exists $u\in L^{\infty}(\Omega)\backslash S(\Omega)$ with $0<\mathcal{E}_{2}(u)<\infty$ and $\mu[u]\neq 0$.

Proof.

(i) Let us assume that $n_{1}=n_{2}=1$ and let us introduce the “roof” function defined on $\mathbf{R}^{2}$ by $v(x_{1},x_{2}):=\min(x_{1},x_{2})$. We have $\partial_{1}\partial_{2}v=(1/\sqrt{2})\mathcal{H}^{1}\hskip 2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule height=0.5pt,width=7.0pt,depth=0.0pt}\,\{x_{1}=x_{2}\}$. We now consider $\Omega=(0,1)^{2}$ and $u(x):=v(x)$ in $\Omega$. We have $\mu[u]=(1/\sqrt{2})\mathcal{H}^{1}\hskip 2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule height=0.5pt,width=7.0pt,depth=0.0pt}\,J$, with $J=\{(t,t):t\in(0,1)\}$ and $\mu[u]$ is a Radon measure with

[TABLE]

Next, using the changes of variable $x=\varepsilon x^{\prime}$, $z=\varepsilon z^{\prime}$ and the 1-homogeneity of $u$, we obtain for $\varepsilon>0$,

[TABLE]

Let us set $w(s):=\min(s,0)$. We have

[TABLE]

and thus

[TABLE]

We claim that $\mathcal{E}_{1,1+\theta}\left(v;\frac{1}{\varepsilon}\Omega\right)=c_{1}/\varepsilon-c_{2}$ for some $c_{1},c_{2}>0$. The idea is that since $w$ is constant in $\mathbf{R}_{+}$, the integrand vanishes when $|z|\leq|x_{1}-x_{2}|$ and since $\operatorname{supp}\rho\subset B_{1}$, the integrand vanishes away from the $\sqrt{2}/2$-neighborhood of the diagonal segment $(1/\varepsilon)J$. Since this segment has length $\sqrt{2}/\varepsilon$, we expect $\mathcal{E}_{1,1+\theta}\left(v;\frac{1}{\varepsilon}\Omega\right)$ to be of order $1/\varepsilon$ and therefore, $\mathcal{E}_{\varepsilon,1+\theta}(u)$ to be of order 1.

Let us note $f(x_{1}-x_{2})$ the integral inside the brackets in (2.2). The function $f$ takes values in $[0,\infty]$, is measurable and even. We first show that $f$ is integrable on $(-1,1)$. For $s\geq 0$ and $z\in\mathbf{R}^{2}$, we have

[TABLE]

Integrating in $s$ over $(0,1)$, we deduce

[TABLE]

Using Fubini and the symmetries of the problem and the above estimate, we get,

[TABLE]

Therefore, $f\in L^{1}(-1,1)$. We also observe from (2.3) that the integral is strictly positive. In summary,

[TABLE]

Let us return to (2.2). Since $\operatorname{supp}\rho\subset B_{1}$, the identity (2.3) shows that we can reduce the integration with respect to $x$ to the set

[TABLE]

Let us introduce the following rectangle which contains $A_{\varepsilon}$,

[TABLE]

We also set $C_{\varepsilon}:=R_{\varepsilon}\backslash\overline{A}_{\varepsilon}$, see Figure 1.

We decompose,

[TABLE]

Obviously the second integral does not depend on $\varepsilon$ and is positive, we note $c_{2}$ its value. For the first integral, we perform the change of variable $s=x_{1}-x_{2}$, $t=(1/2)(x_{1}+x_{2})$ to obtain

[TABLE]

This gives $\mathcal{E}_{1+\theta}\left(v;\frac{1}{\varepsilon}\Omega\right)=c_{1}/\varepsilon-c_{2}$, which leads to

[TABLE]

Sending $\varepsilon$ to 0, we get $\mathcal{E}_{1+\theta}(u)=c_{1}\in(0,\infty)$ (recall (2.4)) and conclude the proof of (i) in the case $n_{1}=n_{2}=1$ and $\Omega=(0,1)^{2}$. We obtain counter-examples for any non empty and bounded two dimensional domain by translation and scaling of this example and in higher dimensions by extending the constructions trivially in the complementary directions.

(ii) Again, we only treat the two-dimensional case $n_{1}=n_{2}=1$, since higher dimensional cases may be obtained by tensorisation. Let $\Omega=(-1,1)^{2}$ and set $u:=\boldsymbol{1}_{\omega}$ with $\omega:=(0,1)^{2}$. Obviously, $\mu=\partial_{1}\partial_{2}u=\delta_{0}$ is a non-trivial finite Radon measure. Let us compute $\mathcal{E}_{2}(u)$. First, we have

[TABLE]

We compute for $0<\varepsilon<1$,

[TABLE]

For the last identity, we have used the radial symmetry of $\rho_{\varepsilon}$ and $\int_{\mathbf{R}^{2}}\rho_{\varepsilon}=1$. We conclude that $\mathcal{E}_{2}(u)=1/(4\pi)\in(0,\infty)$ as required. ∎

In Proposition 3.3 below, we will prove that in the case $\theta\leq 1$, if $\mathcal{E}(u)$ is finite then $\mu[u]$ is a Radon measure. We show that this is no longer true when $\theta>1$.

Proposition 2.3** (Proposition P).**

For every $\theta_{1},\theta_{2}>0$ with $\theta=\theta_{1}+\theta_{2}>1$, there exists $u\in L^{\infty}(\mathbf{R}^{2})$ compactly supported, with $\mathcal{E}_{1+\theta}(u)<\infty$ but for which $\mu[u]=\partial_{1}\partial_{2}u$ is not a finite Radon measure.

Proof.

We will build on the example of Proposition 2.2 (i) and consider the “hat” function defined on $(0,1)^{2}$ as $w(x_{1},x_{2}):=\min(x_{1},x_{2},1-x_{1},1-x_{2})$ and extended by zero outside $(0,1)^{2}$. Arguing as in the proof of Proposition 2.2, we have $0<\mathcal{E}_{1+\theta}(u)<\infty$. For $h,\ell>0$, let $w_{h,\ell}(x):=hw(\ell^{-1}x)$ be defined on $\mathbf{R}^{2}$. We then have

[TABLE]

Let $(h_{k})_{k\geq 1}$, $(\ell_{k})_{k\geq 1}\subset(0,\infty)$ be two decreasing sequences such that

[TABLE]

We also require that there exists some constant $c>0$ such that

[TABLE]

and that

[TABLE]

For instance, the sequences defined by $h_{k}:=k^{-(4\theta+1)/(5\theta)}$ and $\ell_{k}:=k^{-3/4}/2$ satisfy (2.6), (2.7) and (2.8). Next, we build a bounded sequence $(x^{k})_{k\geq 0}\subset\mathbf{R}^{2}$ such that

[TABLE]

For this, we observe that $\sum\ell_{k}\geq(\ell_{1}/h_{1})\sum h_{k}=\infty$. We set $k_{0}=1$ and we define recursively $k_{m+1}$ for $m=0,1,2,\cdots,$ as the unique integer to be such that

[TABLE]

Notice that by (2.8), the sequence $(k_{m})_{m\geq 0}$ is not stationary (and thus converges towards $+\infty$). We then define $L_{0}:=0$ and for $q\geq 1$, $L_{q}:=\sum_{m=0}^{q-1}2\ell_{k_{m}}$. The sequence $(L_{q})_{q\geq 0}$ is increasing and bounded, indeed

[TABLE]

Now, we set for $m\geq 0$ and $k_{m}\leq k<k_{m+1}$,

[TABLE]

By construction the sequence $(x^{k})$ satisfies (2.9) and $(x^{k})\subset(0,L_{\infty})\times(0,1)$. Eventually, we define our candidate (see Figure 2),

[TABLE]

Let us show that $u$ has the desired properties. First, by construction $\operatorname{supp}u\subset[0,1]\times[0,L_{\infty}]$ so that $u$ is compactly supported. Next, from the first identity of (2.5),

[TABLE]

Let us now establish that $\mathcal{E}_{1+\theta}(u)$ is finite. We emphasize that

[TABLE]

so that, from (2.9), the functions $w_{h_{k},\ell_{k}}(\cdot,x^{k})$ have disjoint supports and for $j\neq k$,

[TABLE]

By (2.7) for $\varepsilon>0$ small enough, there exists an integer $k_{\varepsilon}$ such that

[TABLE]

Using $\operatorname{supp}\rho_{\varepsilon}\subset B_{\varepsilon}$ and (2.11), we can write

[TABLE]

where we note

[TABLE]

Let us first bound the remaining term $\mathcal{E}_{\varepsilon,1+\theta}(u_{\varepsilon})$ in (2.13). We notice that $w_{h_{j},\ell_{j}}(\cdot,x^{j})$ is Lipschitz continuous with Lipschitz constant $\|\nabla w_{h_{j},\ell_{j}}\|_{\infty}=h_{j}/\ell_{j}\leq h_{k_{\varepsilon}}/\ell_{k_{\varepsilon}}$ for $j\geq k_{\varepsilon}$. Since these functions have disjoint supports, we conclude that their sum $u_{\varepsilon}$ is Lipschitz continuous with

[TABLE]

Using $|Du_{\varepsilon}(x,z_{l})|\leq(h_{k_{\varepsilon}}/\ell_{k_{\varepsilon}})|z|$ and $|Du_{\varepsilon}(x,z_{1})||Du_{\varepsilon}(x,z_{2})|=0$ if the three points $x$, $x+z_{1}$, $x+z_{2}$ lie in the complement of $\operatorname{supp}u_{\varepsilon}$, we compute,

[TABLE]

where $A_{\varepsilon}(z)$ is the set of points $x\in\mathbf{R}^{2}$ such that at least one of the three points $\{x,x+z_{1},x+z_{2}\}$ belongs to $\operatorname{supp}u_{\varepsilon}$. We have

[TABLE]

This leads to the estimate

[TABLE]

We now pass to the limit in the terms $\mathcal{E}_{\varepsilon,1+\theta}(w_{h_{j},\ell_{j}})$ for $j\leq k_{\varepsilon}$. We have

[TABLE]

As in the proof of Proposition 2.2 (i), we have

[TABLE]

Summing over $1\leq j\leq k_{\varepsilon}$, and sending $\varepsilon\downarrow 0$, we get by monotone convergence theorem,

[TABLE]

Combining this together with (2.13) and (2.14), we obtain $\mathcal{E}_{1+\theta}(u)=\mathcal{E}_{1+\theta}(w)\sum_{j}h_{j}^{\theta}\ell_{j}^{1-\theta}$ and by (2.6) we conclude that $\mathcal{E}_{1+\theta}(u)$ is finite whereas from (2.10), $\mu[u]$ is not a finite measure. ∎

3 The zero energy case

In order to present the main ideas of the proof of Theorem I, we start by considering the simplest possible setting. We restrict ourselves to $n_{1}=n_{2}=1$, $\theta_{1}+\theta_{2}=1$ and work on the torus $\Omega=\mathbf{T}=(\mathbf{R}/\mathbf{Z})^{2}$ to avoid boundary effects (in particular, $\mathbf{T}^{\varepsilon}=\mathbf{T}$). In this periodic setting, we need to distinguish the ambient manifold $\mathbf{T}$ from the space of tangent vectors $X=\mathbf{R}^{2}$, we define:

[TABLE]

With this notation, the definitions of $S(\mathbf{T})$ and of the energy are unchanged.

Proposition 3.1**.**

Let $\theta_{1}$, $\theta_{2}>0$ be such that $\theta_{1}+\theta_{2}=1$ and let $u\in L(\mathbf{T})$ be such that

[TABLE]

then $u\in S(\mathbf{T})$.

Proof.

As noticed in Remark B, we may assume without loss of generality that $u\in L^{\infty}(\Omega)$.

Step 1. In this first step we prove that (3.1) allows us to find a sequence $z^{k}=\varepsilon_{k}(\sigma_{1}+\sigma_{2})$ with $\sigma_{1}\in X_{1}\backslash\{0\}$, $\sigma_{2}\in X_{2}\backslash\{0\}$ and $\varepsilon_{k}\downarrow 0$ such that

[TABLE]

where we have set

[TABLE]

Let us digress slightly and first derive a consequence of (3.2). For this, first notice that

[TABLE]

so that by the triangle inequality we have

[TABLE]

Hence,

[TABLE]

and similarly for $\theta_{2}$ so that since $\theta_{1}+\theta_{2}=1$,

[TABLE]

Therefore, a consequence of  (3.2) would be the maybe more suggestive

[TABLE]

Observe that if $u$ were smooth , the integrand would converge to $|\sigma_{1}|\,|\sigma_{2}|\,|\partial_{1}\partial_{2}u(x)|$ and (3.6) would directly imply $\partial_{1}\partial_{2}u=0$.

We now prove (3.2). Making the change of variable $\tilde{x}=x+z_{1}$, $\tilde{z}=-z_{1}+z_{2}$ in (3.1), we have

[TABLE]

Similarly we get

[TABLE]

and

[TABLE]

Summing these limits and (3.1) we obtain,

[TABLE]

with $q$ given by (3.3). Using polar coordinates and the change of variables $r=\varepsilon s$, we obtain

[TABLE]

Applying Fatou Lemma and then Markov inequality, we may find for $l=1,2$, $\hat{\sigma}_{l}\in X_{l}\backslash\{0\}$ with $\frac{1}{4}\leq|\hat{\sigma}_{l}|^{2}\leq\frac{3}{4}$ and $s\in(0,1)$ such that setting $\sigma:=s(\hat{\sigma}_{1}+\hat{\sigma}_{2})$,

[TABLE]

Passing to a subsequence and noting $z^{k}:=\varepsilon_{k}\sigma$, we get (3.2).

Step 2. Let us show that (3.2) implies $\partial_{1}\partial_{2}u=0$ in $\mathcal{D}^{\prime}(\mathbf{T})$. Let $\varphi\in C^{\infty}(\mathbf{T})$. For every $x\in\mathbf{T}$,

[TABLE]

with uniform convergence in $\mathbf{T}$. Multiplying by $u$, using the dominated convergence theorem and two discrete integration by parts, we compute

[TABLE]

Therefore, $\partial_{1}\partial_{2}u=0$ in the sense of distributions in $\mathbf{T}$.

Step 3. Integrating the relation $\partial_{1}\partial_{2}u=0$ we find that $u(x)=u_{1}(x_{1})+u_{2}(x_{2})$ with $u_{1}$ and $u_{2}$ periodic functions on $\mathbf{R}$. Moreover, since $u$ is bounded, so are $u_{1}$ and $u_{2}$. Let us finally prove that $\partial_{1}u_{1}\equiv 0$ or $\partial_{2}u_{2}\equiv 0$. From (3.2), we have for some $\sigma_{1}\in X_{1}\backslash\{0\}$, $\sigma_{2}\in X_{2}\backslash\{0\}$ and a sequence $\varepsilon_{k}\downarrow 0$,

[TABLE]

so that up to extraction for $l=1$ or $l=2$, there holds

[TABLE]

In particular, since $u_{l}$ is bounded and $0<\theta_{l}<1$,

[TABLE]

Arguing as in (3), we obtain that $\partial_{l}u_{l}\equiv 0$ in the sense of distributions in $\mathbf{R}/\mathbf{Z}$. We conclude that $u\in S(\mathbf{T})$ which ends the proof of the proposition.∎

We turn to the proof of Theorem I, which extends Proposition 3.1 in several directions by considering general space dimensions and powers $\theta_{1},$ $\theta_{2}$.

Let us recall some notation. For $\Omega=\Omega_{1}+\Omega_{2}$ with $\Omega_{1}\subset X_{1}$ and $\Omega_{2}\subset X_{2}$ and a function $u\in L^{1}_{loc}(\Omega)$, the matrix valued distribution $\mu[u]$ is defined as

[TABLE]

where $\dim X_{1}=n_{1}$ and $\dim X_{2}=n_{2}$. For $\theta_{1}$, $\theta_{2}>0$, we also recall that the critical exponent $P(\theta_{1},\theta_{2})$ has been introduced in Definition H and that when $p=P(\theta_{1},\theta_{2})$, we simply write $\mathcal{E}(u)$ for $\mathcal{E}_{p}(u)$.

Remark 3.2**.**

We will use the following inequality to reduce the case $\theta<1$ to the case $\theta=1$. Assume that $\theta<1$ and let us note $\theta^{\prime}_{l}:=\theta_{l}+(1-\theta)/2$ for $l\in\{1,2\}$. We have $\theta^{\prime}:=\theta^{\prime}_{1}+\theta^{\prime}_{2}=1$ and for $u\in L^{\infty}(\Omega)$ and $x,z$ such that $x,x+z\in\Omega$, we have

[TABLE]

so that

[TABLE]

We now prove that the energy $\mathcal{E}(u)$ controls the cross derivatives $\mu[u]=\nabla_{1}\nabla_{2}u$.

Proposition 3.3** (Proposition M).**

Let $u\in L^{1}_{loc}(\Omega)$. We have the following estimates, for every $\varphi\in C^{\infty}_{c}(\Omega,\mathbf{R}^{n_{1}\times n_{2}})$,

[TABLE]

Because of the applications we have in mind in the last part of the paper (and in [GM19]), it will actually be more convenient to derive Proposition 3.3 as a consequence of Lemma 3.4 and Lemma 3.6 below.

We may now state a first lemma which is the extension of Step 1 in the proof of Proposition 3.1 to the more general setting. We recall that we defined $q(x,z)$ in (3.3) as

[TABLE]

We also recall that $(e_{1},\cdots,e_{n_{1}})$ and $(f_{1},\cdots,f_{n_{2}})$ denote orthonormal bases of $X_{1}$ and $X_{2}$.

Lemma 3.4**.**

If $u$ is such that $\mathcal{E}_{p}(u)<\infty$, then there exist sequences $0<r_{k}\leq\varepsilon_{k}$ tending to [math], two numbers $\lambda_{1},\lambda_{2}\in(1/2,\sqrt{3}/2)$ and two rotations $R_{l}\in\operatorname{SO}(X_{l})$ such that

[TABLE]

Proof.

For $l=1,2$, let $\nu_{l}$ denotes the normalized Haar measure on $\operatorname{SO}(X_{l})$ and let $\mathbf{S}(X_{l})$ denote the unit sphere of $X_{l}$. We also set $\nu:=\nu_{1}\otimes\nu_{2}$. We first claim that for every $g\in L^{1}(\mathbf{R}^{n})$, there holds

[TABLE]

To prove (3.11), let us notice first that for every function $v\in L^{1}(\mathbf{S}(X_{1}))$, the integral $\int_{\operatorname{SO}(X_{1})}v(R_{1}e)d\nu_{1}(R_{1})$ does not depend on the choice of $e\in\mathbf{S}(X_{1})$ and therefore

[TABLE]

For $g_{1}\in L^{1}(X_{1})$, we deduce from the above formula and of a decomposition in polar coordinates that

[TABLE]

Combining this with the analog formula for $f\in\mathbf{S}(X_{2})$, $g_{2}\in L^{1}(X_{2})$ and using Fubini, we have for $g\in L^{1}(\mathbf{R}^{n})$,

[TABLE]

Taking $e=e_{i}$ and $f=f_{j}$ in the above identity and summing over $1\leq i\leq n_{1}$, $1\leq j\leq n_{2}$, we obtain (3.11).

Define

[TABLE]

so that arguing as for (3.7) we have

[TABLE]

Using (3.11), polar coordinates $z=r\sigma$, with $r>0$, $\sigma\in\partial B_{1}$ and the change of variables $r=\varepsilon s$, we obtain

[TABLE]

where we note

[TABLE]

Passing to the infimum limit in $\varepsilon$, using Fatou Lemma and then Markov inequality, we find that there exist $R_{1}\in\operatorname{SO}(X_{1})$, $R_{2}\in\operatorname{SO}(X_{2})$, $\sigma\in\partial B_{1}$ with $1/2\leq|\sigma_{1}|,|\sigma_{2}|\leq\sqrt{3}/2$ and $s\in[c,1)$, where $c>0$ only depends on the kernel $\rho$, such that

[TABLE]

Extracting a subsequence $\varepsilon^{\prime}_{k}$ realizing the liminf we find

[TABLE]

Noting $\lambda_{l}:=|\sigma_{l}|$ for $l=1,2$, $\varepsilon_{k}:=2\varepsilon^{\prime}_{k}$ and $r_{k}:=s\varepsilon^{\prime}_{k}$ we conclude the proof of (3.10). ∎

Remark 3.5**.**

With the notation of the lemma, we define the map $A\in\operatorname{GL}(\mathbf{R}^{n})$ by $Az:=\lambda_{1}R_{1}z_{1}+\lambda_{2}R_{2}z_{2}$. Making the change of variables $x=A\hat{x}$ and $\hat{u}(\hat{x})=u(A\hat{x})$ in (3.10), defining $\hat{\varepsilon}_{k}:=\max(\lambda_{1}^{-1},\lambda_{2}^{-1})\varepsilon_{k}$ and observing that $[A^{-1}(\Omega)]^{\hat{\varepsilon}_{k}}\subset A^{-1}(\Omega^{\varepsilon_{k}})$, we find

[TABLE]

with the definition of $\hat{q}$ modeled on the definition of $q$:

[TABLE]

Since, $X_{1}$ and $X_{2}$ are stable by $A$, we have $u\in S(\Omega)\Longleftrightarrow\hat{u}\in S(A^{-1}\Omega)$. Therefore, up to this change of variables, we may always assume that (3.10) holds true with $\lambda_{l}R_{l}=\text{Id}_{X_{l}}$ for $l=1,2$.

Lemma 3.6**.**

Let $u\in L^{1}_{loc}(\Omega)$ and let $(\varepsilon_{k})$, $(r_{k})$ be two sequences with $0<r_{k}\leq\varepsilon_{k}\downarrow 0$. For $i\in\{1,\cdots,n_{1}\}$ and $j\in\{1,\cdots,n_{2}\}$, we define

[TABLE]

(i) If $\theta=1$ or $[\theta<1$ and $\|u\|_{\infty}\leq 1]$ (in both cases, $P(\theta_{1},\theta_{2})=2$), we have for $i\in\{1,\cdots,n_{1}\}$, $j\in\{1,\cdots,n_{2}\}$,

[TABLE]

(ii) Noting $\mu=\mu[u]$, we have for $i\in\{1,\cdots,n_{1}\}$, $j\in\{1,\cdots,n_{2}\}$ and every $\varphi\in C^{\infty}_{c}(\Omega)$,

[TABLE]

Proof.

Part (i). Let us first consider the case $\theta=1$, so that $P(\theta_{1},\theta_{2})=2$. Let $i\in\{1,\cdots,n_{1}\}$, $j\in\{1,\cdots,n_{2}\}$ and let $k\geq 1$ and $x\in\Omega^{\varepsilon_{k}}$. We note $z^{k}=r_{k}(e_{i}+f_{j})$. As in (3.5) we have

[TABLE]

Integrating in $x$ over $\Omega^{\varepsilon_{k}}$ we get (3.12).

In the case $\theta<1$ and $u\in L^{\infty}(\Omega)$, noting $\theta^{\prime}_{l}=\theta_{l}/\theta$ for $l=1,2$, by triangle inequality, we have for every $x\in\Omega^{\varepsilon}$, $z\in B_{\varepsilon}$ and $l\in\{1,2\}$,

[TABLE]

Since $\theta^{\prime}_{1}+\theta^{\prime}_{2}=1$, the case $[\theta<1$ and $\|u\|_{\infty}\leq 1]$ follows form this inequality and from the case $\theta=1$ applied to the pair $(\theta^{\prime}_{1},\theta^{\prime}_{2})$.

Part (ii). Let $u\in L^{1}_{loc}(\Omega)$, $i\in\{1,\cdots,n_{1}\}$, $j\in\{1,\cdots,n_{2}\}$ and $\varphi\in C^{\infty}_{c}(\Omega)$. For $k\geq 1$, we note $z^{k}=r_{k}(e_{i}+f_{j})$. We treat the three cases (a), (b), (c) of (3.13) separately.

Case (a). We proceed exactly as in (3). Using discrete integration by parts, we compute

[TABLE]

This proves the claim.

Case (c). Let us show that this case follows from case (b). Let us assume that $1\leq\theta_{1}\leq\theta_{2}$. By Hölder inequality,

[TABLE]

Applying (3.13.b) with $\theta^{\prime}_{1}=1$, $\theta^{\prime}_{2}={\theta_{2}}/{\theta_{1}}$ and $P(\theta^{\prime}_{1},\theta^{\prime}_{2})=1+\theta_{1}^{\prime}+\theta^{\prime}_{2}=2+{\theta_{2}}/{\theta_{1}}$ yields (3.13.c).

Case (b). From now on we assume $\theta_{1}\leq 1$, $\theta\geq 1$ and $p=P(\theta_{1},\theta_{2})=1+\theta$. Let $x\in\Omega$ and $z\in\mathbf{R}^{n}$ be such that $x\in\Omega^{|z|}$. Arguing as in (3.4), we obtain by triangle inequality,

[TABLE]

and555Remark that this is where we used the hypothesis $\theta_{1}\leq 1$ : in case (c) ($\theta_{1},\theta_{2}>1$), this inequality fails and the present method breaks down.

[TABLE]

Taking the product of the last two inequalities with $(|Du(x+z_{1},z_{2})|^{\theta_{2}}+|Du(x,z_{2})|^{\theta_{2}})^{(\theta-1)/\theta_{2}}$, we get

[TABLE]

We use this estimate in the form

[TABLE]

We now set $\phi(s):=|s|^{\theta-1}s$ for $s\in\mathbf{R}$. Using the estimate

[TABLE]

and (3.14), we have

[TABLE]

Applying this estimate with $z=z^{k}$, integrating in $x\in\Omega^{\varepsilon_{k}}$ and sending $k$ to $\infty$, we have:

[TABLE]

For $v:\Omega\to\mathbf{R}$, $e\in\mathbf{S}(X_{1})\cup\mathbf{S}(X_{2})$, $r>0$ and $x\in\Omega^{r}$, we introduce the discrete derivative,

[TABLE]

With this notation, (3.15) rewrites as

[TABLE]

For smooth functions $u$, this inequality would provide a control on the $L^{1}$-norm of the function $\partial_{e_{i}}\phi(\partial_{f_{j}}u)$. Here, we only assume $u\in L^{1}_{loc}(\Omega)$ and it is difficult to give a meaning to the nonlinear term $\phi(\partial_{f_{j}}u)$. For this reason, we linearize $\phi$ away from [math]. For $\eta>0$, we introduce the function $\phi_{\eta}$ given by

[TABLE]

so that $\phi_{\eta}$ is an odd, Lipschitz continuous function satisfying $0\leq\phi_{\eta}^{\prime}\leq\eta^{1-\theta}\phi^{\prime}$ on $\mathbf{R}_{+}$. As a consequence,

[TABLE]

Thus, from (3.16) we have,

[TABLE]

For $\varphi\in C^{\infty}_{c}(\Omega,\mathbf{R})$, using the dominated convergence theorem and a discrete integration by parts, we compute

[TABLE]

We introduce the decomposition $D_{r_{k}}u(x,f_{j})=\phi_{\eta}(D_{r_{k}}u(x,f_{j}))-\chi_{\eta}(D_{r_{k}}u(x,f_{j}))$ where $\chi_{\eta}(s):=\phi_{\eta}(s)-s$ satisfies

[TABLE]

For $k$ large enough (so that $\operatorname{supp}\varphi\subset\Omega^{\varepsilon_{k}}$), we have

[TABLE]

where we used a discrete integration by parts to treat the first term and the bound $\|\chi_{\eta}\|_{\infty}\leq\eta$ for the second term. Using (3.18) and (3.17), we obtain,

[TABLE]

Eventually, optimizing in $\eta$ by choosing $\eta^{\theta}=\mathcal{F}_{i,j}(u)\|\varphi\|_{\infty}/\|\nabla_{1}\varphi\|_{1}$, we get (3.13.b). ∎

Proof of Proposition 3.3.

For $\theta>1$ the proposition corresponds to (3.13.b) and (3.13.c). For $\theta\leq 1$, the proposition follows from (3.12) and (3.13.a). ∎

We can now show that if $\mathcal{E}(u)=0$ then $u$ depends only on the variables in $X_{1}$ or only on the variables in $X_{2}$.

Theorem 3.7** (Theorem I).**

If $u\in L(\Omega)$ is such that $\mathcal{E}(u)=0$, then $u\in S(\Omega)$.

Proof.

Let $u\in L(\Omega)$ with $\mathcal{E}(u)=0$. As in the proof of Proposition 3.1, we may assume that $u$ is bounded. Applying Lemma 3.4 (see also Remark 3.5) we find sequences $0<r_{k}\leq\varepsilon_{k}\downarrow 0$ such that up to a change of coordinates,

[TABLE]

By Lemma 3.6, we get

[TABLE]

Integrating twice this identity, since $\Omega_{1}$ and $\Omega_{2}$ are connected, there exist two distributions $u_{l}\in\mathcal{D}^{\prime}(\Omega_{l})$, $l\in\{1,2\}$ such that $u=u_{1}\otimes\boldsymbol{1}_{\Omega_{2}}+\boldsymbol{1}_{\Omega_{1}}\otimes u_{2}$ in $\mathcal{D}^{\prime}(\Omega)$. Let $\varphi_{2}\in C^{\infty}_{c}(\Omega_{2},\mathbf{R}_{+})$ with $\int\varphi_{2}=1$. Using test functions of the form $\varphi(x)=\varphi_{1}(x_{1})\varphi_{2}(x_{2})$, we have, since $u$ is bounded,

[TABLE]

We deduce that $u_{1}\in L^{\infty}(\Omega_{1})$ and similarly $u_{2}\in L^{\infty}(\Omega_{2})$. In conclusion, we have two functions $u_{l}\in L^{\infty}(\Omega_{l})$, $l\in\{1,2\}$ with $u(x)=u_{1}(x_{1})+u_{2}(x_{2})$ for $x\in\Omega$.

Since $q(x,r_{k}(e_{i}+f_{j}))\geq|Du(x,r_{k}e_{i})|^{\theta_{1}}|Du(x,r_{k}f_{j})|^{\theta_{2}}$, (3.19) implies that

[TABLE]

It is easy to check from the formula of $P(\theta_{1},\theta_{2})$ of Definition H that $\max(1,\theta_{1})+\max(1,\theta_{2})\leq P(\theta_{1},\theta_{2})$ for $\theta_{1},\theta_{2}>0$. Moreover, for $l=1,2$, we have $|Du|^{\max(1,\theta_{l})}\lesssim|Du|^{\theta_{l}}\|u\|_{\infty}^{(1-\theta_{l})_{+}}$. These inequalities lead to

[TABLE]

Therefore, up to a subsequence, either

[TABLE]

Let us assume without loss of generality that the former holds. Using Hölder inequality, this yields, for $i\in\{1,\cdots,n_{1}\}$,

[TABLE]

We deduce that the distributions $\partial_{e_{i}}u_{1}$ vanish for $i\in\{1,\cdots,n_{1}\}$ and thus (since $\Omega_{1}$ is connected) $u_{1}$ is constant in $\Omega_{1}$. This concludes the proof of the theorem. ∎

Corollary 3.8** (Corollary L).**

For every $p\geq P(\theta_{1},\theta_{2})$, $r>0$, and every measurable function $u$, if

[TABLE]

then $u\in S(\Omega^{r})$.

Proof of Corollary 3.8.

Let $\rho:=\frac{1}{|B_{r}\backslash B_{r/2}|}\chi_{B_{r}\backslash B_{r/2}}$. Then, letting $\varepsilon_{k}:=2^{-k}r$, it is readily seen that (3.20) implies that

[TABLE]

In particular, $\mathcal{E}_{p}(u;\Omega^{r})=0$ and we get $u\in S(\Omega^{r})$ from Theorem 3.7. ∎

In the space of Lipschitz continuous functions, we know from Proposition 2.2 (i) that the critical exponent $p$ is larger than $1+\theta$. We deduce from Theorem 3.7 that, as soon as $\theta_{1}\leq 1$ or $\theta_{2}\leq 1$, this critical exponent is indeed $1+\theta$.

Proposition 3.9** (Proposition E).**

Assume that $\min(\theta_{1},\theta_{2})\leq 1$. Then, for $u\in Lip(\Omega)$, $[\mathcal{E}_{1+\theta}(u)=0]\Longrightarrow u\in S(\Omega)$.

Proof.

Let us assume without loss of generality that $\theta_{1}\leq 1$. Let $u\in Lip(\Omega)$ with $\mathcal{E}_{1+\theta}(u)=0$. If $\theta\geq 1$ then $P(\theta_{1},\theta_{2})=1+\theta$ and by Theorem 3.7, we have $u\in S(\Omega)$. If $\theta<1$, by Lipschitz continuity of $u$, we have, for $x\in\Omega$ and $z\in\mathbf{R}^{n}\backslash\{0\}$ such that $x+z\in\Omega$,

[TABLE]

This yields $\mathcal{E}_{2+\theta_{2}}^{1,\theta_{2}}(u)\leq\|\nabla_{1}u\|_{\infty}^{1-\theta_{1}}\mathcal{E}_{1+\theta}^{\theta_{1},\theta_{2}}(u)=0$. We notice that $P(1,\theta_{2})=2+\theta_{2}$, so that applying Theorem 3.7 with $\theta^{\prime}_{1}=1$, $\theta^{\prime}_{2}=\theta_{2}$, we get again $u\in S(\Omega)$. ∎

4 Quantitative control of the distance to $S(\Omega)$ in terms of $\mathcal{E}$

The aim of this section is to give a quantitative version of Theorem 3.7 by proving that for $u\in L^{\infty}(\Omega)$, $\mathcal{E}(u)$ controls the distance to $S(\Omega)$ in a strong sense. In order to obtain such a strong control, we use the fact that $\mu=\nabla_{1}\nabla_{2}u$ is a measure and thus restrict ourselves to the case $(a)$ of (1.6), i.e. $\theta_{1}+\theta_{2}\leq 1$. We start by investigating the two dimensional case where the proof is simpler and the result stronger.

Theorem 4.1** (Theorem Q).**

Assume that $n_{1}=n_{2}=1$ and $\theta=\theta_{1}+\theta_{2}\leq 1$. Then, for every $u\in L^{\infty}(\Omega)$ with $\|u\|_{\infty}\leq 1$ and $\mathcal{E}(u)<\infty$, there exists $\bar{u}\in S(\Omega)$ such that $u-\bar{u}\in BV(\Omega)\cap L^{\infty}(\Omega)$ with the estimate

[TABLE]

Proof.

To set notation, we assume that $\Omega=(0,\ell_{1})\times(0,\ell_{2})$, so that $(0,0)$ is the bottom left corner of $\Omega$. By Proposition 3.3, $\mu:=\partial_{1}\partial_{2}u$ is a finite Radon measure with $|\mu|(\Omega)\lesssim\mathcal{E}(u)$. For $x\in\Omega$, we set $w(x):=\mu((0,x_{1}]\times(0,x_{2}])$ and notice that

[TABLE]

Since $\partial_{1}\partial_{2}(u-w)=0$, arguing as in the proof of Theorem 3.7, we find two functions $u_{l}\in L^{\infty}((0,\ell_{l}))$, $l\in\{1,2\}$ such that

[TABLE]

Thanks to (4.2), we only need to control $u_{1}$ or $u_{2}$. Let $x_{1}$ and $z_{1}>0$ be such that $0<x_{1}<x_{1}+z_{1}<\ell_{1}$, we have for $x_{2}\in(0,\ell_{2})$,

[TABLE]

In particular, for $x_{2}\in(0,\ell_{2})$,

[TABLE]

Similarly, for $x_{1}\in(0,\ell_{1})$ and $x_{2}$, $z_{2}>0$ such that $0<x_{2}<x_{2}+z_{2}<\ell_{2}$,

[TABLE]

Notice for later use that for $l\in\{1,2\}$,

[TABLE]

Plugging inequalities (4.3) and (4.4) in the definition (1.3) of $\mathcal{E}_{\varepsilon}(u)$, we get,

[TABLE]

Arguing as in the proof of Lemma 3.4, we may select $\sigma_{1}+\sigma_{2}=\sigma\in\partial B_{1}$, with $1/2\leq|\sigma_{l}|\leq\sqrt{3}/2$ for $l=1,2$ and two sequences sequence $(r_{k})$, $(\varepsilon_{k})$ with $0<r_{k}\leq\varepsilon_{k}\downarrow 0$ such that letting $z^{k}:=r_{k}\sigma$,

[TABLE]

Extracting a further subsequence, there exists $l\in\{1,2\}$ such that

[TABLE]

Without loss of generality, we assume $l=1$. Using the definition of $\psi_{1}$ and Fubini, we write

[TABLE]

Dividing by $r_{k}$ and letting $k\uparrow\infty$, we deduce from (4.6),

[TABLE]

We conclude that $u_{1}\in BV(0,\ell_{1})$ with $|\partial_{1}u_{1}|(0,\ell_{1})\lesssim\mathcal{E}(u)+\mathcal{E}(u)^{\frac{1}{2}}$. This implies $\operatorname{osc}(u_{1})\lesssim\,\mathcal{E}(u)+\mathcal{E}(u)^{\frac{1}{2}}$, which together with (4.2) concludes the proof of the theorem. ∎

We now turn to the higher dimensional case $n>2$. In order to obtain the analog of (4.1), we must define the higher dimensional counterpart of $w$. As will be clear from the proofs, the main requirements are that $\nabla_{1}\nabla_{2}w=\mu[u]$ and that for almost every $x=x_{1}+x_{2}\in\Omega$,

[TABLE]

The unique function which satisfies these conditions is given by the formula:

[TABLE]

For $x=x_{1}+x_{2}\in\Omega$, we have the decomposition $u(x)=u_{1}(x_{1})+u_{2}(x_{2})+w(x)$ where $u_{1}$ and $u_{2}$ are explicitly given by:

[TABLE]

We start by establishing some bounds on $u_{1}$, $u_{2}$ and $w$.

Proposition 4.2**.**

We assume that $\Omega_{1}$ and $\Omega_{2}$ are bounded extension domains. Let $u\in L^{\infty}(\Omega)$ and $w$, $u_{1}$, $u_{2}$ be given by (4.8)(4.9). Then, $w\in L^{\infty}(\Omega)$, $u_{l}\in L^{\infty}(\Omega_{l})$ for $l=1,2$ with $u(x)=u_{1}(x_{1})+u_{2}(x_{2})+w(x)$ and we have the estimates

[TABLE]

Moreover, if $\theta\leq 1$ and $\mathcal{E}(u)<\infty$ then $w\in BV(\Omega)$ and denoting $\hat{n}:=\max(n_{1},n_{2})\geq 2$,

[TABLE]

Proof.

The estimates of (4.10) follow from the definitions (4.8), (4.9) and the triangle inequality.

We turn to (4.11). By (3.9) and the definition of $w$, $\nabla_{1}\nabla_{2}w=\mu$ with $|\mu|(\Omega)\lesssim\mathcal{E}(u)\|u\|_{\infty}^{1-\theta}$. It is thus enough to prove

[TABLE]

By density ($\Omega$ is a bounded extension domain) we may assume that $u\in C^{\infty}(\overline{\Omega})$. For every $x_{1}\in\Omega_{1}$, we have $W_{x_{1}}:=\nabla_{1}w(x_{1}+\cdot)\in W^{1,1}(\Omega_{2})$ with,

[TABLE]

By (4.7), we have \mbox{{\scriptsize\diagup}}\int_{\Omega_{2}}W_{x_{1}}\,=0 for every $x_{1}\in\Omega_{1}$ and by Poincaré-Wirtinger inequality in $W^{1,1}(\Omega_{2})$, this yields

[TABLE]

The analog bound on $\nabla_{2}w$ shows that

[TABLE]

We finally establish the $L^{\frac{\hat{n}}{\hat{n}-1}}$ bound. Assume without loss of generality that $\hat{n}=n_{1}\geq n_{2}$. We then have by (4.7) and Poincaré-Wirtinger inequality in $W^{1,1}(\Omega_{1})$ that

[TABLE]

By Minkowski inequality, this leads to

[TABLE]

We now apply the Poincaré-Wirtinger inequality to $\nabla_{1}w(x_{1}+\cdot)$ in $W^{1,1}(\Omega_{2})$ to obtain (since $n_{2}\leq n_{1}$),

[TABLE]

Together with (4.13) this proves (4.12). ∎

Remark 4.3**.**

We point out that the Poincaré-Wirtinger inequality gives a slightly stronger result than (4.11), namely that $\nabla_{1}w\in\mathcal{M}(\Omega_{1},BV(\Omega_{2}))$ (and similarly for $\nabla_{2}w$).

We turn to the higher dimensional analog of Theorem 4.1.

Proposition 4.4**.**

Assume that $\Omega_{1}$ and $\Omega_{2}$ are bounded extension domains and that $\theta\leq 1$. Let $u\in L^{\infty}(\Omega)$ with $\|u\|_{\infty}\leq 1$ and $\mathcal{E}(u)<\infty$. Using the notation $u=u_{1}+u_{2}+w$ of Proposition 4.2, there exist $l\in\{1,2\}$ and $c\in\mathbf{R}$ such that,

[TABLE]

Combining this estimate, Hölder inequality $\|u_{l}-c\|_{L^{\frac{\hat{n}}{\hat{n}-1}}(\Omega)}\lesssim\|u_{l}-c\|_{L^{\frac{n_{l}}{n_{l}-1}}(\Omega_{l})}$ and Proposition 4.2, we deduce the following result.

Theorem 4.5** (Theorem R).**

Let $\Omega_{1}$ and $\Omega_{2}$ be bounded extension domains and $\theta\leq 1$. Let $u\in L^{\infty}(\Omega)$ with $\|u\|_{\infty}\leq 1$ and $\mathcal{E}(u)<\infty$. Noting $\hat{n}:=\max(n_{1},n_{2})$, there exists $\bar{u}\in L^{\infty}(\Omega)\cap S(\Omega)$ such that,

[TABLE]

Proof of Proposition 4.4.

Since $\|u\|_{\infty}\leq 1$, by (4.10) it is enough to establish (4.14) under the additional assumption that

[TABLE]

for some $\eta>0$ only depending on $\Omega$, $n_{1}$, $n_{2}$ and $\rho$.

Step 1. Let $w$, $u_{1}$ and $u_{2}$ be given by (4.8) and (4.9) so that $u(x)=u_{1}(x_{1})+u_{2}(x_{2})+w(x)$. Let us first recall that by (3.9), $\nabla_{1}\nabla_{2}w=\nabla_{1}\nabla_{2}u=\mu$ is a measure with $|\nabla_{1}\nabla_{2}w|(\Omega)=|\mu|(\Omega)\lesssim\mathcal{E}(u)$. Let $(w_{k})$ be a sequence of mollifications of $w$ with $w_{k}\to w$ in $L^{1}(\Omega)$ and almost everywhere, $\nabla w_{k}\to\nabla w$ weakly star in $\mathcal{M}(\Omega)$, $\|\nabla w_{k}\|_{L^{1}(\Omega)}\to|\nabla w|(\Omega)$ and $\|\nabla_{1}\nabla_{2}w_{k}\|_{L^{1}(\Omega)}\to|\mu|(\Omega)$ (which is possible since $\Omega_{1}$ and $\Omega_{2}$ are extension domains). For $k\geq 0$ and $x_{1}\in\Omega_{1}$, we set

[TABLE]

We have $\|\psi_{1}^{k}\|_{L^{1}(\Omega_{1})}=\|\nabla_{2}w_{k}\|_{L^{1}(\Omega)}$ and $\|\nabla_{1}\psi_{1}^{k}\|_{L^{1}(\Omega)}\leq\|\nabla_{1}\nabla_{2}w_{k}\|_{L^{1}(\Omega)}$. Hence, by (4.11)

[TABLE]

Therefore, $\psi_{1}^{k}$ is bounded in $W^{1,1}(\Omega_{1})$ and up to extraction, $\psi_{1}^{k}$ converges in $L^{1}(\Omega_{1})$ to some function $\psi_{1}\in BV(\Omega_{1})$ with

[TABLE]

Let then for $\lambda>0$,

[TABLE]

In the sequel we use that for a.e. $x_{1}\in\Omega_{1,\lambda}$ and every $z_{2}\in X_{2}$ such that $|(\Omega_{2}-z_{2})\cap\Omega_{2}|>0$,

[TABLE]

Now, by co-area formula [AFP00, Theorem 3.40] for almost every $\lambda>0$ the set $\Omega_{1,\lambda}$ is of finite perimeter in $\Omega_{1}$ and denoting by $\partial\Omega_{1,\lambda}$ the measure-theoretic boundary of $\Omega_{1,\lambda}$,

[TABLE]

We thus pick $\lambda_{1}>0$ such that

[TABLE]

We set $\widetilde{\Omega}_{1}:=\Omega_{1,{\lambda_{1}}}$ and define $\omega_{1}:=\Omega_{1}\ \backslash\widetilde{\Omega}_{1}$ (so that $\mathcal{H}^{n_{1}-1}(\partial\Omega_{1,\lambda}\cap\Omega_{1})=\mathcal{H}^{n_{1}-1}(\partial\omega_{1}\cap\Omega_{1})$). We notice for later use that

[TABLE]

Therefore, since $\lambda_{1}\geq\mathcal{E}(u)^{\frac{1}{2}}/4$, choosing $\eta$ small enough in (4.15), we have,

[TABLE]

We define $\psi_{2}$, $\lambda_{2}$, $\widetilde{\Omega}_{2}$ and $\omega_{2}$ similarly.

Step 2. Let $(r_{k})$, $(\varepsilon_{k})$ with $0<r_{k}\leq\varepsilon_{k}\downarrow 0$ be given by Lemma 3.4 (recall Remark 3.5) and such that for every $1\leq i\leq n_{1}$ and $1\leq j\leq n_{2}$,

[TABLE]

Fix for the moment $k\geq 1$, $1\leq i\leq n_{1}$ and $1\leq j\leq n_{2}$ and let $z^{k}:=r_{k}(e_{i}+f_{j})$. We define

[TABLE]

and similarly for $A_{2}^{k}$. Now we write

[TABLE]

and we split the domain of integration as

[TABLE]

In the first subdomain, we use the inequalities $|Du_{1}|(x_{1},z^{k}_{1})\leq 2|Du|(x,z^{k}_{1})$ and $|Du_{2}|(x_{2},z^{k}_{2})\leq 2|Du|(x,z^{k}_{2})$ valid by definition for $x\in A_{1}^{k}\cap A_{2}^{k}$. Then, since $\|u\|_{\infty}\leq 1$, we get

[TABLE]

In the second subdomain, we use the triangle inequality to get for $x\not\in A_{1}^{k}$,

[TABLE]

so that for $x\not\in A_{1}^{k}$, there holds $|Du_{1}(x_{1},z_{1}^{k})|\leq 2|Dw|(x,z_{1}^{k})$. This leads to

[TABLE]

By definition of $\widetilde{\Omega}_{2}$ and (4.17), the inner integral is bounded by $\lambda_{2}\stackrel{{\scriptstyle\eqref{lbdprime}}}{{\leq}}\mathcal{E}(u)^{\frac{1}{2}}/2$. Hence,

[TABLE]

The third term is bounded similarly and we deduce from (4.21),(4.22) and (4.23),

[TABLE]

Recalling the notation $z_{1}^{k}=r_{k}e_{i}$, $z_{2}^{k}=r_{k}f_{j}$ and summing over $i\in\{1,\cdots,n_{1}\}$ and $j\in\{1,\cdots,n_{2}\}$, we see that up to extraction, we have

[TABLE]

Step 3. Let us assume without loss of generality that the first possibility occurs and let us define the function

[TABLE]

We fix again $i\in\{1,\cdots,n_{1}\}$ and use the short-hand notation $z_{1}^{k}:=r_{k}e_{i}$. we may now estimate, $\int_{\Omega_{1}^{\varepsilon_{k}}}|D\tilde{u}_{1}|(x_{1},z_{1}^{k})/r_{k}\,dx_{1}$. For this we use $|D\tilde{u}_{1}(x_{1},z_{1}^{k})|=|Du_{1}(x_{1},z_{1}^{k})|$ when $x_{1},x_{1}+z_{1}^{k}\in\widetilde{\Omega}_{1}$, $D\tilde{u}_{1}(x_{1},z_{1}^{k})=0$ when $x_{1},x_{1}+z_{1}^{k}\in\omega_{1}$ and $|D\tilde{u}_{1}(x_{1},z_{1}^{k})|\leq\|u\|_{\infty}\leq 1$ when $x_{1}$ or $x_{1}+z_{1}^{k}$ belongs to $\tilde{\Omega}_{1}$ but not both. In the later case we thus have $|D\tilde{u}_{1}(x_{1},z_{1}^{k})|\leq|D\boldsymbol{1}_{\omega_{1}}(x_{1},z_{1}^{k})|$. This leads to

[TABLE]

By (4.18), we have $|\nabla_{1}\boldsymbol{1}_{\omega_{1}}|(\Omega_{1})=\mathcal{H}^{n_{1}-1}(\partial\omega_{1}\cap\Omega_{1})\lesssim\mathcal{E}(u)^{\frac{1}{2}}$ and we obtain that $e_{i}\cdot\nabla\widetilde{u}_{1}$ is a measure in $\Omega_{1}$ which satisfies $|e_{i}\cdot\nabla\widetilde{u}_{1}|(\Omega_{1})\lesssim\mathcal{E}(u)^{\frac{1}{2}}$. Summing over $i$ we get $\tilde{u}_{1}\in BV(\Omega_{1})$ with

[TABLE]

Let us note $c$ the mean value of $\tilde{u}_{1}$ (in particular $|c|\leq\|u_{1}\|_{\infty}$). Using the Sobolev injection of $BV(\Omega_{1})$ into $L^{n_{1}/(n_{1}-1)}(\Omega_{1})$, we compute

[TABLE]

By (4.19), we have $|\omega_{1}|\leq|\Omega_{1}|/2$ and thus by the relative isoperimetric inequality in $\Omega_{1}$ (see for instance [Fed69, Lemma 4.5.2]),

[TABLE]

so that our final estimate is

[TABLE]

This concludes the proof of the proposition. ∎

Acknowledgments

We thank V. Millot for very stimulating discussions during the early stages of this research. M. Goldman is partially supported by the ANR SHAPO. B. Merlet is partially supported by the INRIA team RAPSODI and the Labex CEMPI (ANR-11-LABX-0007-01).