Asymptotic behavior of the $W^{1/q,q}$-norm of mollified $BV$ functions and applications to singular perturbation problems

TL;DR

This paper establishes the asymptotic limit of the $W^{1/q,q}$-norm of mollified BV functions, revealing a connection to jump discontinuities, and applies this to analyze singular perturbation problems.

Contribution

It proves a new limit formula for mollified BV functions' Sobolev norms and applies it to singular perturbation analysis, extending prior results by Figalli, Jerison, and Hernández.

Findings

Limit of mollified BV functions' Sobolev norm characterized

Connection between norm asymptotics and jump discontinuities

Applications to singular perturbation problems

Abstract

Motivated by results of Figalli and Jerison and Hern\'andez, we prove the following formula: \begin{equation*} \lim_{\epsilon\to 0^+}\frac{1}{|\ln{\epsilon}|}\big\|\eta_\epsilon*u\big\|^q_{W^{1/q,q}(\Omega)}= C_0\int_{J_u}\Big|u^+(x)-u^-(x)\Big|^qd\mathcal{H}^{N-1}(x), \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a regular domain, $u\in BV(\Omega)\cap L^\infty$, $q>1$ and $\eta_\epsilon(z)=\epsilon^{-N}\eta(z/\epsilon)$ is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.

Equations

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{|\ln{\varepsilon}|}\big{\|}\eta_{\varepsilon}*u\big{\|}^{q}_{W^{1/q,q}(\Omega)}=C_{0}\int_{J_{u}}\Big{|}u^{+}(x)-u^{-}(x)\Big{|}^{q}d\mathcal{H}^{N-1}(x),$$

$$C_{1}P(A)\leq\liminf_{\varepsilon\to 0^{+}}\frac{1}{|\ln{\varepsilon}|}\big{\|}\eta_{\varepsilon}*\chi_{A}\big{\|}^{2}_{H^{1/2}(\mathbb{R}^{N})}\leq\limsup_{\varepsilon\to 0^{+}}\frac{1}{|\ln{\varepsilon}|}\big{\|}\eta_{\varepsilon}*\chi_{A}\big{\|}^{2}_{H^{1/2}(\mathbb{R}^{N})}\leq C_{2}P(A),$$

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{|\ln{\varepsilon}|}\big{\|}\eta_{\varepsilon}*u\big{\|}^{2}_{H^{1/2}(\mathbb{R}^{N})}=C_{0}\int_{J_{u}}\Big{|}u^{+}(x)-u^{-}(x)\Big{|}^{2}d\mathcal{H}^{N-1}(x).$$

$$\lim_{\varepsilon\to 0^{+}}\varepsilon\big{\|}\eta_{\varepsilon}*u\big{\|}^{2}_{H^{1}(\Omega)}=C_{\eta}\int_{J_{u}}\Big{|}u^{+}(x)-u^{-}(x)\Big{|}^{2}d\mathcal{H}^{N-1}(x).$$

$$\lim\limits_{\varepsilon\to 0^{+}}\int\limits_{\Omega}\int\limits_{B_{\varepsilon}(x)\cap\Omega}\frac{1}{\varepsilon^{N}}\,\frac{\big{|}u(y)-u(x)\big{|}^{q}}{|y-x|}dydx=C_{N}\int_{J_{u}}\Big{|}u^{+}(x)-u^{-}(x)\Big{|}^{q}d\mathcal{H}^{N-1}(x),$$

$$C_N:=\frac{1}{N}{\int }_{S^{N-1}}|z_1|d{\mathcal{H}}^{N-1}(z),$$

$$\|u\|_{W^{1/q,q}(\Omega,\mathbb{R}^{d})}:=\Bigg{(}\int_{\Omega}\bigg{(}\int_{\Omega}\frac{\big{|}u(x)-u(y)\big{|}^{q}}{|x-y|^{N+1}}dy\bigg{)}dx\Bigg{)}^{\frac{1}{q}}.$$

$$u_{\varepsilon}(x):=\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}u(y)dy=(\eta_{\varepsilon}*u)(x).$$

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{|\ln{\varepsilon}|}\|u_{\varepsilon}\|^{q}_{W^{1/q,q}(\Omega,\mathbb{R}^{d})}=\\ 2\bigg{|}\int_{\mathbb{R}^{N}}\eta(z)dz\bigg{|}^{q}\Bigg{(}\int_{\mathbb{R}^{N-1}}\frac{dv}{\big{(}\sqrt{1+|v|^{2}}\big{)}^{N+1}}\Bigg{)}\int_{J_{u}\cap\Omega}\Big{|}u^{+}(x)-u^{-}(x)\Big{|}^{q}d\mathcal{H}^{N-1}(x).$$

$$E^{(1)}_{\varepsilon}(v):=\begin{cases}\frac{1}{|\ln{\varepsilon}|}\|v\|^{q}_{W^{1/q,q}(\Omega,\mathbb{R}^{d})}+\frac{1}{\varepsilon}\int_{\Omega}W\big{(}v,x\big{)}dx\quad\quad\text{if}\quad A\cdot\nabla v=0\\ +\infty\quad\quad\text{otherwise},\end{cases}$$

$$E^{(2)}_{\varepsilon}(v):=\begin{cases}\frac{1}{|\ln{\varepsilon}|}\left(\|v\|^{q}_{W^{1/q,q}(\mathbb{R}^{N},\mathbb{R}^{d})}-\|v\|^{q}_{W^{1/q,q}(\mathbb{R}^{N}\setminus\overline{\Omega},\mathbb{R}^{d})}\right)+\frac{1}{\varepsilon}\int_{\Omega}W\big{(}v,x\big{)}dx\quad\text{if}\quad A\cdot\nabla v=0\\ +\infty\quad\quad\text{otherwise},\end{cases}$$

$$\hat{E}_{\varepsilon}(v):=\begin{cases}\varepsilon^{q-1}\|v\|^{q}_{W^{1,q}(\Omega,\mathbb{R}^{d})}+\frac{1}{\varepsilon}\int_{\Omega}W\big{(}v,x\big{)}dx\quad\quad\text{if}\quad A\cdot\nabla v=0\\ +\infty\quad\quad\text{otherwise},\end{cases}$$

$$\tilde{E}_{\varepsilon}(\psi):=\int_{\Omega}\left\{\varepsilon|\nabla^{2}\psi|^{2}+\frac{1}{\varepsilon}\Big{(}1-|\nabla\psi|^{2}\Big{)}^{2}\right\}dx\quad\quad\text{(see \cite[cite]{[\@@bibref{}{adm,ag1,ag2}{}{}]})},$$

$$\big{|}\nabla_{b}W(b,x)\big{|}\leq C_{D}\quad\quad\forall x\in\mathbb{R}^{N},\;\forall\,b\in B_{D}(0).$$

$$\limsup_{\varepsilon\to 0^{+}}\bigg{(}\frac{1}{|\ln{\varepsilon}|}\Big{(}\|\psi_{\varepsilon}\|^{q}_{W^{1/q,q}(\mathbb{R}^{N},\mathbb{R}^{d})}-\|\psi_{\varepsilon}\|^{q}_{W^{1/q,q}(\mathbb{R}^{N}\setminus\overline{\Omega},\mathbb{R}^{d})}\Big{)}+\frac{1}{\varepsilon}\int_{\Omega}W\Big{(}\psi_{\varepsilon}(x),x\Big{)}dx\bigg{)}=\\ \limsup_{\varepsilon\to 0^{+}}\bigg{(}\frac{1}{|\ln{\varepsilon}|}\|\psi_{\varepsilon}\|^{q}_{W^{1/q,q}(\Omega,\mathbb{R}^{d})}+\frac{1}{\varepsilon}\int_{\Omega}W\Big{(}\psi_{\varepsilon}(x),x\Big{)}dx\bigg{)}=\\ \Bigg{(}\int_{\mathbb{R}^{N-1}}\frac{2}{\big{(}\sqrt{1+|v|^{2}}\big{)}^{N+1}}dv\Bigg{)}\int_{J_{u}\cap\Omega}\big{|}u^{+}(y)-u^{-}(y)\big{|}^{q}d\mathcal{H}^{N-1}(y).$$

$${\int }_{\Omega }{\psi }_{\epsilon }(x)dx={\int }_{\Omega }u(x)dx\forall \epsilon >0.$$

$$u_{\varepsilon}(x):=\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}u(y)dy=(\eta_{\varepsilon}*u)(x).$$

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{|\ln{\varepsilon}|}\|u_{\varepsilon}\|^{q}_{W^{1/q,q}(\Omega,\mathbb{R}^{d})}=\\ 2\bigg{|}\int_{\mathbb{R}^{N}}\eta(z)dz\bigg{|}^{q}\Bigg{(}\int_{\mathbb{R}^{N-1}}\frac{1}{\big{(}\sqrt{1+|v|^{2}}\big{)}^{N+1}}dv\Bigg{)}\int_{J_{u}\cap\Omega}\Big{|}u^{+}(x)-u^{-}(x)\Big{|}^{q}d\mathcal{H}^{N-1}(x).$$

$$H_{+}(x,\mathbf{\nu })$$

$$H_{-}(x,\mathbf{\nu })$$

$$H_0(\mathbf{\nu })$$

$$u_{\varepsilon}(x):=\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}u(y)dy=\int_{\mathbb{R}^{N}}\eta(z)u(x+\varepsilon z)dz=\int_{B_{R}(0)}\eta(z)u(x+\varepsilon z)dz.$$

$$\frac{d}{d\varepsilon}u_{\varepsilon}(x):=-\frac{N}{\varepsilon^{N+1}}\int_{\mathbb{R}^{N}}\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}u(y)dy-\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\frac{y-x}{\varepsilon^{2}}\cdot\nabla\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}u(y)dy=\\ -\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\operatorname{div}_{y}\bigg{\{}\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}\frac{y-x}{\varepsilon}\bigg{\}}u(y)dy=\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}\frac{y-x}{\varepsilon}\cdot d\big{[}Du(y)\big{]}.$$

$$\|u_{\varepsilon}\|^{q}_{W^{1/q,q}}=\|u_{\varepsilon}\|^{q}_{W^{1/q,q}(\Omega,\mathbb{R}^{d})}=\int_{\mathbb{R}^{N}}\bigg{(}\int_{\mathbb{R}^{N}}\frac{\big{|}u_{\varepsilon}(x)-u_{\varepsilon}(y)\big{|}^{q}}{|x-y|^{N+1}}\chi_{\Omega}(y)dy\bigg{)}\chi_{\Omega}(x)dx\\ =\int_{\mathbb{R}^{N}}\bigg{(}\int_{\mathbb{R}^{N}}\frac{\big{|}u_{\varepsilon}(x+z)-u_{\varepsilon}(x)\big{|}^{q}}{|z|^{N+1}}\chi_{\Omega}(x+z)\chi_{\Omega}(x)dz\bigg{)}dx,$$

$${\chi }_{\Omega }(x):=\{\begin{array}{l}1\forall x\in \Omega \\ 0\forall x\in {\mathbb{R}}^N\setminus \Omega \end{array}.$$

$$\frac{1}{-\ln{\varepsilon}}\|u_{\varepsilon}\|^{q}_{W^{1/q,q}}=-\frac{1}{\ln{\varepsilon}}\int_{\mathbb{R}^{N}}\bigg{(}\int_{\mathbb{R}^{N}}\frac{\big{|}u_{\varepsilon}(x+z)-u_{\varepsilon}(x)\big{|}^{q}}{|z|^{N+1}}\chi_{\Omega}(x+z)\chi_{\Omega}(x)dz\bigg{)}dx.$$

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{-\ln{\varepsilon}}\|u_{\varepsilon}\|^{q}_{W^{1/q,q}}=\\ -\lim\limits_{\varepsilon\to 0^{+}}\int\limits_{\mathbb{R}^{N}}\Bigg{(}\int\limits_{\mathbb{R}^{N}}\frac{\varepsilon}{|z|^{N+1}}\bigg{(}\frac{d}{d\varepsilon}\Big{(}u_{\varepsilon}(x+z)-u_{\varepsilon}(x)\Big{)}\bigg{)}\cdot\nabla F_{q}\big{(}u_{\varepsilon}(x+z)-u_{\varepsilon}(x)\big{)}\chi_{\Omega}(x+z)\chi_{\Omega}(x)dz\Bigg{)}dx,$$

$$F_q(h):=|h{|}^q\forall h\in {\mathbb{R}}^d.$$

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{-\ln{\varepsilon}}\|u_{\varepsilon}\|^{q}_{W^{1/q,q}}=\\ -\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{\varepsilon}{|z|^{N+1}}\Bigg{\{}\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\bigg{(}\eta\Big{(}\frac{y-(x+z)}{\varepsilon}\Big{)}\frac{y-(x+z)}{\varepsilon}-\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}\frac{y-x}{\varepsilon}\bigg{)}\cdot d\big{[}Du(y)\big{]}\Bigg{\}}\times\\ \times\nabla F_{q}\Bigg{(}\int_{\mathbb{R}^{N}}\eta(\xi)\Big{(}u(x+z+\varepsilon\xi)-u(x+\varepsilon\xi)\Big{)}d\xi\Bigg{)}\chi_{\Omega}(x+z)\chi_{\Omega}(x)dzdx=\\ -\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{\varepsilon}{|z|^{N+1}}\frac{1}{\varepsilon^{N}}\bigg{(}\eta\Big{(}\frac{y-(x+z)}{\varepsilon}\Big{)}\frac{y-(x+z)}{\varepsilon}-\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}\frac{y-x}{\varepsilon}\bigg{)}\times\\ \times\nabla F_{q}\Bigg{(}\int_{\mathbb{R}^{N}}\eta(\xi)\Big{(}u(x+z+\varepsilon\xi)-u(x+\varepsilon\xi)\Big{)}d\xi\Bigg{)}\chi_{\Omega}(x+z)\chi_{\Omega}(x)dzdx\cdot d\big{[}Du(y)\big{]}.$$

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{-\ln{\varepsilon}}\|u_{\varepsilon}\|^{q}_{W^{1/q,q}}=\\ -\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{1}{|z|^{N+1}}\frac{1}{\varepsilon^{N}}\bigg{(}\eta\Big{(}\frac{y-x}{\varepsilon}-z\Big{)}\Big{(}\frac{y-x}{\varepsilon}-z\Big{)}-\eta\Big{(}\frac{y-x}{\varepsilon}\Big{)}\frac{y-x}{\varepsilon}\bigg{)}\times\\ \times\nabla F_{q}\Bigg{(}\int_{\mathbb{R}^{N}}\eta(\xi)\Big{(}u(x+\varepsilon z+\varepsilon\xi)-u(x+\varepsilon\xi)\Big{)}d\xi\Bigg{)}\chi_{\Omega}(x+\varepsilon z)\chi_{\Omega}(x)dzdx\cdot d\big{[}Du(y)\big{]}=\\ -\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{1}{|z|^{N+1}}\bigg{(}\eta\big{(}x-z\big{)}\big{(}x-z\big{)}-\eta\big{(}x\big{)}x\bigg{)}\times\\ \times\nabla F_{q}\Bigg{(}\int_{\mathbb{R}^{N}}\eta(\xi)\Big{(}u(y+\varepsilon z+\varepsilon\xi-\varepsilon x)-u(y+\varepsilon\xi-\varepsilon x)\Big{)}d\xi\Bigg{)}\chi_{\Omega}(y-\varepsilon x+\varepsilon z)\chi_{\Omega}(y-\varepsilon x)dzdx\cdot d\big{[}Du(y)\big{]}.$$

$$\lim_{\varepsilon\to 0^{+}}\frac{1}{-\ln{\varepsilon}}\|u_{\varepsilon}\|^{q}_{W^{1/q,q}}\\ =-\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{1}{|z|^{N+1}}\bigg{(}\eta\big{(}x-z\big{)}\big{(}x-z\big{)}-\eta\big{(}x\big{)}x\bigg{)}\times\\ \times\nabla F_{q}\Bigg{(}\int_{\mathbb{R}^{N}}\Big{(}\eta(\xi-z)-\eta(\xi)\Big{)}u(y+\varepsilon\xi-\varepsilon x)d\xi\Bigg{)}\chi_{\Omega}(y-\varepsilon x+\varepsilon z)\chi_{\Omega}(y-\varepsilon x)dzdx\cdot d\big{[}Du(y)\big{]}\\ =-\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{1}{|z|^{N+1}}\bigg{(}\eta\big{(}x-z\big{)}\big{(}x-z\big{)}-\eta\big{(}x\big{)}x\bigg{)}\times\\ \times\nabla F_{q}\Bigg{(}\int_{\mathbb{R}^{N}}\Big{(}\eta(\xi+x-z)-\eta(\xi+x)\Big{)}u(y+\varepsilon\xi)d\xi\Bigg{)}\chi_{\Omega}(y-\varepsilon x+\varepsilon z)\chi_{\Omega}(y-\varepsilon x)dzdx\cdot d\big{[}Du(y)\big{]}.$$

Full text

[TABLE]

where $\Omega\subset\mathbb{R}^{N}$ is a regular domain, $u\in BV(\Omega)\cap L^{\infty}$, $q>1$ and $\eta_{\varepsilon}(z)=\varepsilon^{-N}\eta(z/\varepsilon)$ is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.

Arkady Poliakovsky 111E-mail: poliakov@math.bgu.ac.il

Department of Mathematics, Ben Gurion University of the Negev,

P.O.B. 653, Be’er Sheva 84105, Israel

1 Introduction

Figalli and Jerison found in [8] a relationship between the perimeter of a set and a fractional Sobolev norm of its characteristic function. More precisely, for the mollifying kernel $\eta_{\varepsilon}(z)=\varepsilon^{-N}\eta(z/\varepsilon)$, where $\eta(z)$ denotes the standard Gaussian in $\mathbb{R}^{N}$, they showed that there exist constants $C_{1}>0$ and $C_{2}>0$ such that for every set $A\subset\mathbb{R}^{N}$ of finite perimeter $P(A)$ we have

[TABLE]

where $\chi_{A}$ is the characteristic function of $A$. More recently, Hernández improved this result in [7] as follows. For $\eta_{\varepsilon}$ as above he showed that there exist a constant $C_{0}>0$ such that for every $u\in BV(\mathbb{R}^{N})\cap L^{\infty}$ we have

[TABLE]

A related result in which the same R.H.S. as in (1.2) appears, was obtained in [13]. More precisely, we showed in [13] that for every radial $\eta\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$ there exists a constant $C=C_{\eta}>0$ such that for every $u\in BV(\Omega,\mathbb{R}^{d})\cap L^{\infty}$ we have

[TABLE]

More recently, we showed in [14] yet another related result:

Theorem 1.1**.**

Let $\Omega\subset\mathbb{R}^{N}$ be an open set with bounded Lipschitz boundary and let $u\in BV(\Omega,\mathbb{R}^{d})\cap L^{\infty}(\Omega,\mathbb{R}^{d})$. Then, for every $q>1$ we have

[TABLE]

with the dimensional constant $C_{N}>0$ defined by

[TABLE]

where we denote $z:=(z_{1},\ldots,z_{N})\in\mathbb{R}^{N}$.

In the present paper we generalize the formula (1.2) in several aspects:

• •

We allow a general mollifying kernel $\eta\in W^{1,1}(\mathbb{R}^{N},\mathbb{R})$ (not only the Gaussian as before),

• •

We allow a general domain $\Omega\subset\mathbb{R}^{N}$, of certain regularity, while previous results required $\Omega=\mathbb{R}^{N}$,

• •

We treat the $W^{1/q,q}(\Omega)$-norm for any $q>1$, while previous results were restricted to the case $q=2$.

Recall that the Gagliardo seminorm $\|u\|_{W^{1/q,q}(\Omega,\mathbb{R}^{d})}$ is given by

[TABLE]

Our first main result is

Theorem 1.2**.**

Let $\Omega\subset\mathbb{R}^{N}$ be an open set and let $u\in BV(\mathbb{R}^{N},\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{N},\mathbb{R}^{d})$ be such that $\|Du\|(\partial\Omega)=0$. For $\eta\in W^{1,1}(\mathbb{R}^{N},\mathbb{R})$, every $x\in\mathbb{R}^{N}$ and every $\varepsilon>0$ define

[TABLE]

Then, for any $q>1$ we have

[TABLE]

Theorem 1.2 enables us to prove an upper bound, in the limit $\varepsilon\to 0^{+}$, for the following singular perturbation functionals with differential constraints:

• (i)

[TABLE]

for $v:\Omega\to\mathbb{R}^{d}$;

• (ii)

[TABLE]

for $v:\mathbb{R}^{N}\to\mathbb{R}^{d}$.

In both cases $A:\mathbb{R}^{d\times N}\to\mathbb{R}^{l}$ is a linear operator (possibly trivial). The most important particular cases are the following:

• (a)

$A\equiv 0$ (i.e., without any prescribed differential constraint),

• (b)

$d=N$, $l=N^{2}$ and A\cdot\nabla v\equiv\text{curl}\,v:=\big{\{}\partial_{k}v_{j}-\partial_{j}v_{k}\big{\}}_{1\leq k,j\leq N},

• (c)

$l=d$ and $A\cdot\nabla v\equiv\operatorname{div}v$.

The $\Gamma$-limit of the functional (1.9) in the $L^{p}$-topology when $A\equiv 0$, $q=2$, $N=1$ and $W$ is a double-well potential was found by Alberti, Bouchitté and Seppecher [1]. The result was generalized to any dimension $N\geq 1$, for the functional (1.10), by Savin and Valdinoci [15].

Note that the functional (1.9) resembles the energy functional in the following singular perturbation problem:

[TABLE]

that attracted a lot of attention by many authors, starting from Modica and Mortola [10], Modica [9], Sternberg [16] and others, who studied the basic special case of (1.11) with $A\equiv 0$, $q=2$ and $W$ being a double-well potential. The $\Gamma$ limit of (1.11) with $A\equiv 0$, $q=2$ and a general $W\in C^{0}$ that does not depend on $x$, was found by Ambrosio in [2]. As an example with a nontrivial differential constraint we mention the Aviles-Giga functional, that appear in various applications. It is defined for scalar functions $\psi$ by

[TABLE]

and the objective is to study the $\Gamma$-limit, as $\varepsilon\to 0^{+}$. This can be seen as a special case of (1.11) if we set $v:=\nabla\psi$ and let $A\cdot\nabla v\equiv\text{curl}\,v$, $q=2$ and $W(v,x)=(1-|v|^{2})^{2}$.

Our second result provides an upper bound for the energies (1.9)-(1.10):

Theorem 1.3**.**

Let $\Omega\subset\mathbb{R}^{N}$ be an open set and let $W:\mathbb{R}^{d}\times\mathbb{R}^{N}\to\mathbb{R}$ be a Borel measurable nonnegative function, continuous and continuously differentiable w.r.t. the first argument, such that $W(0,\cdot)\in L^{1}(\Omega,\mathbb{R})$. Assume further that for every $D>0$ there exists $C:=C_{D}>0$ such that

[TABLE]

Let $u\in BV(\mathbb{R}^{N},\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{N},\mathbb{R}^{d})$ be such that W\big{(}u(x),x\big{)}=0 a.e. in $\Omega$, $\|Du\|(\partial\Omega)=0$, and $A\cdot Du=0$ in $\mathbb{R}^{N}$, where $A:\mathbb{R}^{d\times N}\to\mathbb{R}^{l}$ is a prescribed linear operator (possibly trivial). Then, for any $q>1$ there exists a sequence of functions \big{\{}\psi_{\varepsilon}\big{\}}_{\varepsilon>0}\subset C^{\infty}(\mathbb{R}^{N},\mathbb{R}^{d})\cap W^{1,1}(\mathbb{R}^{N},\mathbb{R}^{d})\cap W^{1,\infty}(\mathbb{R}^{N},\mathbb{R}^{d}) such that $A\cdot D\psi_{\varepsilon}=0$ in $\mathbb{R}^{N}$, $\psi_{\varepsilon}(x)\to u(x)$ strongly in $L^{p}(\mathbb{R}^{N},\mathbb{R}^{d})$ for every $p\geq 1$, and

[TABLE]

Moreover, in the case $A\equiv 0$ we can choose $\psi_{\varepsilon}$ to satisfy also

[TABLE]

Unfortunately, the upper bound found in Theorem 1.3 is not sharp in the most general case with a nontrivial prescribed differential constraint. For example, in the particular case of (1.9) with $N=2$, $A\cdot\nabla v\equiv\text{curl}\,v$, $q>3$ and $W(v,x)=(1-|v|^{2})^{2}$, the functional on the R.H.S. of (1.14) is not lower semicontinuous, hence cannot be the $\Gamma$-limit (see [3]). However, we still hope that the result of the above theorem could provide the sharp upper bound in some cases with $A=0$. Indeed, the $\Gamma$-limit, computed in [1] for the special case of (1.9) with $A\equiv 0$, $q=2$, $N=1$ and $W$ being a double well potential, coincides with the upper bound found in Theorem 1.3. Moreover, since the functional in (1.10) is superior to the functional in (1.9), the $\Gamma$-limit, found in [15] (see also [12]) for the energy (1.10) in any dimension $N\geq 1$ with $A\equiv 0$, $q=2$ and $W$ being a double well potential, coincides again with our upper bound.

The paper is organized as follows. In section 2 we prove our two main results. For the convenience of the reader, in the Appendix we recall some known results on $BV$ functions, needed for the proofs.

2 Proof of the main results

Proposition 2.1**.**

Let $q>1$, $\Omega\subset\mathbb{R}^{N}$ be an open set and $u\in BV(\mathbb{R}^{N},\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{N},\mathbb{R}^{d})$ be such that $\|Du\|(\partial\Omega)=0$. Let $\eta\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$ and for every $x\in\mathbb{R}^{N}$ and every $\varepsilon>0$ define

[TABLE]

Then,

[TABLE]

Proof.

We start with some notations. For every $\boldsymbol{\nu}\in S^{N-1}$ and $x\in\mathbb{R}^{N}$ set

[TABLE]

Let $R>0$ be such that $\operatorname{supp}\eta\subset B_{R}(0)$. For every $x\in\mathbb{R}^{N}$ and every $\varepsilon>0$ we rewrite (2.1) as:

[TABLE]

By (2.6) we have

[TABLE]

Moreover, by (1.6) we have

[TABLE]

where

[TABLE]

Thus,

[TABLE]

Since $-\ln{\varepsilon}\to+\infty$ as $\varepsilon\to 0^{+}$, applying L’Hôpital’s rule to the expression in (2.10) yields

[TABLE]

where $F_{q}\in C^{1}(\mathbb{R}^{d},\mathbb{R})$ is defined by

[TABLE]

Thus, by (2.11), (2.6) and (2.7) we get

[TABLE]

Changing variable, $z/\varepsilon\to z$, in the integration on the R.H.S. of (2.13) gives

[TABLE]

Therefore,

[TABLE]

On the other hand, by (3.1) in the Appendix, for every $x,z\in\mathbb{R}^{N}$ and $\mathcal{H}^{N-1}$-a.e. $y\in\mathbb{R}^{N}$ we have

[TABLE]

with $H_{\pm}(x,\boldsymbol{\nu})$ as defined in (2.3) and (2.4). Thus, since $\|Du\|(\partial\Omega)=0$, by (2.16) and the Dominated Convergence Theorem we obtain:

[TABLE]

It follows that

[TABLE]

where we used in the last step the fact that $\int_{\mathbb{R}^{N}}\eta(\xi+x-z)d\xi=\int_{\mathbb{R}^{N}}\eta(\xi+x)d\xi$. Next, by (2.18) and (2.12) we infer that

[TABLE]

where $G_{q}(\rho)\in C^{1}(\mathbb{R},\mathbb{R})$ is defined by

[TABLE]

and $H_{0}(\boldsymbol{\nu})$ is defined in (2.5). Therefore,

[TABLE]

Introducing the notation

[TABLE]

allows us to rewrite (2.21) as

[TABLE]

The change of variables $w/|\tau|\to v$ in the R.H.S. of (2.23) gives

[TABLE]

where $D_{N}$ is the dimensional constant given by

[TABLE]

Then we rewrite (2.24) as

[TABLE]

Integration by parts of (2.26) and using (2.20) give

[TABLE]

Therefore, applying L’Hôpital’s rule in (2.27), using (2.20), we deduce that

[TABLE]

Changing variables of integration we rewrite (2.28) as

[TABLE]

Applying Newton-Leibniz formula in (2.29) and using (2.20) we obtain that

[TABLE]

and (2.2) follows. ∎

Corollary 2.1**.**

Let $q>1$ and let $\Omega\subset\mathbb{R}^{N}$ be an open set. Assume $W:\mathbb{R}^{d}\times\mathbb{R}^{N}\to\mathbb{R}$ is a Borel measurable function such that, $W(0,\cdot)\in L^{1}(\Omega,\mathbb{R})$ and for every $D>0$ there exists $C:=C_{D}>0$ such that

[TABLE]

Let $u\in BV(\mathbb{R}^{N},\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{N},\mathbb{R}^{d})$ be such that $\|Du\|(\partial\Omega)=0$ and W\big{(}u(x),x\big{)}=0 a.e. in $\Omega$. Let $\eta\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$ be such that $\int_{\mathbb{R}^{N}}\eta(z)dz=1$ and $\operatorname{supp}\eta\subset B_{R}(0)$. For every $\rho>0$ set

[TABLE]

Finally, for every $x\in\mathbb{R}^{N}$ and every $\varepsilon>0$ define

[TABLE]

Then,

[TABLE]

Proof.

Since $\int_{\mathbb{R}^{N}}\eta_{\rho}(z)dz=1$, applying Proposition 2.1, first for $\mathbb{R}^{N}$, then for $\mathbb{R}^{N}\setminus\overline{\Omega}$, and finally for $\Omega$, yields, for every $\rho>0$,

[TABLE]

where $D_{N}$ is the constant defined in (2.25). On the other hand, since W\big{(}u(x),x\big{)}=0 a.e. in $\Omega$ and $u\in L^{\infty}$, by (2.31) we get that

[TABLE]

for some constant $C>0$, independent of $\varepsilon$ and $\rho$. Thus, taking into account the following well known uniform bound from the theory of $BV$ functions,

[TABLE]

we obtain that

[TABLE]

By (2.38) and (2.35) we finally derive (2.34). ∎

Proof of Theorem 1.3.

Let $\eta,\eta_{\rho}$ and $u_{\rho,\varepsilon}$ be defined as in Corollary 2.1. Then $u_{\rho,\varepsilon}\in C^{\infty}(\mathbb{R}^{N},\mathbb{R}^{d})\cap W^{1,1}(\mathbb{R}^{N},\mathbb{R}^{d})\cap W^{1,\infty}(\mathbb{R}^{N},\mathbb{R}^{d})$ and by Corollary 2.1 we have

[TABLE]

Clearly, for every $x\in\mathbb{R}^{N}$ we have $A\cdot\nabla u_{\rho,\varepsilon}(x)=0$ and $u_{\rho,\varepsilon}(x)\to u(x)$ strongly in $L^{p}(\mathbb{R}^{N},\mathbb{R}^{d})$ as $\varepsilon\to 0^{+}$ for every fixed $\rho$ and $p$. Therefore, by the above and by (2.39) we can complete the proof of the first assertion of the theorem using a standard diagonal argument.

It remains to show the second assertion of the theorem, namely, that in the case $A\equiv 0$ we can construct $\psi_{\varepsilon}$ satisfying the additional condition (1.15). Let $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N},\mathbb{R})$ be such that $\int_{\Omega}\varphi(x)dx=1$. Define

[TABLE]

where

[TABLE]

In particular,

[TABLE]

and $\lim_{\varepsilon\to 0^{+}}c_{\varepsilon,\rho}=0$. On the other hand, since W\big{(}u(x),x\big{)}=0 a.e. in $\Omega$, $W(b,x)$ is nonnegative and $W(b,x)$ is differentiable with respect to the $b$ variable, we have

[TABLE]

Thus, since $u\in L^{\infty}$, by (2.40) we get that

[TABLE]

On the other hand, taking into account (2.37) and using the Dominated Convergence Theorem and (2.43), we obtain that

[TABLE]

Using (2.45) in (2.44) yields

[TABLE]

Plugging (2.46) into (2.39) we get that

[TABLE]

Moreover, $\tilde{u}_{\rho,\varepsilon}\to u$ strongly in $L^{p}(\mathbb{R}^{N},\mathbb{R}^{d})$ as $\varepsilon\to 0^{+}$ for every fixed $\rho$ and $p$. Therefore, by the above and (2.47) we complete again the proof by a standard diagonal argument. ∎

The next lemma is needed for the proof of Theorem 1.2 (in the general case $\eta\in W^{1,1}$).

Lemma 2.1**.**

Let $\Omega\subset\mathbb{R}^{N}$ be an open set and let $u\in BV(\mathbb{R}^{N},\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{N},\mathbb{R}^{d})$. For $\eta\in W^{1,1}(\mathbb{R}^{N},\mathbb{R})$, every $x\in\mathbb{R}^{N}$ and every $\varepsilon>0$ define

[TABLE]

Then, for every $q>1$ and for every $\varepsilon\in(0,1)$ we have

[TABLE]

where $\omega_{N-1}$ denotes the surface area of the unit ball in $\mathbb{R}^{N}$.

Proof.

Assume first that $\eta(z)\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$. Then, by (2.48) we have

[TABLE]

By (2.48) and (2.50) we get that

[TABLE]

Next, for every $\varepsilon\in(0,1)$ we have

[TABLE]

On the other hand, (2.51) yields

[TABLE]

Thus, inserting (2.53) into (2.52) we deduce that

[TABLE]

Inserting (2.48) into (2.54) and using the second inequality in (2.51) we infer,

[TABLE]

Taking into account the following well known uniform bound from the theory of $BV$ functions:

[TABLE]

we rewrite (2.55) as

[TABLE]

Computing the integrals on the R.H.S. of (2.57) yields (2.49) in the case $\eta\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$.

Next consider the general case $\eta\in W^{1,1}(\mathbb{R}^{N},\mathbb{R})$. Thanks to the density of $C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$ in $W^{1,1}(\mathbb{R}^{N},\mathbb{R})$, there exists a sequence \big{\{}\eta_{n}\big{\}}_{n=1}^{\infty}\subset C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R}) such that

[TABLE]

Thus if we define

[TABLE]

then

[TABLE]

On the other hand, since we proved (2.49) for the case $\eta_{n}\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$, for every $q>1$, for every $n=1,2,\ldots$ and for every $\varepsilon\in(0,1)$ we have:

[TABLE]

Letting $n$ go to infinity in (2.61), using (2.58) in the R.H.S. and (2.60) together with Fatou’s Lemma in the L.H.S., we obtain (2.49) in the general case $\eta\in W^{1,1}(\mathbb{R}^{N},\mathbb{R})$. ∎

Proof of Theorem 1.2.

In the case $\eta\in C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$ the result follows by Proposition 2.1. Next consider the general case $\eta\in W^{1,1}(\mathbb{R}^{N},\mathbb{R})$. As before, by the density of $C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R})$ in $W^{1,1}(\mathbb{R}^{N},\mathbb{R})$, there exists a sequence \big{\{}\eta_{n}\big{\}}_{n=1}^{\infty}\subset C^{\infty}_{c}(\mathbb{R}^{N},\mathbb{R}) such that

[TABLE]

Next, as before, define

[TABLE]

Defining $u_{n,\varepsilon}$ as in (2.59) we get by Proposition 2.1, for all $n\geq 1$ (see (2.25)),

[TABLE]

and then

[TABLE]

On the other hand, by Lemma 2.1, for all $n\geq 1$ and every $\varepsilon\in(0,1/e)$ we have

[TABLE]

Thus, by the triangle inequality we get, for every $n\geq 1$ and every $\varepsilon\in(0,1/e)$,

[TABLE]

Then, by (2.67) and (2.64), for all $n\geq 1$ we obtain:

[TABLE]

Letting $n$ go to infinity in (2.68), using (2.65), the definition of $\bar{L}$ in (2.65) and the fact that $\lim_{n\to+\infty}H_{n}=0$, we finally deduce (1.8). ∎

3 Appendix:

Some known results on BV-spaces

In what follows we present some known definitions and results on BV-spaces; some of them were used in the previous sections. We rely mainly on the book [4] by Ambrosio, Fusco and Pallara.

Definition 3.1**.**

Let $\Omega$ be a domain in $\mathbb{R}^{N}$ and let $f\in L^{1}(\Omega,\mathbb{R}^{m})$. We say that $f\in BV(\Omega,\mathbb{R}^{m})$ if the following quantity is finite:

[TABLE]

Definition 3.2**.**

Let $\Omega$ be a domain in $\mathbb{R}^{N}$. Consider a function $f\in L^{1}_{loc}(\Omega,\mathbb{R}^{m})$ and a point $x\in\Omega$.

i) We say that $x$ is an approximate continuity point of $f$ if there exists $z\in\mathbb{R}^{m}$ such that

[TABLE]

In this case we denote $z$ by $\tilde{f}(x)$. The set of approximate continuity points of $f$ is denoted by $G_{f}$.

ii) We say that $x$ is an approximate jump point of $f$ if there exist $a,b\in\mathbb{R}^{m}$ and $\boldsymbol{\nu}\in S^{N-1}$ such that $a\neq b$ and

[TABLE]

where $\chi(a,b,\boldsymbol{\nu})$ is defined by

[TABLE]

The triple $(a,b,\boldsymbol{\nu})$, uniquely determined, up to a permutation of $(a,b)$ and a change of sign of $\boldsymbol{\nu}$, is denoted by $(f^{+}(x),f^{-}(x),\boldsymbol{\nu}_{f}(x))$. We shall call $\boldsymbol{\nu}_{f}(x)$ the approximate jump vector and we shall sometimes write simply $\boldsymbol{\nu}(x)$ if the reference to the function $f$ is clear. The set of approximate jump points is denoted by $J_{f}$. A choice of $\boldsymbol{\nu}(x)$ for every $x\in J_{f}$ determines an orientation of $J_{f}$. At an approximate continuity point $x$, we shall use the convention $f^{+}(x)=f^{-}(x)=\tilde{f}(x)$.

Theorem 3.1** (Theorems 3.69 and 3.78 from [4]).**

*Consider an open set $\Omega\subset\mathbb{R}^{N}$ and $f\in BV(\Omega,\mathbb{R}^{m})$. Then:

i) $\mathcal{H}^{N-1}$-a.e. point in $\Omega\setminus J_{f}$ is a point of approximate continuity of $f$.

ii) The set $J_{f}$ is $\sigma$-$\mathcal{H}^{N-1}$-rectifiable Borel set, oriented by $\boldsymbol{\nu}(x)$. I.e., the set $J_{f}$ is $\mathcal{H}^{N-1}$ $\sigma$-finite, there exist countably many $C^{1}$ hypersurfaces $\{S_{k}\}^{\infty}_{k=1}$ such that \mathcal{H}^{N-1}\Big{(}J_{f}\setminus\bigcup\limits_{k=1}^{\infty}S_{k}\Big{)}=0, and for $\mathcal{H}^{N-1}$-a.e. $x\in J_{f}\cap S_{k}$, the approximate jump vector $\boldsymbol{\nu}(x)$ is normal to $S_{k}$ at the point $x$.

iii) \big{[}(f^{+}-f^{-})\otimes\boldsymbol{\nu}_{f}\big{]}(x)\in L^{1}(J_{f},d\mathcal{H}^{N-1}).*

Theorem 3.2** (Theorems 3.92 and 3.78 from [4]).**

Consider an open set $\Omega\subset\mathbb{R}^{N}$ and $f\in BV(\Omega,\mathbb{R}^{m})$. Then, the distributional gradient $Df$ can be decomposed as a sum of two Borel regular finite matrix-valued measures $\mu_{f}$ and $D^{j}f$ on $\Omega$,

[TABLE]

where

[TABLE]

is called the jump part of $Df$ and

[TABLE]

is a sum of the absolutely continuous and the Cantor parts of $Df$. The two parts $\mu_{f}$ and $D^{j}f$ are mutually singular to each other. Moreover, $\mu_{f}(B)=0$ for any Borel set $B\subset\Omega$ which is $\mathcal{H}^{N-1}$ $\sigma$-finite.