Sobolev regularity for convex functionals on BD

Abstract

We establish the first Sobolev regularity and uniqueness results for minimisers of autonomous, convex variational integrals of linear growth which depend on the symmetric rather than the full gradient. This extends the results available in the literature for the BV-setting to the case of functionals whose full gradients are a priori not known to exist as finite matrix-valued Radon measures.

Appendix

1.1 Extensions of Theorem 1.2 to nonautonomous problems

Let us now briefly comment on the situation where f has additional x-dependence. If $f\in {\text {C}}^{2}({\overline{\Omega }}\times {\mathbb {R}}_{{\text {sym}}}^{n\times n})$ is an integrand that satisfies essentially the assumptions of [9, Ass. 4.22], that is, f satisfies (1.2) uniformly in x together with

$$\begin{aligned} {\left\{ \begin{array}{ll} \sup _{x\in {\overline{\Omega }}}\sup _{\xi \in {\mathbb {R}}_{{\text {sym}}}^{n\times n}}\max \{|{\text {D}}_{\xi }f(x,\xi )|,|{\text {D}}_{x}^{2}{\text {D}}_{\xi }f(x,\xi )|,|{\text {D}}_{x}{\text {D}}_{\xi }f(x,\xi )|\}<\infty ,\\ \lambda \frac{|\xi |^{2}}{(1+|z|^{2})^{\mu /2}}\le \langle {\text {D}}_{\xi }^{2}f(z)\xi ,\xi \rangle \le \Lambda \frac{|\xi |^{2}}{(1+|z|^{2})^{1/2}},\\ |\langle {\text {D}}_{x}{\text {D}}_{\xi }^{2}f(x,\xi )\eta ,\eta '\rangle |\le C(|\langle {\text {D}}_{\xi }^{2}f(x,\xi )\eta ,\eta '\rangle | + |\eta |\,|\eta '|/(1+|\xi |^{2})) \end{array}\right. } \end{aligned}$$

(4.1)

for all $x\in {\overline{\Omega }},\eta ,\eta ',\xi ,z\in {\mathbb {R}}_{{\text {sym}}}^{n\times n}$, then the results of this paper carry over in a straightforward manner to the situation of interest; in fact, as we work with finite differences, these assumptions can even be weakened, but this is left to the interested reader; also see the discussion in [8, App. C]. If the smoothness of the x-dependence is diminished, a merger of the arguments outlined in this work with [37] leads to the correspondingly modified theorems.

1.2 Proofs of auxiliary results

In this section, we provide the proofs of minor auxiliary results used in the main body of the paper. We begin with the

Proof of Proposition 2.2

Since the embedding we aim for is of local nature, it suffices to give estimates for maps $u\in {\text {C}}_{c}^{\infty }({\mathbb {R}}^{n};{\mathbb {R}}^{n})$ which are compactly supported in a given ball ${\text {B}}={\text {B}}(x_{0},R)$; the full statement then follows by smooth approximation and is left to the reader. Let $0<\alpha <1$ be given. We have for all $0<s<1$

$$\begin{aligned}{}[u]_{{\text {W}}^{s,\frac{n}{n-1+s}}({\mathbb {R}}^{n};{\mathbb {R}}^{n})}&\le C(n,s)\Vert \varvec{\varepsilon }(u)\Vert _{{\text {L}}^{1}({\mathbb {R}}^{n};{\mathbb {R}}_{{\text {sym}}}^{n\times n})}, \end{aligned}$$

since the symmetric gradient operator is elliptic and cancelling, cf. [52, Thm. 8.1, Prop. 6.4]. Now, for all $h\in {\mathbb {R}}^{n}$ with $|h|<1$ sufficiently small, $u(x+h)$ is supported in ${\text {B}}(x_{0},2R)$. Then, by Hölder’s inequality and some $\alpha<t<1$,

$$\begin{aligned} \int _{{\mathbb {R}}^{n}}|u(x+h)-u(x)|{\text {d}}x&\le c(R,n,t)\Big (\int _{{\text {B}}(x_{0},2R)}|u(x+h)-u(x)|^{\frac{n}{n-1+t}}{\text {d}}x\Big )^{\frac{n-1+t}{n}} \\&\le C(R,n,t)\Vert u\Vert _{{\text {B}}_{\frac{n}{n-1+t},\infty }^{t}({\mathbb {R}}^{n};{\mathbb {R}}^{n})}|h|^{t}\\&\le C(R,n,t)\Vert u\Vert _{{\text {W}}^{t,\frac{n}{n-1+t}}({\mathbb {R}}^{n};{\mathbb {R}}^{n})}|h|^{t} \\&\le C(R,n,t)\Vert \varvec{\varepsilon }(u)\Vert _{{\text {L}}^{1}({\mathbb {R}}^{n};{\mathbb {R}}_{{\text {sym}}}^{n\times n})}|h|^{t}. \end{aligned}$$

But since $t>\alpha$ and $|h|<1$, we obtain $u\in {\text {B}}_{1,\infty }^{t}({\mathbb {R}}^{n};{\mathbb {R}}^{n})$ and standard inclusions between Besov and Sobolev-Slobodeckjiĭ spaces imply that $[u]_{{\text {W}}^{\alpha ,1}({\mathbb {R}}^{n};{\mathbb {R}}^{n})}\le C \Vert \varvec{\varepsilon }(u)\Vert _{{\text {L}}^{1}({\mathbb {R}}^{n};{\mathbb {R}}_{{\text {sym}}}^{n\times n})}$ and the proof is complete. $\square$

Proof of Lemma 2.4

By [1, Lemma 2.2], for every $-\frac{1}{2}< \gamma <0$ and $\mu \ge 0$ there exists a constant $c=c(M)>0$ such that for all $\xi ,\eta \in {\mathbb {R}}^{M}$ there holds

$$\begin{aligned} (2\gamma +1)|\xi -\eta |\le \frac{|(\mu ^{2}+|\xi |^{2})^{\gamma }\xi -(\mu ^{2}+|\eta |^{2})^{\gamma }\eta |}{(\mu ^{2}+|\xi |^{2}+|\eta |^{2})^{\gamma }}\le \frac{c(M)}{2\gamma +1}|\xi -\eta |. \end{aligned}$$

Applying this with $\mu =1$ and $\gamma =(1-\alpha )/2$ yields the claim as $-\frac{1}{2}<\gamma <0$ if and only if $1<\alpha <2$.

Now let $\xi \in {\mathbb {R}}^{M}$ with $|\xi |\ge 1$ and let $1<\alpha <2$. Then $(1-\alpha )/2<0$. Hence, since $t\mapsto t^{(1-\alpha )/2}$ is monotonically decreasing on $(0,\infty )$,

$$\begin{aligned} |\xi | \ge 1&\Rightarrow 2|\xi |^{2} \ge 1+|\xi |^{2} \Rightarrow 2^{\frac{1-\alpha }{2}}|\xi |^{1-\alpha }\le (1+|\xi |^{2})^{\frac{1-\alpha }{2}}\\&\Rightarrow |\xi |^{2-\alpha }\le 2^{\frac{\alpha -1}{2}}|V_{\alpha }(\xi )| {\mathop {\le }\limits ^{1<\alpha <2}}\sqrt{2}|V_{\alpha }(\xi )|. \end{aligned}$$

Since $1<\alpha <2$ and $|\xi |\ge 1$, we have $|\xi |^{2-\alpha }\le |\xi |$ and thus $\min \{|\xi |,|\xi |^{2-\alpha }\}\le \sqrt{2}|V_{\alpha }(\xi )|$ in this case. Now, if $|\xi |<1$, then

$$\begin{aligned} 2^{\frac{1-\alpha }{2}}\le (1+|\xi |^{2})^{\frac{1-\alpha }{2}}\Rightarrow 2^{\frac{1-\alpha }{2}}|\xi |\le |V_{\alpha }(\xi )|\Rightarrow |\xi |\le \sqrt{2}|V_{\alpha }(\xi )|, \end{aligned}$$

and hence $\min \{|\xi |,|\xi |^{2-\alpha }\}\le \sqrt{2}|V_{\alpha }(\xi )|$ holds, too. Therefore $\min \{|\xi |,|\xi |^{2-\alpha }\}\le \sqrt{2}|V_{\alpha }(\xi )|$ holds for all $\xi \in {\mathbb {R}}^{M}$. Lastly if the measurable map $u:\Omega \rightarrow {\mathbb {R}}^{M}$ is such that $V_{\alpha }(u)\in {\text {L}}^{p}(\Omega ;{\mathbb {R}}^{m})$, then we have

$$\begin{aligned} \int _{\Omega }|u|^{(2-\alpha )p}{\text {d}}x&= \int _{\Omega \cap \{|u|\le 1\}}|u|^{(2-\alpha )p}{\text {d}}x + \int _{\Omega \cap \{|u|> 1\}}|u|^{(2-\alpha )p}{\text {d}}x \\&\le {\mathscr {L}}^{n}(\Omega )+c(p)\int _{\Omega }|V_{\alpha }(u)|^{p}{\text {d}}x \end{aligned}$$

The proof is complete. $\square$

1.3 Relaxation

As mentioned in the introduction, we now give justification for some results used in the main part of the paper. The primary aim of this section is to explain formula (1.6) and the existence of generalised minima. We thus recap the requisite foundational theory of functions of measures as exposed in [3, Section 2.6], see also [4, 21].

1.3.1 Convex functions of measures

Given $m\in {\mathbb {N}}$, let $f:{\mathbb {R}}^{m}\rightarrow {\mathbb {R}}_{\ge 0}{\mathbb {R}}$ be a convex function of linear growth, i.e., f satisfies (1.2) with the obvious modifications. In this situation, it can be shown that $f^{\infty }$ defined by (1.5) is well-defined, convex and positively 1-homogeneous. Let $\mu$ be an ${\mathbb {R}}^{m}$-valued Radon measure of finite total variation on an open and bounded set $\Omega \subset {\mathbb {R}}^{n}$. We denote

$$\begin{aligned} \mu =\mu ^{a}+\mu ^{s}=\frac{{\text {d}}\mu }{{\text {d}}{\mathscr {L}}^{n}}{\mathscr {L}}^{n}+ \frac{{\text {d}}\mu }{{\text {d}}|\mu ^{s}|}|\mu ^{s}| \end{aligned}$$

its Radon–Nikodým decomposition into its absolutely continuous and singular parts $\mu ^{a},\mu ^{s}$ with respect to Lebesgue measure, where $|\mu ^{s}|$ denotes the total variation measure of $\mu ^{s}$. We then define a new Radon measure $f[\mu ]$ by

$$\begin{aligned} f[\mu ](A):=\int _{A}f\left( \frac{{\text {d}}\mu }{{\text {d}}{\mathscr {L}}^{n}}\right) {\text {d}}{\mathscr {L}}^{n}+\int _{A}f^{\infty }\left( \frac{{\text {d}}\mu }{{\text {d}}|\mu ^{s}|}\right) {\text {d}}|\mu ^{s}|, \qquad A\in {\mathscr {B}}(\Omega ), \end{aligned}$$

(4.2)

where ${\mathscr {B}}(\Omega )$ denotes the Borel-$\sigma$-algebra on $\Omega$. We note that, by positive 1-homogeneity of $f^{\infty }$, this gives rise to a well-defined measure indeed. Linking this to the area functional as required for the definition of area-strict convergence, for a given map $u\in {\text {BD}}(\Omega )$, we have with $f:=\sqrt{1+|\cdot |^{2}}$ that $\sqrt{1+|{\text {E}}u|^{2}}(\Omega ):=f[{\text {E}}u](\Omega )$.

We turn to formula (1.6) for the relaxed functional as given for ${\text {BV}}$-functions in [31] and find by a straightforward applications of the results of Goffman and Serrin [32] that, given an open and bounded Lipschitz subset $\Omega$ of ${\mathbb {R}}^{n}$ together with a Dirichlet datum $u_{0}\in {\text {LD}}(\Omega )$, we have

$$\begin{aligned} \overline{{\mathfrak {F}}}_{u_{0}}[u] = \inf \left\{ \liminf _{k\rightarrow \infty }{\mathfrak {F}}[u_{k}]:\; \begin{array}{c} (u_{k})\subset {\mathscr {D}}_{u_{0}}:=u_{0}+{\text {LD}}_{0}(\Omega ) \\ u_{k}\rightarrow u\;\text {in}\;{\text {L}}^{1}(\Omega ;{\mathbb {R}}^{n}) \end{array}\right\} . \end{aligned}$$

(4.3)

We pick a ball ${\text {B}}={\text {B}}(z,R)$ with $\Omega \Subset {\text {B}}$. By surjectivity of the trace operator ${\text {Tr}}:{\text {LD}}(U)\rightarrow {\text {L}}_{{\mathcal {H}}^{n-1}}^{1}(\partial U;{\mathbb {R}}^{n})$ on bounded Lipschitz subsets U of ${\mathbb {R}}^{n}$ (see Sect. 2.2) that there exists $v_{0}\in {\text {LD}}({\text {B}}{\setminus }{\overline{\Omega }})$ such that ${\text {Tr}}(v_{0})|_{\partial {\text {B}}}=0$ and ${\text {Tr}}(v_{0})|_{\partial \Omega }={\text {Tr}}(u_{0})|_{\partial \Omega }$ ${\mathcal {H}}^{n-1}$—a.e. on $\partial {\text {B}}$ or $\partial \Omega$, respectively.

Given $u\in {\text {BD}}(\Omega )$, we put

$$\begin{aligned} {\widetilde{u}}(x):={\left\{ \begin{array}{ll} u(x)&{}\;\text {for}\;x\in \Omega ,\\ v_{0}(x)&{}\;\text {for}\;x\in {\text {B}}{\setminus }{\overline{\Omega }}. \end{array}\right. } \end{aligned}$$

(4.4)

Then there holds ${\widetilde{u}}\in {\text {BD}}({\text {B}})$, and by the interior trace theorem as recalled in Sect. 2.2 we have

We insert this expression for $\mu ={\text {E}}{\widetilde{u}}$ with $A={\text {B}}$ into (4.2) (which equally holds for measures $\mu$ on ${\mathbb {R}}^{n}$ subject to the obvious modifications) and obtain

$$\begin{aligned} f[{\text {E}}{\widetilde{u}}]({\text {B}})&= \int _{\Omega }f({\mathscr {E}}u){\text {d}}{\mathscr {L}}^{n}+\int _{\Omega }f\left( \frac{{\text {d}}{\text {E}}u}{{\text {d}}|{\text {E}}^{s}u|}\right) {\text {d}}|{\text {E}}^{s}u| \nonumber \\&+\quad \int _{\partial \Omega }f^{\infty }\left( {\text {Tr}}(v_{0}-u) \odot \nu _{\partial \Omega }\right) {\text {d}}{\mathcal {H}}^{n-1} + \int _{{\text {B}}{\setminus }{\overline{\Omega }}}f({\mathscr {E}}v_{0}){\text {d}}{\mathscr {L}}^{n}. \end{aligned}$$

(4.5)

If we then aim for minimising $f[{\text {E}}{\widetilde{u}}]({\text {B}})$ over all $u\in {\text {BD}}(\Omega )$, we see by constancy of the very last term of the preceding expression that it does not affect the minimiser $v\in {\text {BD}}(\Omega )$, and thus a function $v\in {\text {BD}}(\Omega )$ minimises $f[{\text {E}}{\widetilde{u}}]({\text {B}})$ if and only if it minimises the relaxed functional given by (1.6).

We conclude this section by recalling two results due to Reshetnyak concerning the (lower semi-)continuity of convex functions of measures in the version as given in [8].

Proposition 4.1

(Reshetnyak, [39]) Let $m\in {\mathbb {N}}$, $\Omega \subset {\mathbb {R}}^{n}$ open and let $(\mu _{k})$ be a sequence of ${\mathbb {R}}^{m}$-valued Radon measures of finite total variation which converges to a ${\mathbb {R}}^{m}$-valued Radon measure of finite total variation $\mu$ on $\Omega$ in the weak*-sense. Moreover, assume that all measures $\mu _{k}$ and $\mu$ take values in some closed convex cone $K\subset {\mathbb {R}}^{m}$. Then the following holds:

• (a) Lower Semicontinuity. If ${\widetilde{f}}:K \rightarrow [0,\infty ]$ is a lower semicontinuous function, then there holds

  $$\begin{aligned} \int _{\Omega }{\widetilde{f}}\left( \frac{{\text {d}}\mu }{{\text {d}}|\mu |}\right) {\text {d}}|\mu | \le \liminf _{k\rightarrow \infty }\int _{\Omega }{\widetilde{f}}\left( \frac{{\text {d}}\mu _{k}}{{\text {d}}|\mu _{k}|}\right) {\text {d}}|\mu _{k}|. \end{aligned}$$

• (b) If $\mu _{k}\rightarrow \mu$ strictly³ as $k\rightarrow \infty$ and ${\widetilde{f}}:K \rightarrow [0,\infty )$ is a continuous and 1-homogeneous function, then there holds

  $$\begin{aligned} \int _{\Omega }{\widetilde{f}}\left( \frac{{\text {d}}\mu }{{\text {d}}|\mu |}\right) {\text {d}}|\mu | = \lim _{k\rightarrow \infty }\int _{\Omega }{\widetilde{f}}\left( \frac{{\text {d}}\mu _{k}}{{\text {d}}|\mu _{k}|}\right) {\text {d}}|\mu _{k}|. \end{aligned}$$

1.3.2 Generalised minima: existence and characterisations

We now pass on to the verification of (4.3) and establish the existence of generalised minima.

Proposition 4.2

Let $\Omega$ be an open and bounded Lipschitz subset of ${\mathbb {R}}^{n}$. Given a convex integrand $f:{\mathbb {R}}_{{\text {sym}}}^{n\times n}\rightarrow {\mathbb {R}}$ with (1.2) and a boundary datum $u_{0}\in {\text {LD}}(\Omega )$, define $\overline{{\mathfrak {F}}}_{u_{0}}$ by (1.6). Then there exists a generalised minimiser of ${\mathfrak {F}}$ in the sense of (1.7).

Moreover, the following are equivalent for $u\in {\text {BD}}(\Omega )$:

• (a) u is a generalised minimiser in the sense of (1.7).

• (b) u is the weak*-limit of an ${\mathfrak {F}}$-minimising sequence $(u_{k})\subset {\mathscr {D}}_{u_{0}}(:= u_{0}+{\text {LD}}_{0}(\Omega ))$.

• (c) u is the strong ${\text {L}}^{1}$-limit of an ${\mathfrak {F}}$-minimising sequence $(u_{k})\subset {\mathscr {D}}_{u_{0}}$.

Proof

We begin with a preparatory remark. We choose an open and bounded Lipschitz subset ${\widetilde{\Omega }}\subset {\mathbb {R}}^{n}$ with $\Omega \Subset {\widetilde{\Omega }}$. Given $u_{0}\in {\text {LD}}(\Omega )$, by surjectivity of the trace operator on ${\text {LD}}$ (see Sect. 2.2), we may extend $u_{0}$ by some $v_{0}\in {\text {LD}}({\text {B}}{\setminus }{\overline{\Omega }})$ to a function $\widetilde{u_{0}}\in {\text {LD}}_{0}({\widetilde{\Omega }})$. We now invoke the straightforward generalisation of [9, Chpt. 2.3.1] whose proof we leave to the interested reader:

Given ${\widetilde{\Omega }}$ and $u_{0}$ as above, let $u\in {\text {BD}}(\Omega )$ and denote its extension to ${\widetilde{\Omega }}$ via $\widetilde{u_{0}}$ by ${\widetilde{u}}$. Then there exists $(u_{k})\subset u_{0}+{\text {C}}_{c}^{\infty }(\Omega ;{\mathbb {R}}^{n})$ such that $\widetilde{u_{k}}\rightarrow {\widetilde{u}}$ area-strictly in ${\widetilde{\Omega }}$ as $k\rightarrow \infty$, where $\widetilde{u_{k}},{\widetilde{u}}$ denote the extensions of $u_{k},u$ to ${\widetilde{\Omega }}$ by ${\widetilde{u}}$, respectively.

We turn to the actual proof, and choose ${\widetilde{\Omega }}\equiv {\text {B}}$ as above before (4.5).

Step 1. Existence of a generalised minimiser. By (1.2) we have $m:=\inf _{u\in {\text {BD}}(\Omega )}f[{\text {E}}{\widetilde{u}}]({\text {B}})>-\infty$ and so there exists a sequence $(u_{k})\subset {\text {BD}}(\Omega )$ and $v\in {\text {BD}}({\text {B}})$ such that $\widetilde{u_{k}}{\mathop {\rightharpoonup }\limits ^{*}}v$ in ${\text {BD}}({\text {B}})$ as $k\rightarrow \infty$. By Proposition 4.1(a), $f[{\text {E}}v]({\text {B}})\le \liminf _{k\rightarrow \infty }f[{\text {E}}\widetilde{u_{k}}]({\text {B}})=m$. Since $\widetilde{u_{k}}|_{{\text {B}}{\setminus }{\overline{\Omega }}}=v_{0}$, $v|_{{\text {B}}{\setminus }{\overline{\Omega }}}=v_{0}$ and so we conclude from (4.5) that $u:=v|_{\Omega }$ is a generalised minimiser in the sense of (1.7). Now, since ${\mathscr {D}}_{u_{0}}\subset {\text {BD}}(\Omega )$ and $\overline{{\mathfrak {F}}}_{u_{0}}|_{{\mathscr {D}}_{u_{0}}}={\mathfrak {F}}|_{{\mathscr {D}}_{u_{0}}}$, we have $\inf _{{\text {BD}}(\Omega )}\overline{{\mathfrak {F}}}_{u_{0}}\le \inf _{{\mathscr {D}}_{u_{0}}}{\mathfrak {F}}$. On the other hand, let $u\in {\text {GM}}({\mathfrak {F}};u_{0})$ and apply the above area-strict approximation strategy to obtain a sequence $(u_{k})\subset u_{0}+{\text {C}}_{c}^{\infty }(\Omega ;{\mathbb {R}}^{n})$ such that $\widetilde{u_{k}}\rightarrow {\widetilde{u}}$ area-strictly as $k\rightarrow \infty$. Then we obtain by (4.1)—as the ultimate term on the right side of (4.5) is constant—

$$\begin{aligned} \overline{{\mathfrak {F}}}_{u_{0}}[u]&= f[{\text {E}}{\widetilde{u}}]({\text {B}})-\int _{{\text {B}}{\setminus }{\overline{\Omega }}}f({ \mathscr {E}}v_{0}){\text {d}}x \nonumber \\&= \lim _{k\rightarrow \infty }f[{\text {E}}\widetilde{u_{k}}]({\text {B}})- \int _{{\text {B}}{\setminus }{\overline{\Omega }}}f({\mathscr {E}}v_{0}){\text {d}}x = \lim _{k\rightarrow \infty }{\mathfrak {F}}[u_{k}] \ge \inf _{{\mathscr {D}}_{u_{0}}}{\mathfrak {F}}. \end{aligned}$$

(4.6)

Altogether, we have therefore established the absence of gaps, i.e.,

$$\begin{aligned} \min _{{\text {BD}}(\Omega )}\overline{{\mathfrak {F}}}_{u_{0}}=\inf _{ {\text {BD}}(\Omega )}\overline{{\mathfrak {F}}}_{u_{0}}=\inf _{{\mathscr {D}}_{u_{0}}}{ \mathfrak {F}}. \end{aligned}$$

(4.7)

Step 2. Proof of the claimed equivalences. Ad (a)$\Rightarrow$(b) and (a)$\Rightarrow$(c). Let $u\in {\text {GM}}({\mathfrak {F}};u_{0})$ and choose an area-strictly approximating sequence $(u_{k})\subset u_{0}+{\text {C}}_{c}^{\infty }(\Omega ;{\mathbb {R}}^{n})$ as indicated. Then, employing formula (4.5) with ${\widetilde{\Omega }}\equiv {\text {B}}$, we obtain $f[{\text {E}}\widetilde{u_{k}}]({\text {B}})\rightarrow f[{\text {E}}{\widetilde{u}}]({\text {B}})$ by virtue of Proposition 4.1. By constancy of the ultimate term in (4.5) and the fact that area-strict convergence implies both ${\text {L}}^{1}$- and weak*-convergence, we conclude by means of (4.7). Ad (b)$\Rightarrow$(c). This follows trivially as weak*-convergence implies strong ${\text {L}}^{1}$-convergence. Ad (c)$\Rightarrow$(a). Let $(u_{k})\subset {\mathscr {D}}_{u_{0}}$ be an ${\mathfrak {F}}$-minimising sequence. By (4.3), we obtain for all $v\in {\text {BD}}(\Omega )$

$$\begin{aligned} \overline{{\mathfrak {F}}}_{u_{0}}[u] \le \liminf _{k\rightarrow \infty }{\mathfrak {F}}[u_{k}] = \inf _{{\mathscr {D}}_{u_{0}}}{\mathfrak {F}} {\mathop {=}\limits ^{(4.7)}} \min _{{\text {BD}}(\Omega )}\overline{{\mathfrak {F}}}_{u_{0}}. \end{aligned}$$

(4.8)

The proof is complete. $\square$