The problems that we want to study with ultrafunctions have the following form: minimize a given functional *J* on $V(\mathrm{\Omega})$ subjected to certain restrictions (e.g., some boundary constrictions, or a minimization on a proper vector subspace of $V(\mathrm{\Omega})$). This kind of problems can be studied in ultrafunctions theory by means of a modification of the Faedo–Galerkin method, based on standard approximations by finite-dimensional spaces. The following is a (maybe even too) general formulation of this idea.

#### Theorem 4.1.

*Let $W\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}\mathrm{\ne}\mathrm{\varnothing}$ be a vector subspace of $V\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$. Let*

$\begin{array}{cc}\hfill \mathcal{\mathcal{F}}=\{f:V(\mathrm{\Omega})\to \mathbb{R}\mid & \mathit{\text{for all}}E\mathit{\text{finite-dimensional vector subspaces of}}W(\mathrm{\Omega}),\hfill \\ & \mathit{\text{there exists}}u\in Ef(u)=\underset{v\in E}{\mathrm{min}}f(v)\}.\hfill \end{array}$

*Then every $F\mathrm{\in}{\mathcal{F}}^{\mathrm{*}}$ has a minimizer in ${W}_{\mathrm{\Lambda}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$.*

#### Proof.

Let $F={lim}_{\lambda \uparrow \mathrm{\Lambda}}{f}_{\lambda}$, with ${f}_{\lambda}\in \mathcal{\mathcal{F}}$ for every $\lambda \in \U0001d50f$. By hypothesis, for every $\lambda \in \U0001d50f$, there exists

${u}_{\lambda}\in {W}_{\lambda}:-\mathrm{Span}(W\cap \lambda )$

that minimizes ${f}_{\lambda}$ on ${W}_{\lambda}$. Then $u={lim}_{\lambda \uparrow \mathrm{\Lambda}}{u}_{\lambda}$ minimizes *F* on ${lim}_{\lambda \uparrow \mathrm{\Lambda}}{W}_{\lambda}={W}_{\mathrm{\Lambda}}$ as, if $v={lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}\in {W}_{\lambda}(\mathrm{\Omega})$, then, for every $\lambda \in \U0001d50f$, we have that ${f}_{\lambda}({v}_{\lambda})\le {f}_{\lambda}({u}_{\lambda})$, hence,

$F(v)=\underset{\lambda \uparrow \mathrm{\Lambda}}{lim}{f}_{\lambda}({v}_{\lambda})\le \underset{\lambda \uparrow \mathrm{\Lambda}}{lim}{f}_{\lambda}({u}_{\lambda})=F(u).\mathit{\u220e}$

For applications, the following particular case of Theorem 4.1 is particularly relevant.

#### Corollary 4.2.

*Let $f\mathit{}\mathrm{(}\xi \mathrm{,}u\mathrm{,}x\mathrm{)}$ be coercive in ξ on every finite-dimensional subspace of $V\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ and for every $x\mathrm{\in}\mathrm{\Omega}$. Let $F\mathit{}\mathrm{(}u\mathrm{)}\mathrm{:-}\mathrm{\u2a16}f\mathit{}\mathrm{(}\mathrm{\nabla}\mathit{}u\mathrm{,}u\mathrm{,}x\mathrm{)}\mathit{}d\mathit{}x$. Then ${F}^{\mathrm{\circ}}$ has a minimum on ${V}_{\mathrm{\Lambda}}$.*

#### Proof.

Just notice that $F\in \mathcal{\mathcal{F}}$, in the notations of Theorem 4.1.
∎

Theorem 4.1 provides a general existence result. However, such a general result poses two questions: the first is how wild such generalized solutions can be; the second is if this method produces new generalized solutions for problems that already have classical ones.

The answer to these questions depends on the problem that is studied. However, regarding the second question, we have the following result, which strengthens Theorem 4.1:

#### Theorem 4.3.

*Let $F\mathrm{:}{V}_{\mathrm{\Lambda}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}\mathrm{\to}{\mathrm{R}}^{\mathrm{*}}$, $F\mathrm{=}{\mathrm{lim}}_{\lambda \mathrm{\uparrow}\mathrm{\Lambda}}\mathit{}{F}_{\lambda}$. For every $\lambda \mathrm{\in}\mathrm{L}$, let*

${M}_{\lambda}:-\{u\in {V}_{\lambda}(\mathrm{\Omega}):{F}_{\lambda}(u)=\underset{v\in {V}_{\lambda}}{\mathrm{min}}{F}_{\lambda}(v)\}.$

*Assume that ${\mathrm{lim}}_{\lambda \mathrm{\uparrow}\mathrm{\Lambda}}\mathit{}{M}_{\lambda}\mathrm{\ne}\mathrm{\varnothing}$. Then*

${M}_{\mathrm{\Lambda}}:-\{u\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega}):F(u)=\underset{v\in {V}_{\mathrm{\Lambda}}}{\mathrm{min}}F(v)\}=\underset{\lambda \uparrow \mathrm{\Lambda}}{lim}{M}_{\lambda}\ne \mathrm{\varnothing}.$

#### Proof.

${M}_{\mathrm{\Lambda}}\subseteq {lim}_{\lambda \uparrow \mathrm{\Lambda}}{M}_{\lambda}$:
Let $v={lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}\in {M}_{\mathrm{\Lambda}}$, and let $u={lim}_{\lambda \uparrow \mathrm{\Lambda}}{u}_{\lambda}\in {lim}_{\lambda \uparrow \mathrm{\Lambda}}{M}_{\lambda}$. As $F(v)\le F(u)$, there is a qualified set *Q* such that, for every $\lambda \in Q$, ${F}_{\lambda}({v}_{\lambda})\le F({u}_{\lambda})$. But then ${v}_{\lambda}\in {M}_{\lambda}$, for every $\lambda \in Q$, hence $v={lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}\in {lim}_{\lambda \uparrow \mathrm{\Lambda}}{M}_{\lambda}$.

${M}_{\mathrm{\Lambda}}\supseteq {lim}_{\lambda \uparrow \mathrm{\Lambda}}{M}_{\lambda}$:
Let $u={lim}_{\lambda \uparrow \mathrm{\Lambda}}{u}_{\lambda}\in {lim}_{\lambda \uparrow \mathrm{\Lambda}}{M}_{\lambda}$. Let $v={lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$. Let

$Q=\{\lambda \in \U0001d50f:{u}_{\lambda}\in {M}_{\lambda}\}.$

Then *Q* is qualified and, for every $\lambda \in Q$, ${F}_{\lambda}({u}_{\lambda})\le {F}_{\lambda}({v}_{\lambda})$. Therefore $F(u)\le F(v)$, and so $u\in {M}_{\mathrm{\Lambda}}$.
∎

The following easy consequences of Theorem 4.3 hold:

#### Corollary 4.4.

*In the same notations of Theorem 4.3, let us now assume that there exists $k\mathrm{\in}\mathrm{N}$ such that $\mathrm{|}{M}_{\lambda}\mathrm{|}\mathrm{\le}k$ for every $\lambda \mathrm{\in}\mathrm{L}$. Then $\mathrm{|}{M}_{\mathrm{\Lambda}}\mathrm{|}\mathrm{\le}k$.*

#### Proof.

This holds, as the hypothesis on $|{M}_{\lambda}|$ trivially entails that $|{lim}_{\lambda \uparrow \mathrm{\Lambda}}{M}_{\lambda}|\le k$.
∎

#### Corollary 4.5.

*In the same notations of Theorem 4.3, let us now assume that $F\mathrm{=}{J}^{\mathrm{*}}$, where $J\mathrm{:}V\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}\mathrm{\to}\mathrm{R}$. Let
*

$M:-\{v\in V(\mathrm{\Omega}):v=\underset{w\in V(\mathrm{\Omega})}{\mathrm{min}}J(w)\}.$

*Assume that $M\mathrm{\ne}\mathrm{\varnothing}$. Then the following facts are equivalent:*

*In particular, if $u\mathrm{\in}M$, then ${u}^{\mathrm{*}}$ minimizes **F*.

#### Proof.

$(1)\Rightarrow (2)$
Let $u\in M$. Let $Q(u):-\{\lambda \in \U0001d50f:u\in \lambda \}$. Then, for every $\lambda \in Q(u)$,

$v\in {M}_{\lambda}\iff J(v)=J(u)\u27f9v\in M,$

hence ${M}_{\lambda}\subseteq M$ for every $\lambda \in Q(u)$, which is qualified, and so ${lim}_{\lambda \uparrow \mathrm{\Lambda}}{M}_{\lambda}\subseteq {M}^{*}\cap {V}_{\mathrm{\Lambda}}$, and we conclude by Theorem 4.3.

$(2)\Rightarrow (1)$
By definition,

$u\in {M}^{*}\iff F(u)=\underset{v\in {[V(\mathrm{\Omega})]}^{*}}{\mathrm{min}}F(v),$

hence, if $u\in {M}^{*}\cap {V}_{\lambda}(\mathrm{\Omega})$, it trivially holds that *u* minimizes *F*.
∎

#### Corollary 4.6.

*In the same hypotheses and notations of Corollary 4.3, let us assume that $M\mathrm{=}\mathrm{\{}{u}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{u}_{n}\mathrm{\}}$ is finite. Then **v* minimizes *F* in ${V}_{\mathrm{\Lambda}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ if, and only if, there exists $u\mathrm{\in}M$ such that ${u}^{\mathrm{*}}\mathrm{=}v$.

#### Proof.

Just remember that ${S}^{\circ}=\{{s}^{*}:s\in S\}$ for every finite set *S*, and that

${M}^{\sigma}=\{{u}^{*}:u\in M\}\subseteq V(\mathrm{\Omega})\subseteq {V}_{\mathrm{\Lambda}}(\mathrm{\Omega}).\mathit{\u220e}$

In general, one might not have minima, but minimization sequences could still exist. In this case, we have the following result (in which, for every $\rho \in {\mathbb{R}}^{*}$, we set ${\mathrm{st}}_{\mathbb{R}}(\rho )=-\mathrm{\infty}$ if, and only if, ρ is a negative infinite number). Notice that in the following result we are not assuming the continuity of *J* with respect to any topology on $V(\mathrm{\Omega})$, in general.

#### Theorem 4.7.

*Let $V\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ be a Banach space, let $J\mathrm{:}V\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}\mathrm{\to}\mathrm{R}$ and let ${\mathrm{inf}}_{u\mathrm{\in}V\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}}\mathit{}J\mathit{}\mathrm{(}u\mathrm{)}\mathrm{=}m\mathrm{\in}\mathrm{R}\mathrm{\cup}\mathrm{\{}\mathrm{-}\mathrm{\infty}\mathrm{\}}$. The following facts hold:
*

(1)

${J}^{*}(v)\ge m$
* for every *
$v\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$.

(2)

*There exists *
$v\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$
* such that *
${\mathrm{st}}_{\mathbb{R}}({J}^{*}(v))=m$.

(3)

*If *
$v\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$
* is a minimum of *
${J}^{*}:{V}_{\mathrm{\Lambda}}(\mathrm{\Omega})\to {\mathbb{R}}^{*}$
* then *
${J}^{*}(v)\ge {\mathrm{st}}_{\mathbb{R}}({J}^{*}(v))=m$.

(4)

*Let *
${\{{u}_{n}\}}_{n\in \mathbb{N}}$
* be a minimizing sequence that converges to *
$u\in V(\mathrm{\Omega})$
* in some topology *
τ
*. Then there exists *
$v\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$
* such that *
${\mathrm{st}}_{\tau}(v)=u$
* and *
${J}^{*}(v)\ge {\mathrm{st}}_{\mathbb{R}}({J}^{*}(v))=m$
*. Moreover, if *
${w}^{\circ}+\psi $
* is the canonical splitting of *
*v*
*, then*

–

*if *
τ
* is the topology of pointwise convergence, then *
$w=v$
* and *
$w(x)=u(x)$
* for every *
$x\in \mathrm{\Omega}$
*;*

–

*if *
τ
* is the topology of pointwise convergence a.e., then *
$w=v$
* and *
$w(x)=u(x)$
* a.e. in *
$x\in \mathrm{\Omega}$
*;*

–

*if *
τ
* is the topology of weak convergence, then *
$w(x)=u(x)$
* for every *
$x\in \mathrm{\Omega}$
* and *
${\u3008\psi ,{\phi}^{*}\u3009}^{*}\sim 0$
* for every *
φ
* in the dual of *
$V(\mathrm{\Omega})$
*;*

–

*if *
τ
* is the topology associated with a norm *
$\parallel \cdot \parallel $
* and, moreover, *
${\{{u}_{n}\}}_{n}$
* converges pointwise to *
*u*
*, then *
$w=u$
* and *
${\parallel \psi \parallel}^{*}\sim 0$.

(5)

*If all minimizing sequences of *
*J*
* converge to *
$u\in V(\mathrm{\Omega})$
* in some topology *
τ
* and *
*v*
* is a minimum of the functional *
${J}^{*}:{V}_{\mathrm{\Lambda}}(\mathrm{\Omega})\to {\mathbb{R}}^{*}$
*, then *
${\mathrm{st}}_{\tau}(v)=u$
* and *
${J}^{*}(v)\ge {\mathrm{st}}_{\mathbb{R}}({J}^{*}(v))=m$.

#### Proof.

(1) Let $v={lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}$. Since $m={inf}_{u\in V(\mathrm{\Omega})}J(u)$, we have that $J({v}_{\lambda})\ge m$ for every $\lambda \in \mathrm{\Lambda}$, hence ${J}^{*}(v)\ge m$.

(2) By (1) it suffices to show that ${\mathrm{st}}_{\mathbb{R}}({J}^{*}(v))=m$. Let ${\{{u}_{n}\}}_{n\in \mathbb{N}}$ be a minimizing sequence for *J*. For every $\lambda \in \U0001d50f$, let ${v}_{\lambda}:-{u}_{|\lambda |}$. Let $v:-{lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}$. We claim that *v* is the desired ultrafunction.

To prove that ${\mathrm{st}}_{\mathbb{R}}({J}^{*}(v))={lim}_{n\to +\mathrm{\infty}}J({u}_{n})=m$, we just have to observe that, by our definition of the net ${\{{v}_{\lambda}\}}_{\lambda}$, it follows that^{4}

$\underset{n\to +\mathrm{\infty}}{lim}J({u}_{n})={\mathrm{st}}_{\mathbb{R}}\left(\underset{\lambda \uparrow \mathrm{\Lambda}}{lim}J({v}_{\lambda})\right),$

and we conclude as ${lim}_{\lambda \uparrow \mathrm{\Lambda}}J({v}_{\lambda})={J}^{*}(v)$ by definition.

(3) Let $v={lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}$, and let $w\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$ be such that ${\mathrm{st}}_{\mathbb{R}}({J}^{*}(w))=m$. Then $m\le {J}^{*}(v)$ by (1), whilst ${\mathrm{st}}_{\mathbb{R}}({J}^{*}(v))\le {\mathrm{st}}_{\mathbb{R}}({J}^{*}(w))=m$. Hence, $\mathrm{st}({J}^{*}(v))=m$, as desired.

(4) Let *v* be given as in point (2). Let us show that ${\mathrm{st}}_{V(\mathrm{\Omega})}(v)=u$; let $A\in \tau $ be an open neighborhood of *u*. As ${\{{u}_{n}\}}_{n}$ converges to *u*, there exists $N>0$ such that, for every $m>N$, ${u}_{n}\in A$. Let $\mu \in \U0001d50f$ be such that $|\mu |>N$. Then, for every

$\lambda \in {Q}_{\mu}:-\{\lambda \in \U0001d50f:\mu \subseteq \lambda \},{v}_{\lambda}\in A,$

and as ${Q}_{\mu}$ is qualified, this entails that $v\in {A}^{*}$. Since this holds for every *A* neighborhood of *u*, we deduce that ${\mathrm{st}}_{\tau}(v)=u$, as desired.

Now, let $u={w}^{\circ}+\psi $ be the splitting of *u*.

If τ is the pointwise convergence, ${\mathrm{st}}_{\tau}(v)(x)=u(x)$ for every $x\in \mathrm{\Omega}$, hence, by Definition 3.9, we have that the singular set of *u* is empty and that $w(x)=u(x)$ for every $x\in \mathrm{\Omega}$, as desired. A similar argument works in the case of the pointwise convergence a.e.

If τ is the weak convergence topology, then ${\mathrm{st}}_{\tau}(v)=u$ means that ${\u3008v,{\phi}^{*}\u3009}^{*}\sim \u3008u,\phi \u3009$ for every φ in the dual of $V(\mathrm{\Omega})$. Now, let *S* be the singular set of *u*. We claim that $S=\mathrm{\varnothing}$. If not, let $x\in S$ and let $\phi ={\delta}_{x}$. Then ${\u3008v,{\phi}^{*}\u3009}^{*}=v(x)$ is infinite, whilst $\u3008u,{\delta}_{x}\u3009=u(x)$ is finite, which is absurd. Henceforth, for every $x\in \mathrm{\Omega}$, we have that $\psi (x)=0$. But

$\u3008u,\phi \u3009\sim {\u3008v,{\phi}^{*}\u3009}^{*}={\u3008{w}^{\circ}+\psi ,{\phi}^{*}\u3009}^{*}={\u3008{w}^{\circ},{\phi}^{*}\u3009}^{*}+{\u3008\psi ,{\phi}^{*}\u3009}^{*}=\u3008w,\phi \u3009+{\u3008\psi ,{\phi}^{*}\u3009}^{*},$

hence, ${\mathrm{st}}_{\tau}(\psi )=u-w$. As $\psi (x)=0$ for all $x\in \mathrm{\Omega}$, this means that $u(x)=w(x)$ for every $x\in \mathrm{\Omega}$. Then

$\u3008u,\phi \u3009+{\u3008\psi ,{\phi}^{*}\u3009}^{*}=\u3008w,\phi \u3009+{\u3008\psi ,{\phi}^{*}\u3009}^{*}=\u3008v,\phi \u3009\sim \u3008u,\phi \u3009,$

and so ${\u3008\psi ,{\phi}^{*}\u3009}^{*}\sim 0$.

Finally, if τ is the strong convergence with respect to a norm $\parallel \cdot \parallel $ and ${\{{u}_{n}\}}_{n}$ converges pointwise to *u*, then, by what we proved above, we have that $v(x)\sim u(x)$ for every $x\in \mathrm{\Omega}$, hence $u(x)\sim w(x)$ for every $x\in \mathrm{\Omega}$, which means $u=w$ as both $u,w\in V(\mathrm{\Omega})$. Then $\parallel \psi \parallel =\parallel u-{w}^{\circ}\parallel =\parallel u-{v}^{\circ}\parallel +\parallel {v}^{\circ}-{w}^{\circ}\parallel \sim 0$.

(5) Let $v={lim}_{\lambda \uparrow \mathrm{\Lambda}}{v}_{\lambda}$. By point (2), the only claim to prove is that ${\mathrm{st}}_{\tau}(v)=u$. We distinguish two cases:

Case 1: ${J}^{*}(v)\sim r\in \mathbb{R}$. As we noticed in point (2), it must be $r=m$. By contrast, let us assume that ${\mathrm{st}}_{\tau}(v)\ne u$. In this case, there exists an open neighborhood *A* of *u* such that the set

$Q:-\{\lambda \in \U0001d50f:{v}_{\lambda}\notin A\}$

is qualified. For every $n\in \mathbb{N}$, let

${Q}_{n}:-\{\lambda \in \U0001d50f:|J({v}_{\lambda})-r|<\frac{1}{n}\}\cap Q.$

Every ${Q}_{n}$ is qualified, hence nonempty. For every $n\in \mathbb{N}$, let ${\lambda}_{n}\in {Q}_{n}$. Finally, set ${u}_{n}:-{v}_{{\lambda}_{n}}$. By construction, ${lim}_{n\in \mathbb{N}}J({u}_{n})=m$. This means that ${\{{u}_{n}\}}_{n\in \mathbb{N}}$ is a minimizing sequence, hence, it converges to *u* in the topology τ, and this is absurd as, for every $n\in \mathbb{N}$, by construction, ${u}_{n}\notin A$. Henceforth, ${\mathrm{st}}_{\tau}(v)=u$.

Case 2: ${J}^{*}(v)\sim -\mathrm{\infty}$. As we noticed in the proof of point (2), in this case $m=-\mathrm{\infty}$. Let us assume that ${\mathrm{st}}_{V(\mathrm{\Omega})}(v)\ne u$. Then there exists an open neighborhood *A* of *u* such that the set

$Q:-\{\lambda \in \U0001d50f:{v}_{\lambda}\notin A\}$

is qualified. For every $n\in \mathbb{N}$, let

${Q}_{n}=\{\lambda \in \U0001d50f:J({v}_{\lambda})<-n\}\cap Q$

and let ${\lambda}_{n}\in {Q}_{n}$. Finally, let ${u}_{n}:-{v}_{{\lambda}_{n}}$. Then $J({u}_{n})<-n$ for every $n\in \mathbb{N}$, hence ${\{{u}_{n}\}}_{n\in \mathbb{N}}$ is a minimizing sequence, and so it must converge to *u*. However, by construction, ${u}_{n}\notin A$ for every $n\in \mathbb{N}$, which is absurd.
∎

#### Example 4.8.

Let $\mathrm{\Omega}=(0,1)$, let

$V(\mathrm{\Omega})=\{u:\mathrm{\Omega}\to \mathbb{R}\mid u\text{is the restriction to}\mathrm{\Omega}\text{of a piecewise}{\mathcal{\mathcal{C}}}^{1}([0,1])\text{function}\}$

and let $J:V(\mathrm{\Omega})\to \mathbb{R}$ be the functional

$J(u):-{\int}_{\mathrm{\Omega}}{u}^{2}(x)dx+{\int}_{\mathrm{\Omega}}{\left({({u}^{\prime})}^{2}-1\right)}^{2}dx.$

It is easily seen that ${inf}_{u\in V(\mathrm{\Omega})}J(u)=0$, and that the minimizing sequences of *J* converge pointwise and strongly in the ${L}^{2}$ norm to 0, but $J(0)=1$.

Let $v\in {V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$ be the minimum of ${J}^{*}:{V}_{\mathrm{\Lambda}}(\mathrm{\Omega})$. From points (4) and (5) of Theorem 4.7, we deduce that $0<{J}^{*}(v)\sim 0$, that ${\mathrm{st}}_{V(\mathrm{\Omega})}(v)=0$ and that the canonical decomposition of *v* is $v={0}^{\circ}+\psi $, with $\psi =0$ for every $x\in \mathrm{\Omega}$ and ${\int}_{{\mathrm{\Omega}}^{*}}^{*}{\psi}^{2}dx\sim 0$. Moreover, as ${J}^{*}(\psi )=0$, we also have that ${\int}_{{\mathrm{\Omega}}^{*}}^{*}{({({\psi}^{\prime})}^{2}-1)}^{2}dx\sim 0$.

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