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I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.

Recapitulation

for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E_0 \cup \bigcup_{k\in\mathbb N} E_k$ with $\mathcal H^m(E_0)=0$ and $E_k\subseteq F_k$ for some $\mathcal C^1$-manifolds $F_k$ of dimension $m$, and a non-negativ function $\theta\in L^1_{\text{loc}}(\mathcal H^m|_E)$ such that $\mu=\theta \mathcal H^m|_E$. This is a characterisation of $m$-recitifiable varifolds. The first variation $\delta\mu$ of a varifold $\mu$ is for $\eta\in\mathcal C^1_c(\mathbb R^n;\mathbb R^n)$ given by $$\delta\mu(\eta)=\int div_\mu\eta\,d\mu,$$ where $ div_\mu(\eta)(x) = \sum_{i=1}^n \tau_i^T(x)\cdot (D\eta)\cdot \tau_i(x)$ where $\tau_i(x)$ is a orthogonal basis of the tangentspace of $\mu$ in $x$, which coinsidence $\mu$-almost everywhere with $T_xF_i$ for $x\in E_i\subseteq F_i$ as above. So $div_\mu\eta(x)$ is just the divergence in the manifold $F_i$, with $x\in E_i\subseteq F_i$.

We say $\mu$ has an locally bounded first variation, if for all $\Omega'\subseteq \Omega$ there exists $c(\Omega')<\infty$ such that $$ \delta\mu(\eta) \le C(\Omega',\Omega) \Vert \eta\Vert_{L^\infty(\Omega)} \qquad\forall\;\eta\in\mathcal C^1_c(\Omega'). $$ See for more explanation for example http://eom.springer.de/G/g130040.htm.

For a $\mathcal C^2$-manifold $M$ in $\mathbb R^n$ with mean curvature $H_M$ the first variation is $$ \delta M(\eta)=-\int_M H_M \cdot \eta \,dvol_M -\int_{\partial M} \tau_0 \cdot \eta \,dvol_{\partial M} \qquad\forall\;\eta\in\mathcal C_c^1(\mathbb R^n)$$ with the inner normal $\tau_0\in T_xM\cap(T_x\partial M)^\bot$ and where the mean curvature is the trace of the second fundamental form $A$ by the meaning of $H_M(x)=\sum_{i=1}^m A_x(\tau_i,\tau_i)$ in the normal space of $M$. As obviouse in this case the first variation is locally bounded.

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.

Recapitulation

for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E_0 \cup \bigcup_{k\in\mathbb N} E_k$ with $\mathcal H^m(E_0)=0$ and $E_k\subseteq F_k$ for some $\mathcal C^1$-manifolds $F_k$ of dimension $m$, and a non-negativ function $\theta\in L^1_{\text{loc}}(\mathcal H^m|_E)$ such that $\mu=\theta \mathcal H^m|_E$. This is a characterisation of $m$-recitifiable varifolds. The first variation $\delta\mu$ of a varifold $\mu$ is for $\eta\in\mathcal C^1_c(\mathbb R^n;\mathbb R^n)$ given by $$\delta\mu(\eta)=\int div_\mu\eta\,d\mu,$$ where $ div_\mu(\eta)(x) = \sum_{i=1}^n \tau_i^T(x)\cdot D\eta(x)\cdot \tau_i(x)$ where $\tau_i(x)$ is a orthogonal basis of the tangentspace of $\mu$ in $x$, which coinsidence $\mu$-almost everywhere with $T_xF_i$ for $x\in E_i\subseteq F_i$ as above. So $div_\mu\eta(x)$ is just the divergence in the manifold $F_i$, with $x\in E_i\subseteq F_i$.

We say $\mu$ has an locally bounded first variation, if for all $\Omega'\subseteq \Omega$ there exists $c(\Omega')<\infty$ such that $$ \delta\mu(\eta) \le C(\Omega',\Omega) \Vert \eta\Vert_{L^\infty(\Omega)} \qquad\forall\;\eta\in\mathcal C^1_c(\Omega'). $$ See for more explanation for example http://eom.springer.de/G/g130040.htm.

For a $\mathcal C^2$-manifold $M$ in $\mathbb R^n$ with mean curvature $H_M$ the first variation is $$ \delta M(\eta)=-\int_M H_M \cdot \eta \,dvol_M -\int_{\partial M} \tau_0 \cdot \eta \,dvol_{\partial M} \qquad\forall\;\eta\in\mathcal C_c^1(\mathbb R^n)$$ with the inner normal $\tau_0\in T_xM\cap(T_x\partial M)^\bot$ and where the mean curvature is the trace of the second fundamental form $A$ by the meaning of $H_M(x)=\sum_{i=1}^m A_x(\tau_i,\tau_i)$ in the normal space of $M$. As obviouse in this case the first variation is locally bounded.

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.

Recapitulation

for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E_0 \cup \bigcup_{k\in\mathbb N} E_k$ with $\mathcal H^m(E_0)=0$ and $E_k\subseteq F_k$ for some $\mathcal C^1$-manifolds $F_k$ of dimension $m$, and a non-negativ function $\theta\in L^1_{\text{loc}}(\mathcal H^m|_E)$ such that $\mu=\theta \mathcal H^m|_E$. This is a characterisation of $m$-recitifiable varifolds. The first variation $\delta\mu$ of a varifold $\mu$ is for $\eta\in\mathcal C^1_c(\mathbb R^n;\mathbb R^n)$ given by $$\delta\mu(\eta)=\int div_\mu\eta\,d\mu,$$ where $ div_\mu(\eta)(x) = \sum_{i=1}^n \tau_i^T(x)\cdot (D\eta)\cdot \tau_i(x)$ where $\tau_i(x)$ is a orthogonal basis of the tangentspace of $\mu$ in $x$, which coinsidence $\mu$-almost everywhere with $T_xE_i$ for $x\in E_i$ as above. So $div_\mu\eta(x)$ is just the divergence in the manifold $F_i$, with $x\in E_i\subseteq F_i$.

We say $\mu$ has an locally bounded first variation, if for all $\Omega'\subseteq \Omega$ there exists $c(\Omega')<\infty$ such that $$ \delta\mu(\eta) \le C(\Omega',\Omega) \Vert \eta\Vert_{L^\infty(\Omega)} \qquad\forall\;\eta\in\mathcal C^1_c(\Omega'). $$ See for more explanation for example http://eom.springer.de/G/g130040.htm.

For a $\mathcal C^2$-manifold $M$ in $\mathbb R^n$ with mean curvature $H_M$ the first variation is $$ \delta M(\eta)=-\int_M H_M \cdot \eta \,dvol_M -\int_{\partial M} \tau_0 \cdot \eta \,dvol_{\partial M} \qquad\forall\;\eta\in\mathcal C_c^1(\mathbb R^n)$$ with the inner normal $\tau_0\in T_xM\cap(T_x\partial M)^\bot$ and where the mean curvature is the trace of the second fundamental form $A$ by the meaning of $H_M(x)=\sum_{i=1}^m A_x(\tau_i,\tau_i)$ in the normal space of $M$. As obviouse in this case the first variation is locally bounded.

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.

Recapitulation

for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E_0 \cup \bigcup_{k\in\mathbb N} E_k$ with $\mathcal H^m(E_0)=0$ and $E_k\subseteq F_k$ for some $\mathcal C^1$-manifolds $F_k$ of dimension $m$, and a non-negativ function $\theta\in L^1_{\text{loc}}(\mathcal H^m|_E)$ such that $\mu=\theta \mathcal H^m|_E$. This is a characterisation of $m$-recitifiable varifolds. The first variation $\delta\mu$ of a varifold $\mu$ is for $\eta\in\mathcal C^1_c(\mathbb R^n;\mathbb R^n)$ given by $$\delta\mu(\eta)=\int div_\mu\eta\,d\mu,$$ where $ div_\mu(\eta)(x) = \sum_{i=1}^n \tau_i^T(x)\cdot (D\eta)\cdot \tau_i(x)$ where $\tau_i(x)$ is a orthogonal basis of the tangentspace of $\mu$ in $x$, which coinsidence $\mu$-almost everywhere with $T_xF_i$ for $x\in E_i\subseteq F_i$ as above. So $div_\mu\eta(x)$ is just the divergence in the manifold $F_i$, with $x\in E_i\subseteq F_i$.

We say $\mu$ has an locally bounded first variation, if for all $\Omega'\subseteq \Omega$ there exists $c(\Omega')<\infty$ such that $$ \delta\mu(\eta) \le C(\Omega',\Omega) \Vert \eta\Vert_{L^\infty(\Omega)} \qquad\forall\;\eta\in\mathcal C^1_c(\Omega'). $$ See for more explanation for example http://eom.springer.de/G/g130040.htm.

For a $\mathcal C^2$-manifold $M$ in $\mathbb R^n$ with mean curvature $H_M$ the first variation is $$ \delta M(\eta)=-\int_M H_M \cdot \eta \,dvol_M -\int_{\partial M} \tau_0 \cdot \eta \,dvol_{\partial M} \qquad\forall\;\eta\in\mathcal C_c^1(\mathbb R^n)$$ with the inner normal $\tau_0\in T_xM\cap(T_x\partial M)^\bot$ and where the mean curvature is the trace of the second fundamental form $A$ by the meaning of $H_M(x)=\sum_{i=1}^m A_x(\tau_i,\tau_i)$ in the normal space of $M$. As obviouse in this case the first variation is locally bounded.

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.

Edited: Recapitulation, for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E_0 \cup \bigcup_{k\in\mathbb N} E_k$ with $\mathcal H^m(E_0)=0$ and $E_k\subseteq F_k$ for some $\mathcal C^1$-manifolds $F_k$ of dimension $m$, and a non-negativ function $\theta\in L^1_{\text{loc}}(\mathcal H^m|_E)$ such that $\mu=\theta \mathcal H^m|_E$. This is a characterisation of $m$-recitifiable varifolds. The first variation $\delta\mu$ of a varifold $\mu$ is for $\eta\in\mathcal C^1_c(\mathbb R^n;\mathbb R^n)$ given by $$\delta\mu(\eta)=\int div_\mu\eta\,d\mu,$$ where $ div_\mu(\eta)(x) = \sum_{i=1}^n \tau_i^T (D\eta) \tau_i$ where $\tau_i$ is a orthogonal basis of the tangentspace of $\mu$ in $x$, which coinsidence $\mu$-almost everywhere with $T_xE_i$ for $x\in E_i$ as above.

We say $\mu$ has an locally bounded first variation, if for all $\Omega'\subseteq \Omega$ there exists $c(\Omega')<\infty$ such that $$ \delta\mu(\eta) \le C(\Omega',\Omega) \Vert \eta\Vert_{L^\infty(\Omega)} \qquad\forall\;\eta\in\mathcal C^1_c(\Omega'). $$ See for more explanation for example http://eom.springer.de/G/g130040.htm.

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.

Recapitulation

for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E_0 \cup \bigcup_{k\in\mathbb N} E_k$ with $\mathcal H^m(E_0)=0$ and $E_k\subseteq F_k$ for some $\mathcal C^1$-manifolds $F_k$ of dimension $m$, and a non-negativ function $\theta\in L^1_{\text{loc}}(\mathcal H^m|_E)$ such that $\mu=\theta \mathcal H^m|_E$. This is a characterisation of $m$-recitifiable varifolds. The first variation $\delta\mu$ of a varifold $\mu$ is for $\eta\in\mathcal C^1_c(\mathbb R^n;\mathbb R^n)$ given by $$\delta\mu(\eta)=\int div_\mu\eta\,d\mu,$$ where $ div_\mu(\eta)(x) = \sum_{i=1}^n \tau_i^T(x)\cdot (D\eta)\cdot \tau_i(x)$ where $\tau_i(x)$ is a orthogonal basis of the tangentspace of $\mu$ in $x$, which coinsidence $\mu$-almost everywhere with $T_xE_i$ for $x\in E_i$ as above. So $div_\mu\eta(x)$ is just the divergence in the manifold $F_i$, with $x\in E_i\subseteq F_i$.

We say $\mu$ has an locally bounded first variation, if for all $\Omega'\subseteq \Omega$ there exists $c(\Omega')<\infty$ such that $$ \delta\mu(\eta) \le C(\Omega',\Omega) \Vert \eta\Vert_{L^\infty(\Omega)} \qquad\forall\;\eta\in\mathcal C^1_c(\Omega'). $$ See for more explanation for example http://eom.springer.de/G/g130040.htm.

For a $\mathcal C^2$-manifold $M$ in $\mathbb R^n$ with mean curvature $H_M$ the first variation is $$ \delta M(\eta)=-\int_M H_M \cdot \eta \,dvol_M -\int_{\partial M} \tau_0 \cdot \eta \,dvol_{\partial M} \qquad\forall\;\eta\in\mathcal C_c^1(\mathbb R^n)$$ with the inner normal $\tau_0\in T_xM\cap(T_x\partial M)^\bot$ and where the mean curvature is the trace of the second fundamental form $A$ by the meaning of $H_M(x)=\sum_{i=1}^m A_x(\tau_i,\tau_i)$ in the normal space of $M$. As obviouse in this case the first variation is locally bounded.