Finite element discretization of local minimization schemes for rate-independent evolutions

Abstract

This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in Efendiev and Mielke (J Convex Anal 13(1):151–167, 2006), which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions for mesh size tending to zero exist and are so-called parametrized solutions of the continuous problem. The discrete problems are solved by means of a mass lumping scheme for the non-smooth dissipation functional in combination with a semi-smooth Newton method. A numerical test indicates the efficiency of this approach. In addition, we compared the local minimization scheme with a time stepping scheme for global energetic solutions, which shows that both schemes yield different solutions with differing time discontinuities.

Appendices

A Proof of Lemma 3.2

We start the proof of Lemma 3.2 with the following result, which is a direct consequence of the characterization of $z^{\tau ,h}_k$ as a minimizer of (3.1). (This is actually the only point, where one uses that $z^{\tau ,h}_k$ is a minimizer and not only a stationary point satisfying (3.3)–(3.6).)

Lemma A.1

(Local energy-inequality) For all $h,\tau >0$ and all $k \in {\mathbb {N}}$, the inequality

$$\begin{aligned} {\mathcal {I}}\left( t_{k}^{\tau ,h},z_{k}^{\tau ,h}\right) + {\mathcal {R}}_h\left( z_{k}^{\tau ,h}-z_{k-1}^{\tau ,h}\right) \le {\mathcal {I}}\left( t_{k-1}^{\tau ,h},z_{k-1}^{\tau ,h}\right) + \int _{t_{k-1}^{\tau ,h}}^{t_{k}^{\tau ,h}} \partial _t{\mathcal {I}}\left( s,z_{k}^{\tau ,h}\right) d s \end{aligned}$$

(A.1)

is valid.

Proof

By optimality of $z_{k}^{\tau ,h}$ and feasibility of $z_{k-1}^{\tau ,h}$ for the minimization problem in (3.1), we obtain

$$\begin{aligned} {\mathcal {I}}\left( t_{k-1}^{\tau ,h},z_{k}^{\tau ,h}\right) + {\mathcal {R}}_h\left( z_{k}^{\tau ,h}-z_{k-1}^{\tau ,h}\right) \le {\mathcal {I}}\left( t_{k-1}^{\tau ,h},z_{k-1}^{\tau ,h}\right) \end{aligned}$$

(A.2)

Adding ${\mathcal {I}}(t_{k}^{\tau ,h},z_{k}^{\tau ,h})$ on both sides then gives the result. $\square$

The local energy inequality is now used to derive the uniform bounds on energy and dissipation in Lemma 3.2.

Proof (of Lemma 3.2)

Again we suppress the superscript $\tau , h$ in the proof to shorten the notation, except for $z_0^{\tau ,h}$ in order to avoid confusion with the initial value. Estimating the right hand side in (A.1) by employing (2.4) gives

$$\begin{aligned}&{\mathcal {I}}(t_{k},z_{k}) + {\mathcal {R}}_h\left( z_{k}-z_{k-1}\right) \\&\qquad \le {\mathcal {I}}(t_{k-1},z_{k-1}) + \int _{t_{k-1}}^{t_{k}} \mu ({\mathcal {I}}(t_{k-1},z_{k})+\beta ) \exp (\mu (s-t_{k-1}))d s \\&\qquad = {\mathcal {I}}(t_{k-1},z_{k-1}) + ({\mathcal {I}}(t_{k-1},z_{k})+\beta )( \exp (\mu (t_{k}-t_{k-1}))-1). \end{aligned}$$

Using the non-negativity of ${\mathcal {R}}_h$ by (2.16) in combination with (A.2) yields ${\mathcal {I}}(t_{k-1},z_{k}) \le {\mathcal {I}}(t_{k-1},z_{k-1})$ so that

$$\begin{aligned} \begin{aligned}&{\mathcal {I}}(t_{k},z_{k}) + {\mathcal {R}}_h\left( z_{k}-z_{k-1}\right) \\&\qquad \le {\mathcal {I}}(t_{k-1},z_{k-1}) + ({\mathcal {I}}(t_{k-1},z_{k-1})+\beta )( \exp (\mu (t_{k}-t_{k-1}))-1) \end{aligned} \end{aligned}$$

(A.3)

is obtained. By exploiting once again ${\mathcal {R}}_h \ge 0$, this implies

$$\begin{aligned} {\mathcal {I}}(t_{k},z_{k}) \le ({\mathcal {I}}(t_{k-1},z_{k-1}) + \beta ) \exp (\mu (t_{k} - t_{k-1}))-\beta \end{aligned}$$

such that induction over k already gives the desired result for the energy:

$$\begin{aligned} \begin{aligned} {\mathcal {I}}(t_{k},z_{k})&\le \left( {\mathcal {I}}(0,z_0^{\tau ,h})+\beta \right) \prod _{j=1}^k \exp \left( \mu (t_{j}-t_{j-1})\right) -\beta \\&\le \left( {\mathcal {I}}(0,z_0^{\tau ,h})+\beta \right) \exp (\mu t_{k}) - \beta . \end{aligned} \end{aligned}$$

(A.4)

To include the dissipation in the estimate, we sum up (A.3) to obtain

$$\begin{aligned}&{\mathcal {I}}(t_{k},z_{k}) + \sum _{j=1}^k {\mathcal {R}}_h\left( z_{j}-z_{j-1}\right) \\&\qquad \le {\mathcal {I}}(0,z_0^{\tau ,h}) + \sum _{j=1}^k ({\mathcal {I}}(t_{j-1},z_{j-1})+\beta )(\exp (\mu (t_{j}-t_{j-1}))-1). \end{aligned}$$

Inserting (A.4) and adding $\beta$ on both sides, we finally obtain

$$\begin{aligned}&{\mathcal {I}}(t_{k},z_{k}) + \sum _{j=1}^k {\mathcal {R}}_h\left( z_{j}-z_{j-1}\right) + \beta \\&\quad \le \left( {\mathcal {I}}(0,z_0^{\tau ,h}) + \beta \right) + \sum _{j=1}^k \left( {\mathcal {I}}(0,z_0^{\tau ,h})+\beta \right) \exp (\mu t_{j-1})(\exp (\mu (t_{j}-t_{j-1}))-1) \\&\qquad = ({\mathcal {I}}(0,z_0^{\tau ,h})+\beta ) \exp (\mu t_{k}) \le \left( {\mathcal {I}}(0,z_0^{\tau ,h})+\beta \right) \exp (\mu T), \end{aligned}$$

which is the claimed estimate. $\square$

B Proof of the discrete energy identity in Lemma 3.8

We follow the lines of [6] and start with the proof of (3.41). From (3.3), (3.4), and the binomial formula, we infer

$$\begin{aligned} {\text {dist}}_{{\mathcal {V}}^*}\{-\varPi _h^*D_z{\mathcal {I}}\left( t_{k-1}^{\tau ,h},z_{k}^{\tau ,h}\right) , \partial ({\mathcal {R}}_h \circ \varPi _h)(0)\} = \lambda _k^{\tau ,h}\tau . \end{aligned}$$

(B.1)

For arbitrary $k\in \{1, \ldots , N-1\}$, we thus deduce from (3.3) and (3.37) that

$$\begin{aligned} 0&= \lambda _k^{\tau ,h}\left( ||z_{k}^{\tau ,h}-z_{k-1}^{\tau ,h}||_{\mathbb {V}}-\tau \right) \\&= \lambda _k^{\tau ,h}\tau \left( 1 - ||\hat{z}_{\tau ,h}^\prime (s)||_{\mathbb {V}}\right) \\&= \hat{t}_{\tau ,h}^\prime (s) {\text {dist}}_{{\mathcal {V}}^*}\left\{ -\varPi _h^*D_z{\mathcal {I}}\left( \underline{t}_{\tau ,h}(s),\overline{z}_{\tau ,h}(s)\right) ,\partial ({\mathcal {R}}_h\circ \varPi _h)(0)\right\} , \quad \forall \, s\in \left[ s_{k-1}^{\tau ,h}, s_k^{\tau ,h}\right) , \end{aligned}$$

where we employed the definition of the constant interpolants in (3.35). This gives (3.41) for almost all $s\in (0,s^{\tau ,h}_{N-1})$. As seen at the end of the proof of Lemma 3.7, it holds $\lambda ^{\tau ,h}_N = 0$ so that (B.1) implies the assertion for $s\in (s^{\tau ,h}_{N-1}, S_{\tau ,h})$.

Next we turn to the discrete energy-identity. Since the affine interpolants in (3.34) are by construction elements of $W^{1,\infty }((0,S_{\tau ,h}))$ and $W^{1,\infty }((0,S_{\tau ,h});{\mathcal {Z}})$, respectively, and due to ${\mathcal {I}}\in C^1([0,T] \times {\mathcal {Z}})$ by assumption, the chain rule is applicable and gives for $s \in [s_{k-1}^{\tau ,h},s_k^{\tau ,h})$ that

$$\begin{aligned}&\frac{d }{d s} {\mathcal {I}}(\hat{t}_{\tau ,h}(s),\hat{z}_{\tau ,h}(s)) \\&\quad =\partial _t {\mathcal {I}}(\hat{t}_{\tau ,h}(s),\hat{z}_{\tau ,h}(s)) \hat{t}^\prime _{\tau ,h}(s) + \left\langle D_z{\mathcal {I}}(\hat{t}_{\tau ,h}(s),\hat{z}_{\tau ,h}(s)) , \hat{z}_{\tau ,h}^\prime (s) \right\rangle _{{\mathcal {Z}}^*,{\mathcal {Z}}} \\&\quad =\partial _t {\mathcal {I}}(\hat{t}_{\tau ,h}(s),\hat{z}_{\tau ,h}(s)) \hat{t}^\prime _{\tau ,h}(s) + \frac{1}{s_{k}^{\tau ,h}- s_{k-1}^{\tau ,h}}\left\langle D_z{\mathcal {I}}(\underline{t}_{\tau ,h}(s),\overline{z}_{\tau ,h}(s)) , z_{k}^{\tau ,h}-z_{k-1}^{\tau ,h} \right\rangle _{{\mathcal {Z}}^*,{\mathcal {Z}}} \\&\qquad + \left\langle D_z{\mathcal {I}}(\hat{t}_{\tau ,h}(s),\hat{z}_{\tau ,h}(s)) - D_z{\mathcal {I}}(\underline{t}_{\tau ,h}(s),\overline{z}_{\tau ,h}(s)) , \hat{z}_{\tau ,h}^\prime (s) \right\rangle _{{\mathcal {Z}}^*,{\mathcal {Z}}} \, . \end{aligned}$$

From (3.5), we have in combination with the 1-homogeneity of ${\mathcal {R}}_h$ that

$$\begin{aligned}&- \frac{1}{s_{k}^{\tau ,h}- s_{k-1}^{\tau ,h}} \left\langle D_z{\mathcal {I}}(\underline{t}_{\tau ,h}(s),\overline{z}_{\tau ,h}(s)) , z_{k}^{\tau ,h}-z_{k-1}^{\tau ,h} \right\rangle _{{\mathcal {Z}}^*,{\mathcal {Z}}} \\&\quad = \frac{1}{s_{k}^{\tau ,h}- s_{k-1}^{\tau ,h}} \left( {\mathcal {R}}\left( z_{k}^{\tau ,h}-z_{k-1}^{\tau ,h}\right) + \tau {\text {dist}}_{{\mathcal {V}}^*}\left\{ -\varPi _h^*D_z{\mathcal {I}}\left( t_{k-1}^{\tau ,h},z_{k}^{\tau ,h}\right) ,\partial ({\mathcal {R}}_h\circ \varPi _h)(0)\right\} \right) \\&\quad = {\mathcal {R}}_h(\hat{z}^\prime _{\tau ,h}) + {\text {dist}}_{{\mathcal {V}}^*}\left\{ -\varPi _h^*D_z{\mathcal {I}}\left( t_{k-1}^{\tau ,h},z_{k}^{\tau ,h}\right) ,\partial ({\mathcal {R}}_h\circ \varPi _h)(0)\right\} ). \end{aligned}$$

Note that, in case of $k<N$, the last equation results from $s_{k}^{\tau ,h}- s_{k-1}^{\tau ,h}= \tau$, while, for the case $k=N$, it follows from $\lambda _N^{\tau ,h}= 0$ and (B.1), similarly to above. By taking into account the definition of $r_{\tau ,h}$ in (3.40), integration over $(\sigma _1,\sigma _2)$ then yields (3.39).

It remains to estimate $r_{\tau ,h}$. To this end, first observe that the definition of the affine and constant interpolants in (3.34) and (3.35) implies for every $k\in \{1, \ldots , N\}$ and every $s \in [s_{k-1}^{\tau ,h},s_k^{\tau ,h})$ that

$$\begin{aligned} \hat{z}_{\tau ,h}(s) - \overline{z}_{\tau ,h}(s) = \left( s -s_{k}^{\tau ,h}\right) \hat{z}_{\tau ,h}^\prime (s) \, \text { and } \, \hat{t}_{\tau ,h}(s) - \underline{t}_{\tau ,h}(s) = \left( s -s_{k-1}^{\tau ,h}\right) \hat{t}_{\tau ,h}^\prime (s) , \end{aligned}$$

which is frequently used in the following estimates. Now, let $k \in \{1, \ldots , N\}$ and $s\in [s_{k-1}^{\tau ,h},s_k^{\tau ,h})$ be arbitrary. Then, by inserting the concrete form of ${\mathcal {I}}$ into the definition of $r_{{\tau ,h}}$ in (3.40) and employing the coercivity of A together with $(s-s_k^{\tau ,h})<0$, we arrive at

$$\begin{aligned} \begin{aligned} r_{\tau ,h}(s)&= \left( s-s_k^{\tau ,h}\right) \left\langle A(\hat{z}_{\tau ,h}^\prime (s)) , \hat{z}_{\tau ,h}^\prime (s) \right\rangle _{{\mathcal {Z}}^*,{\mathcal {Z}}} \\&\qquad + \left\langle D_z{\mathcal {F}}(\hat{z}_{\tau ,h}(s)) - D_z{\mathcal {F}}(\overline{z}_{\tau ,h}(s)) , \frac{\hat{z}_{\tau ,h}(s) - \overline{z}_{\tau ,h}(s)}{(s-s_k^{\tau ,h})} \right\rangle _{{\mathcal {V}}^*,{\mathcal {V}}} \\&\qquad - \left\langle \ell \left( \hat{t}_{\tau ,h}(s)\right) - \ell (\underline{t}_{\tau ,h}(s)) , \hat{z}_{\tau ,h}^\prime (s) \right\rangle _{{\mathcal {V}}^*,{\mathcal {V}}} \\&\qquad \le \alpha \left( s-s_k^{\tau ,h}\right) ||\hat{z}_{\tau ,h}^\prime (s)||_{\mathcal {Z}}^2 \\&\qquad + \frac{1}{\left|s-s_k^{\tau ,h}\right|} \, \left|\left\langle D_z{\mathcal {F}}(\overline{z}_{\tau ,h}(s)) - D_z{\mathcal {F}}(\hat{z}_{\tau ,h}(s)) , \overline{z}_{\tau ,h}(s)- \hat{z}_{\tau ,h}(s) \right\rangle _{{\mathcal {V}}^*,{\mathcal {V}}}\right| \\&\qquad + \left|\hat{t}_{\tau ,h}(s) - \underline{t}_{\tau ,h}(s)\right| \, ||\ell ||_{C^1([0,T],{\mathcal {V}}^*)} ||\hat{z}_{\tau ,h}^\prime (s)||_{\mathcal {V}}. \end{aligned} \end{aligned}$$

(B.2)

We apply Lemma 3.5 with $\varepsilon = \alpha /2$ to the second term on the right hand side to obtain

$$\begin{aligned} \begin{aligned}&\frac{1}{\left|s-s_k^{\tau ,h}\right|} \, \left|\left\langle D_z{\mathcal {F}}(\overline{z}_{\tau ,h}(s)) - D_z{\mathcal {F}}(\hat{z}_{\tau ,h}(s)) , \overline{z}_{\tau ,h}(s)- \hat{z}_{\tau ,h}(s) \right\rangle _{{\mathcal {V}}^*,{\mathcal {V}}}\right| \\&\quad \le \frac{\alpha }{2}\, \left|s-s_k^{\tau ,h}\right|\, ||\hat{z}_{\tau ,h}^\prime (s)||_{\mathcal {Z}}^2 + C_\alpha \, \left|s-s_k^{\tau ,h}\right|\, {\mathcal {R}}_h(\hat{z}_{\tau ,h}^\prime (s))\, ||\hat{z}_{\tau ,h}^\prime (s)||_{\mathbb {V}}\end{aligned} \end{aligned}$$

where we also used the positive homogeneity of ${\mathcal {R}}_h$. By inserting this in (B.2) and using again that $(s-s_k^{\tau ,h})<0$, one deduces

$$\begin{aligned} r_{\tau ,h}(s) \le C \big ( {\mathcal {R}}_h(\hat{z}_{\tau ,h}^\prime (s)) + \hat{t}_{\tau ,h}^\prime (s) \, ||\ell ||_{C^1([0,T],{\mathcal {V}}^*} \big ) (s-s_{k-1}^{\tau ,h}) \, ||\hat{z}_{\tau ,h}^\prime (s)||_{\mathbb {V}}\, . \end{aligned}$$

Integrating and exploiting the definition of $\hat{z}_{\tau ,h}$ and $\hat{t}_{\tau ,h}$, respectively, then yields

$$\begin{aligned}&\int _{\sigma _1}^{\sigma _2} r_{\tau ,h}(s) \mathrm {d}s \\&\quad \le \sum _{i=1}^{N} \int _{s_{i-1}^{\tau ,h}}^{s_i^{\tau ,h}} C \Big ( {\mathcal {R}}_h(z_{i}^{\tau ,h}-z_{i-1}^{\tau ,h}) {+} (t_{i}^{\tau ,h} {-} t_{i-1}^{\tau ,h}) \, ||\ell ||_{C^1([0,T],{\mathcal {V}}^*)} \Big ) \frac{s_i^{\tau ,h}-s}{(s_i^{\tau ,h}- s_{i-1}^{\tau ,h})^2} \, ||z_{i}^{\tau ,h}-z_{i-1}^{\tau ,h}||_{\mathbb {V}}\, \mathrm {d}s\\&\quad = \sum _{i=1}^{N} \frac{1}{2}\, C \Big ({\mathcal {R}}_h(z_{i}^{\tau ,h}-z_{i-1}^{\tau ,h}) + (t_{i}^{\tau ,h} - t_{i-1}^{\tau ,h}) \, ||\ell ||_{C^1([0,T],{\mathcal {V}}^*)} \Big ) ||z_{i}^{\tau ,h}-z_{i-1}^{\tau ,h}||_{\mathbb {V}}\\&\quad \le C\, \tau \Big ( T + \sum _{i=1}^{N} {\mathcal {R}}_h(z_{i}^{\tau ,h}-z_{i-1}^{\tau ,h}) \Big ). \end{aligned}$$

Thanks to Lemma 3.2, the bracket on the right hand side is bounded independent of $\tau$ and h so that (3.42) is proven, too. $\square$

C Auxiliary results from convex analysis

In this section, we collect some useful properties of ${\mathcal {R}}$ and ${\mathcal {R}}_h$, respectively. We start with the following lemma, whose proof is straight forward and therefore omitted:

Lemma C.1

Let ${\mathcal {W}}$ be a normed vector space and ${\mathcal {J}}: {\mathcal {W}}\rightarrow {\mathbb {R}}$ a convex and positive 1-homogeneous functional. Then it holds

$$\begin{aligned}&\partial {\mathcal {J}}(v) \subset \partial {\mathcal {J}}(0) \quad \forall v \in {\mathcal {W}} \end{aligned}$$

(C.1)

$$\begin{aligned}&\xi \in \partial {\mathcal {J}}(0) \; \Longleftrightarrow \; {\mathcal {J}}(w) \ge \left\langle \xi , w \right\rangle \quad \forall w \in {\mathcal {W}} \end{aligned}$$

(C.2)

$$\begin{aligned}&\partial {\mathcal {J}}(v) = \{ \xi \in \partial {\mathcal {J}}(0) \, : \, {\mathcal {J}}(v) = \left\langle \xi , v \right\rangle \} \end{aligned}$$

(C.3)

$$\begin{aligned}&{\mathcal {J}}^*(\xi ) = I_{\partial {\mathcal {J}}(0)}(\xi ) \quad \forall \xi \in {\mathcal {W}}^* \end{aligned}$$

(C.4)

where $I_{\partial {\mathcal {J}}(0)}$ denotes the indicator functional of $\partial {\mathcal {J}}(0)$.

Remark 4

Since only convexity and positive homogeneity is required for Lemma C.1 to hold, we may apply the above results to ${\mathcal {R}}$, its approximation ${\mathcal {R}}_h$, and ${\mathcal {R}}_h \circ \varPi _h$, considered as operators on ${\mathcal {V}}$ as well as ${\mathcal {Z}}$.

As in the proof of Lemma 3.1, we abbreviate ${\mathcal {R}}_{\tau ,h} = {\mathcal {R}}_h \circ \varPi _h + I_\tau$, where $I_\tau$ is as defined in (3.8). Analogously, we set ${\mathcal {R}}^h_\tau := {\mathcal {R}}_h + I_\tau (v)$.

Lemma C.2

For every $\eta \in {\mathcal {V}}^*$, there holds

$$\begin{aligned} ({\mathcal {R}}_{\tau ,h})^*(\eta ) = \tau {\text {dist}}_{{\mathcal {V}}^*}\left\{ \eta ,\varPi _h^* \partial {\mathcal {R}}_h(0)\right\} , \end{aligned}$$

(C.5)

where ${\text {dist}}_{{\mathcal {V}}^*}\{\eta ,\varPi _h^* \partial {\mathcal {R}}_h(0)\} = \inf \{||\eta -\varPi _h^* w||_{{\mathbb {V}}^{-1}} \, : \, w \in \partial {\mathcal {R}}_h(0)\}$ and $||\eta ||_{{\mathbb {V}}^{-1}}^2 = \left\langle \eta , {\mathbb {V}}^{-1} \eta \right\rangle$.

Proof

We use the inf-convolution formula (see [1, Prop. 3.4]), which is applicable, since ${\mathcal {R}}_h \circ \varPi _h$ is continuous. This gives

$$\begin{aligned} \left( {\mathcal {R}}_h \circ \varPi _h+I_\tau \right) ^*(\eta ) = \inf _{w\in {\mathcal {V}}^*}\left( ({\mathcal {R}}_h\circ \varPi _h)^*(\eta )+ I_\tau ^*(w-\eta )\right) . \end{aligned}$$

(C.6)

For $I_\tau ^*$, direct calculation leads to

$$\begin{aligned} I_\tau ^{*}(\eta ) = \tau ||\eta ||_{{\mathbb {V}}^{-1}}. \end{aligned}$$

(C.7)

To calculate the conjugate functional of $({\mathcal {R}}_h\circ \varPi _h)^*$, note that by the linearity of $\varPi _h$ the composition ${\mathcal {R}}_h \circ \varPi _h$ is again convex and 1-homogeneous. Therefore, Lemma C.1 gives $({\mathcal {R}}_h \circ \varPi _h)^*(\eta ) = I_{\partial ({\mathcal {R}}_h\circ \varPi _h)(0)}(\eta )$. The chain-rule for subdifferentials yields $\partial ({\mathcal {R}}_h \circ \varPi _h)(0) = \varPi _h^* \partial {\mathcal {R}}_h(0)$ so that we obtain

$$\begin{aligned} ({\mathcal {R}}_h \circ \varPi _h)^{*}(\eta ) = I_{\partial ({\mathcal {R}}_h\circ \varPi _h)(0)}(\eta ) = I_{\varPi _h^* \partial {\mathcal {R}}_h(0)}(\eta ). \end{aligned}$$

Inserting this together with (C.7) in (C.6) finally yields

$$\begin{aligned} ({\mathcal {R}}_h\circ \varPi _h + I_\tau )^*(\eta ) = \inf _{w \in \varPi _h^* \partial {\mathcal {R}}_h(0)} \{\tau ||\eta -w||_{{\mathbb {V}}^{-1}} \} = \tau {\text {dist}}_{{\mathcal {V}}^*}\left( \eta , \varPi _h^* \partial {\mathcal {R}}_h(0)\right) , \end{aligned}$$

which is (C.5). $\square$

Lemma C.3

Let $v \in {\mathcal {V}}$ be arbitrary. Then, $\xi \in {\mathcal {V}}^*$ is an element of $\partial I_\tau (v)$, iff there exists a multiplier $\lambda \in {\mathbb {R}}$ such that $\xi = \lambda {\mathbb {V}}v$ and

$$\begin{aligned} ||v||_{\mathbb {V}}\le \tau , \quad \lambda (||v||_{\mathbb {V}}- \tau ) = 0, \quad \lambda \ge 0. \end{aligned}$$

Proof

According to a classical result of convex analysis in combination with (C.7), it holds

$$\begin{aligned} \xi \in \partial I_\tau (v) \quad \Longleftrightarrow \quad I_\tau (v)+I_\tau ^*(\xi ) = \left\langle \xi , v \right\rangle \quad \Longleftrightarrow \quad \left\{ \;\begin{aligned}&||v||_{\mathbb {V}}\le \tau \\&\tau ||\xi ||_{{\mathbb {V}}^{-1}} = \left\langle \xi , v \right\rangle \end{aligned}\right. \end{aligned}$$

(C.8)

Now, the Cauchy–Schwarz-Inequality implies $\left\langle \xi , v \right\rangle = \left\langle {\mathbb {V}}({\mathbb {V}}^{-1}\xi ) , v \right\rangle \le ||\xi ||_{{\mathbb {V}}^{-1}} ||v||_{\mathbb {V}} \le \tau ||\xi ||_{{\mathbb {V}}^{-1}}$ so that the equivalence in (C.8) can only hold if ${\mathbb {V}}^{-1}\xi = \lambda v$ for some $\lambda \in {\mathbb {R}}$. Inserting this into (C.8), we conclude that $\lambda \ge 0$. Moreover, if $||v||_{\mathbb {V}}< \tau$, then $\xi = 0$ so that $\lambda$ fulfills also $\lambda (||v||_{\mathbb {V}}- \tau ) = 0$ as claimed. $\square$