Theorems Stability of Minimizing Movements under Gamma-Convergence Attributions

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Statement

Let $(X,d)$ be a complete metric space. For each $k\in\mathbb N$, let $\mathcal E_k:X\to(-\infty,\infty]$ be a proper lower semicontinuous functional, and let $\mathcal E:X\to(-\infty,\infty]$ be a proper lower semicontinuous functional. Let $|\partial\mathcal E_k|:X\to[0,\infty]$ and $|\partial\mathcal E|:X\to[0,\infty]$ denote their metric slopes.

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Assume that there exists $M\in\mathbb R$ such that $\mathcal E_k[x]\ge M$ for every $k\in\mathbb N$ and every $x\in X$, and $\mathcal E[x]\ge M$ for every $x\in X$.

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Assume that, for every $\tau>0$ and every sequence $(z_k)_{k\in\mathbb N}\subset X$ with $z_k\to z$ in $X$, the functionals $F_{k,\tau,z_k}:X\to(-\infty,\infty]$ defined by

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\begin{align*} F_{k,\tau,z_k}(x):=\mathcal E_k[x]+\frac{1}{2\tau}d^2(x,z_k) \end{align*}

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Gamma-converge, with respect to the metric topology of $X$, to the functional $F_{\tau,z}:X\to(-\infty,\infty]$ defined by

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\begin{align*} F_{\tau,z}(x):=\mathcal E[x]+\frac{1}{2\tau}d^2(x,z). \end{align*}

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Assume the following compactness condition. For every $C>0$ and every $T>0$, there exists a compact set $K_{C,T}\subset X$ such that, whenever $k\in\mathbb N$, $\tau\in(0,T]$, $N\in\mathbb N$, $N\tau\le T$, and $x_0,\dots,x_N\in X$ satisfy

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\begin{align*} x_{j+1}\in\operatorname*{argmin}_{x\in X}\left\{\mathcal E_k[x]+\frac{1}{2\tau}d^2(x,x_j)\right\} \end{align*}

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for every $j\in\{0,\dots,N-1\}$,

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\begin{align*} \mathcal E_k[x_0]\le C, \end{align*}

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and

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\begin{align*} \sum_{j=0}^{N-1}\frac{d^2(x_{j+1},x_j)}{\tau}\le C, \end{align*}

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then $x_j\in K_{C,T}$ for every $j\in\{0,\dots,N\}$.

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Assume the Sandier--Serfaty lower semicontinuity condition: if $(k_n)_{n\in\mathbb N}\subset\mathbb N$ satisfies $k_n\to\infty$, if $(y_n)_{n\in\mathbb N}\subset X$ satisfies $y_n\to y$ in $X$, and if

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\begin{align*} \sup_{n\in\mathbb N}\mathcal E_{k_n}[y_n]<\infty \end{align*}

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and

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\begin{align*} \sup_{n\in\mathbb N}|\partial\mathcal E_{k_n}|(y_n)<\infty, \end{align*}

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then

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\begin{align*} \mathcal E[y]\le \liminf_{n\to\infty}\mathcal E_{k_n}[y_n] \end{align*}

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and

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\begin{align*} |\partial\mathcal E|(y)\le \liminf_{n\to\infty}|\partial\mathcal E_{k_n}|(y_n). \end{align*}

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Assume also that $|\partial\mathcal E|$ is a strong upper gradient for $\mathcal E$.

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Let $(\tau_k)_{k\in\mathbb N}\subset(0,\infty)$ satisfy $\tau_k\downarrow0$, and let $(x_{k,0})_{k\in\mathbb N}\subset X$ satisfy $x_{k,0}\to x_0$ in $X$ and

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\begin{align*} \mathcal E_k[x_{k,0}]\to\mathcal E[x_0]<\infty. \end{align*}

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For each $k\in\mathbb N$, assume that a sequence $(x_{k,j})_{j\ge0}\subset X$ exists and satisfies

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\begin{align*} x_{k,j+1}\in\operatorname*{argmin}_{x\in X}\left\{\mathcal E_k[x]+\frac{1}{2\tau_k}d^2(x,x_{k,j})\right\} \end{align*}

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for every $j\ge0$.

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Define $\bar x_k:[0,\infty)\to X$ by $\bar x_k(0):=x_{k,0}$ and, for $j\in\mathbb N$, by

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\begin{align*} \bar x_k(t):=x_{k,j}\qquad\text{for }(j-1)\tau_k<t\le j\tau_k. \end{align*}

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For $r\ge0$, define $\bar x_k(r-\tau_k):=x_{k,0}$ when $0\le r\le\tau_k$, and define it by the preceding interpolation when $r>\tau_k$.

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Assume in addition that the schemes admit discrete slope control in the following sense. For every $T>0$, there exist Borel maps $\tilde x_k:[0,T]\to X$ and Borel functions $g_k:[0,T]\to[0,\infty]$ such that

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\begin{align*} g_k(r)\ge |\partial\mathcal E_k|(\tilde x_k(r)) \end{align*}

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for $\mathcal L^1$-a.e. $r\in[0,T]$,

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\begin{align*} \lim_{k\to\infty}\sup_{0\le r\le T}d(\tilde x_k(r),\bar x_k(r))=0, \end{align*}

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and

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\begin{align*} \sup_{k\in\mathbb N}\mathcal E_k[\tilde x_k(r)]<\infty \end{align*}

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for $\mathcal L^1$-a.e. $r\in[0,T]$.

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Finally, assume that for every $T>0$ there is a sequence $\varepsilon_k(T)\to0$ such that, for every $0\le s\le t\le T$,

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\begin{align*} \mathcal E_k[\bar x_k(t)]+\frac12\int_s^t \left(\frac{d(\bar x_k(r),\bar x_k(r-\tau_k))}{\tau_k}\right)^2\,d\mathcal L^1(r)+\frac12\int_s^t g_k^2(r)\,d\mathcal L^1(r)\le \mathcal E_k[\bar x_k(s)]+\varepsilon_k(T). \end{align*}

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Then there exist a subsequence of $(k)_{k\in\mathbb N}$, not relabelled, and an absolutely continuous curve $x:[0,\infty)\to X$ such that, for every $T>0$,

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\begin{align*} \bar x_k(t)\to x(t) \end{align*}

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in $X$ for every $t\in[0,T]$. Moreover $x(0)=x_0$, and $x$ is a curve of maximal slope for $\mathcal E$ with respect to $|\partial\mathcal E|$: for every $0\le s\le t<\infty$,

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\begin{align*} \mathcal E[x(t)]+\frac12\int_s^t |x'|^2(r)\,d\mathcal L^1(r)+\frac12\int_s^t |\partial\mathcal E|^2(x(r))\,d\mathcal L^1(r)\le \mathcal E[x(s)]. \end{align*}

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