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If you want the smallest, try $$x = -LambertW(-\epsilon)/\epsilon = 1+\epsilon+{\frac{3}{2}}{\epsilon}^{2}+{\frac{8}{3}}{\epsilon}^{3}+{\frac{125}{24}}{\epsilon}^{4} +O \left( {\epsilon}^{5} \right)$$

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The ABP estimate indeed holds in your setting. The key is that the concave envelope of $u$ is in $C^{1,\,1}$, so the area formula is valid for its gradient. Assuming for simplicity that $L = \Delta$, that $\Omega = B_1$ and that $\sup_{\partial B_1} u = 0$, the way I would argue is: Let $\Gamma$ be the concave envelope (the infimum of linear functions ...

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The equations of motion cannot be written as $$\ddot{r}_{ij} = f_{ij}(r_{12},\ldots, r_{n-1,n}, \dot{r}_{12},\ldots, \dot{r}_{n-1,n})$$ Consider $n$ equally massive particles arranged equidistantly around a circle. If they are all initially stationary, the system will collapse to the centre of the circle. But if they are rotating at the correct speed, ...

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To expand on my comment: suppose you have $U$ not simply connected, and $\omega$ closed but not exact, and suppose there exists a closed loop $\gamma: [0,1]\to U$ such that $\omega(\dot{\gamma})$ is signed. (This is in particular the case with the "example" in your comment, where you can take $\gamma$ to be any circle centered at the origin.) Suppose \$\...

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There are many well-known ODE systems from various fields of science that can be transformed into the Lorenz system or generalized Lorenz system after a change of variables. After such a transformation the parameters of the original system may become negative in terms of the Lorenz system. The reason for looking for such a transformation is simple: to ...

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