### Asymptotics of the spectral function for the Steklov problem in a family of sets with fractal boundaries.

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We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ${u}_{t}=\Delta u$, ${v}_{t}=\Delta v$ in $\Omega \times (0,T)$; fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta}={u}^{{p}_{11}}{v}^{{p}_{12}}$, $\frac{\partial v}{\partial \eta}={u}^{{p}_{21}}{v}^{{p}_{22}}$ on $\partial \Omega \times (0,T)$, where $\Omega $ is a bounded smooth domain in ${\mathbb{R}}^{d}$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation $(U,V)$. We prove that if $U$ blows up in finite time then $V$ can fail to blow up if and only if ${p}_{11}\>1$ and ${p}_{21}\<2({p}_{11}-1)$, which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover,...

We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a $p$-Laplacian system. We prove...

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations , in Ω x (0,); fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta}={u}^{{p}_{11}}{v}^{{p}_{12}}$, $\frac{\partial v}{\partial \eta}={u}^{{p}_{21}}{v}^{{p}_{22}}$ on ∂Ω x (0,), where is a bounded smooth domain in ${\mathbb{R}}^{d}$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation . We prove that if blows up in finite time then can fail to blow up if and only if > 1 and < 2( - 1) , which is the same...

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