This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian,

$$\begin{aligned} \frac{\partial v}{\partial t} - \Delta _1 u \ni 0 \hbox { in } (0, \infty ) \times \Omega , \quad v\in \gamma (u), \end{aligned}$$

and initial data in \(L^1(\Omega )\), where \(\Omega \) is a bounded smooth domain in \({\mathbb {R}}^N\) and \(\gamma \) is a maximal monotone graph in \({\mathbb {R}}\times {\mathbb {R}}\). We prove that, under certain assumptions on the graph \(\gamma \), there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.

中文翻译：

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1-拉普拉斯算子的双非线性方程

本文关注一类 1-拉普拉斯双非线性方程的诺伊曼问题，

$$\begin{对齐} \frac{\partial v}{\partial t} - \Delta _1 u \ni 0 \hbox { in } (0, \infty ) \times \Omega , \quad v\in \gamma (u), \end{对齐}$$

和\(L^1(\Omega )\)中的初始数据 ，其中\(\Omega \)是\({\mathbb {R}}^N\)中的有界平滑域 ，而 \(\gamma \)是\({\mathbb {R}}\times {\mathbb {R}}\)中的最大单调图。我们证明，在图 \(\gamma \)的某些假设下，解存在且唯一。此外，我们证明这些解与全变分流的诺伊曼问题的解一致。我们证明这样的假设是必要的。