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The total variation flow in \(\mathbb R^N\). (English) Zbl 1036.35099

This paper consists of two parts. In the first part, the authors prove an existence and uniqueness result for a total variation flow \(u_t = \text{div} ( \frac{Du}{| Du| } )\) in \({\mathbb R}^N\) with initial data \(u_0 \in \text{L}^1_{loc}({\mathbb R}^N)\). In the second part, they study the equation \(\text{div} ( \frac{Du}{| Du| } ) = \lambda_{\Omega} u\) on a bounded set \(\Omega\) of finite perimeter in \({\mathbb R}^2\), with a certain constant \(\lambda_{\Omega}\). They give a condition such that the characteristic function \(\chi_{\Omega}\) of \(\Omega\) is a solution of this equation when \(\Omega\) is connected.

MSC:

35K65 Degenerate parabolic equations
35K15 Initial value problems for second-order parabolic equations
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