Calculus of Variations and Geometric Measure Theory
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G. Bellettini - V. Caselles - M. Novaga

Explicit solutions of the eigenvalue problem $- {\rm div} \left( Du/ \vert Du \vert \right) = u$

created on 11 Jun 2003
modified by belletti on 23 Dec 2005


Published Paper

Inserted: 11 jun 2003
Last Updated: 23 dec 2005

Journal: SIAM J. Math. Anal.
Volume: 36
Pages: 1095-1129
Year: 2005


In this paper we compute explicit solutions of the eigenvalue problem $-{\rm div} (Du /\vert D u\vert) = u$ in $\R^2$, in particular explicit solutions whose truncatures are in $W^{1,1}_{{\rm loc}}(\R^2)$, and piecewise constant explicit solutions. The solutions of the above eigenvalue problem describe the asymptotic behavior of solutions of the minimizing total variation flow, and allow to construct some explicit solutions of it. As an application, we construct explicit solutions of the denoising problem in image processing.

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