*Published Paper*

**Inserted:** 11 jun 2003

**Last Updated:** 23 dec 2005

**Journal:** SIAM J. Math. Anal.

**Volume:** 36

**Pages:** 1095-1129

**Year:** 2005

**Abstract:**

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.