Explicit Solutions of the Eigenvalue Problem $div \left(\frac Du\vert Du \vert \right)=u$ in $R^2$

Abstract

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