The Dirichlet-to-Neumann operator associated with the 1-Laplacian and evolution problems

Daniel Hauer and José M. Mazón

Abstract

In this paper, we present the first insights about the Dirichlet-to-Neumann operator in \(L^{1}\) associated with the \(1\)-Laplace operator or total variational flow operator. This operator is the main object, for example, in studying inverse problems related to image processing, but also admits important relation to geometry. We show that this operator can be represented by the sub-differential in \(L^1\times L^{\infty}\) of a convex, homogeneous, and continuous functional on \(L^{1}\). This is quite surprising since it implies a type of stability or compactness result even though the singular Dirichlet problem governed \(1\)-Laplace operator by the might have infinitely many weak solutions if the given boundary data is not continuous. As an application, we obtain well-posedness and long-time stability of solutions of a singular coupled elliptic-parabolic initial boundary-value problem.

Keywords: Sub-differential, nonlinear semigroups, \(L^1\) regularity theory, total variational flow, \(1\)-Laplacian, least gradient, Dirichlet-to-Neumann operator.

AMS Subject Classification: Primary 35K65; secondary 35J25, 35J92, 35B40.