On total variation flow in L1 with L∞ exponent and Neumann boundary conditions

Abstract

We consider the case of the gradient time flow corresponding to a minimization problem with L∞ variable exponent of the gradient in BV space, with L1 data, and Neumann boundary conditions. The flow for total variation, as defined in Rudin, Osher, and Fatemi, was extensively studied in Andreuet. al. including the L1data case using the theory of nonlinear semigroups. Due to the homogeneous structure of the total variation term in their work, regularity and subsequent qualitative results are obtained there. In Andreuet. al. they also obtained results for entropy and mild solutions for the flow with a quasilinear term φ(x, p) with continuity assumptions in x. We also use the method of nonlinear semigroupshere to prove existence of a mild solution u(t) ∈ L1((0, ∞), BV((Ω)) to the quasilinear total variation flow with initial condition and L1 penalty, but only assume the quasilinear term, φ(x, p), to have exponentq(x) in L∞without a continuity assumption. Additionally, we use a different definition of mild solu...