The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

Abstract

In this paper, we obtain existence and uniqueness of strong solutions to the inhomogenous Neumann initial-boundary problem for a parabolic PDE which arises as a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution are proved by means of a suitable pseudoparabolic relaxed approximation of the equation and the corresponding passage to the limit. Our main result is monotonicity in time of the positive and negative singular parts of the distributional space derivative for bounded variation initial data. Sufficient conditions for instantaneous $$L^1$$ L 1 -$$W^{1,1}_{{\mathrm{loc}}}$$ W loc 1 , 1 or BV-$$W^{1,1}_{{\mathrm{loc}}}$$ W loc 1 , 1 regularizing effects are also discussed.

Local estimates for vectorial Rudin–Osher–Fatemi type problems in one dimension

“…Then, by Theorem 1.1, for all N, j ∈ N, it is true that v N j ∈ BV (U, R n ) and This generalizes an analogous result for the scalar 1D total variation flow from [8,9]. We note also a recent paper [33], where similar estimate was obtained for more general parabolic equations in the scalar 1D case m = n = 1. Moreover, the authors provide conditions for instantaneous regularization BV → W 1,1 loc and L 1 → W 1,1 loc .…”