p-Parabolic approximation of total variation flow solutions (original) (raw)

On the definition of solution to the total variation flow

2021

∂tu− div(|Du|Du) = 0, 1 < p < ∞, as p → 1. A Sobolev space is the natural function space in the existence and regularity theories for a weak solution to the parabolic p-Laplace equation, see the monograph by DiBenedetto [12]. The corresponding function space for the total variation flow is functions of bounded variation and in that case the weak derivative of a function is a vector valued Radon measure. A standard definition of weak solution to the parabolic p-Laplace equation is based on integration by parts, but it is not immediately clear what is the corresponding definition of weak solution to the total variation flow. One possibility is to apply the so-called Anzellotti pairing [6]. This approach has been applied for the total variation flow, for example, in the monograph by Andreu, Caselles and Mazón [5]. For the parabolic p-Laplace equation, it is also possible to consider solutions to the parabolic variational inequality

The Dirichlet Problem for the Total Variation Flow

Journal of Functional Analysis, 2001

We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L 1 for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.

A variational principle for gradient flows

Mathematische Annalen, 2004

We verify-after appropriate modifications-an old conjecture of Brezis-Ekeland ([3], [4]) concerning the feasibility of a global variational approach to the problems of existence and uniqueness of gradient flows for convex energy functionals. Our approach is based on a concept of "self-duality" inherent in many parabolic evolution equations, and motivated by Bolza-type problems in the classical calculus of variations. The modified principle allows to identify the extremal value-which was the missing ingredient in [3]-and so it can now be used to give variational proofs for the existence and uniqueness of solutions for the heat equation (of course) but also for quasi-linear parabolic equations, porous media, fast diffusion and more general dissipative evolution equations.

The Minimizing Total Variation Flow with Measure Initial Conditions

Communications in Contemporary Mathematics, 2004

In this paper we obtain existence and uniqueness of solutions for the Cauchy problem for the minimizing total variation flow when the initial condition is a Radon measure in ℝN. We study limit solutions obtained by weakly approximating the initial measure μ by functions in L1(ℝN). We are able to characterize limit solutions when the initial condition μ=h+μs, where h∈L1(ℝN)∩L∞(ℝN), and μs=αℋk⌊ S,α≥0,k is an integer and S is a k-dimensional manifold with bounded curvatures. In case k

Variational solutions to the total variation flow on metric measure spaces

2021

We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincaré inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian spaces and functions of bounded variation to prove a necessary and sufficient condition for a variational solution to be continuous at a given point.

A variational method for a class of parabolic PDES

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2011

In this manuscript we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. Assuming the initial condition to be a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Then De Giorgi's interpolation method reveals to be a powerful tool for proving convergence of our algorithm. Finally we show uniqueness and stability in L 1 of our solutions.

Variational parabolic capacity

Discrete and Continuous Dynamical Systems, 2015

We establish a variational parabolic capacity in a context of degenerate parabolic equations of p-Laplace type, and show that this capacity is equivalent to the standard capacity. As an application, we compute capacities of several explicit sets.

Minimizing total variation flow

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000

We prove existence and uniqueness of weak solutions for the minimizing Total Variation ow with initial data in L 1 . W e prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t ! 1 . We also prove that local maxima strictly decrease with time, in particular, at zones immediately decrease their level. We display some numerical experiments illustrating these facts.

A Variational Inequality for Discontinuous Solutions of Degenerate Parabolic Equations

2000

The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. We study in the framework for functions of bounded variation, the well-posedness of the Beltrami flow in the one-dimensional case. We prove existence and uniqueness of the weak solution using lower semi-continuity results for convex functions of measures. The solution

On a variational approach to the Navier-Stokes Equations

2020

 (i) ∂v ∂t + divx (v ⊗ v) +∇xp = ν∆xv + f ∀(x, t) ∈ Ω× (0, T ) , (ii) divx v = 0 ∀(x, t) ∈ Ω× (0, T ) , (iii) v = 0 ∀(x, t) ∈ ∂Ω× (0, T ) , (iv) v(x, 0) = v0(x) ∀x ∈ Ω . Here v = v(x, t) : Ω×(0, T ) → R is an unknown velocity, p = p(x, t) : Ω×(0, T ) → R is an unknown pressure, associated with v, ν > 0 is a given constant viscosity, f : Ω× (0, T ) → R is a given force field and v0 : Ω → R N is a given initial velocity. The existence of weak solution to (1.1) satisfying the Energy inequality was first proved in the celebrating works of Leray (1934). There are many different procedures for constructing weak solutions (see Leray [9],[10] (1934); Kiselev and Ladyzhenskaya [8] (1957); Shinbrot [12] (1973)). The most common methods are based on the so called Faedo-Galerkin approximation process. Application of Faedo-Galerkin method for (1.1) was first considered by Hopf in [7]. We also refer to Masuda [11] for the problem in higher dimension. In this paper we present a variational met...