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Minimizing total variation flow
 Differential and Integral Equations
, 2001
"... (Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We 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 ..."
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Cited by 54 (8 self)
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(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We 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 →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitzcontinuous boundary ∂Ω. We are interested in the problem ∂u Du = div(
The Dirichlet Problem for the Total Variation Flow
, 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 L¹ for entropy solutions. To prove the existence we u ..."
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Cited by 27 (9 self)
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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 L¹ 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.
ftp ejde.math.txstate.edu (login: ftp) NONLINEAR NEUMANN PROBLEMS ON BOUNDED LIPSCHITZ DOMAINS
"... Abstract. We prove existence and uniqueness, up to a constant, of an entropy solution to the nonlinear and non homogeneous Neumann problem − div[a(., ∇u)] + β(u) = f in Ω ∂u + γ(τu) = g on ∂Ω. ..."
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Abstract. We prove existence and uniqueness, up to a constant, of an entropy solution to the nonlinear and non homogeneous Neumann problem − div[a(., ∇u)] + β(u) = f in Ω ∂u + γ(τu) = g on ∂Ω.
Nonlinear Parabolic Problems with Neumanntype Boundary Conditions and L 1data
"... In this paper, we study existence, uniqueness and stability questions for the nonlinear parabolic equation: ∂u − △pu + α(u) = f ∂t in]0, T[×Ω, with Neumanntype boundary conditions and initial data in L 1. Our approach is based essentially on the time discretization technique by Euler forward schem ..."
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In this paper, we study existence, uniqueness and stability questions for the nonlinear parabolic equation: ∂u − △pu + α(u) = f ∂t in]0, T[×Ω, with Neumanntype boundary conditions and initial data in L 1. Our approach is based essentially on the time discretization technique by Euler forward scheme.