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36
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 43 (6 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 use th ..."
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Cited by 21 (7 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.
Uniqueness and Maximal Regularity for Nonlinear Elliptic Systems of nLaplace Type With . . .
"... ..."
NonLinear Elliptic Systems With MeasureValued Right Hand Side
"... .  We prove existence of a solution u for the nonlinear elliptic system \Gamma div oe(x; u; Du) = ¯ in D 0(\Omega\Gamma ; u = 0 on @\Omega where ¯ is Radon measure on\Omega with finite mass. In particular we show that if the coercivity rate of oe lies in the range (1; 2 \Gamma 1 n ] then u ..."
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Cited by 9 (3 self)
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.  We prove existence of a solution u for the nonlinear elliptic system \Gamma div oe(x; u; Du) = ¯ in D 0(\Omega\Gamma ; u = 0 on @\Omega where ¯ is Radon measure on\Omega with finite mass. In particular we show that if the coercivity rate of oe lies in the range (1; 2 \Gamma 1 n ] then u is approximately differentiable and the equation holds with Du replaced by ap Du. The proof relies on an approximation of ¯ by smooth functions f k and a compactness result for the corresponding solutions u k . This follows from a detailed analysis of the Young measure fffi u(x)\Omega x g generated by the sequence f(u k ; Du k )g and the divcurl type inequality h x ; oe(x; u; \Delta)\Deltai ¯ oe(x)h x ; \Deltai for the weak limit ¯ oe of the sequence foe(\Delta; u k ; Du k )g. Contents 1 Introduction 2 2 A brief review of Young measures 6 3 Refined convergence results for 1 ! p 2 \Gamma 1 n 8 4 Approximate solutions and a priori bounds 12 5 A divcurl inequality 15 6 Compactne...
Renormalized Entropy Solutions For Quasilinear Anisotropic Degenerate Parabolic Equations
 SIAM J. MATH. ANAL
, 2003
"... We prove the well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L¹ data. This paper complements the work by Chen and Perthame [19], who developed a pure L¹ theory based on the notion of kin ..."
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Cited by 9 (5 self)
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We prove the well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L¹ data. This paper complements the work by Chen and Perthame [19], who developed a pure L¹ theory based on the notion of kinetic solutions.
Existence of a Solution to a Coupled Elliptic System
, 1994
"... We study here an elliptic system. Using the L 1 theory for elliptic operators, we prove existence of a solution by a fixed point technique. Keywords : Nonlinear elliptic system, Schauder Theorem, L 1 data, Electrochemical modelling. 1 Introduction While studying the modelling of an electrochemi ..."
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Cited by 8 (2 self)
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We study here an elliptic system. Using the L 1 theory for elliptic operators, we prove existence of a solution by a fixed point technique. Keywords : Nonlinear elliptic system, Schauder Theorem, L 1 data, Electrochemical modelling. 1 Introduction While studying the modelling of an electrochemical engineering problem [7], [8], we came across a coupled system between temperature and electrical potential. One of the coupling terms arises from the "Joule effect", i.e. the production of heat by electrical current. This term may be encountered in other physical problems, see for instance [5]. Here, in an attempt to prove existence of the solution to the electrochemical problem, we study the following simplified problem: \Gammadiv(oe(x; u(x))DOE(x)) = f(x; u(x)); x 2\Omega ; (1) OE(x) = 0; x 2 @\Omega ; (2) \Gammadiv((x; u(x))Du(x)) = oe(x; u(x))DOE(x):DOE(x); x 2\Omega ; (3) u(x) = 0; x 2 @\Omega ; (4) 1 Laboratoire de Math'ematiques, Universit'e de Savoie, F73376 Le Bourge...
The CalderónZygmund theory for elliptic problems with measure data
, 2007
"... Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density propertie ..."
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Cited by 8 (4 self)
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Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable CalderónZygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.
A Uniqueness Result for Quasilinear Elliptic Equations with Measures as Data
, 2001
"... We prove here a uniqueness result for Solutions Obtained as the Limit of Approximations of quasilinear elliptic equations with different kinds of boundary conditions and measures as data. 1 Introduction 1.1 Notations In this paper,\Omega is a bounded domain in R N (N 2), with a Lipschitz contin ..."
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Cited by 4 (3 self)
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We prove here a uniqueness result for Solutions Obtained as the Limit of Approximations of quasilinear elliptic equations with different kinds of boundary conditions and measures as data. 1 Introduction 1.1 Notations In this paper,\Omega is a bounded domain in R N (N 2), with a Lipschitz continuous boundary. The unit normal to @\Omega outward to\Omega is denoted by n. We denote by x \Delta y the usual Euclidean product of two vectors (x; y) 2 R N \Theta R N ; the associated Euclidean norm is written j:j. The Lebesgue measure of a measurable subset E in R N is denoted by jEj; oe is the Lebesgue measure on @\Omega (i.e. the (N\Gamma1)dimensional Hausdorff measure). \Gamma d and \Gamma f are measurable subsets of @\Omega such that @\Omega = \Gamma d [ \Gamma f and oe(\Gamma d " \Gamma f ) = 0. For q 2 [1; +1], we denote by q 0 the conjugate exponent of q (i.e. q 0 = q=(q \Gamma 1)). W 1;q is the usual Sobolev space, endowed with the norm jjujj W 1;q (\Omega\Gamm...
and mass comparison for degenerate nonlinear parabolic and related elliptic equations
 Adv. Nonlinear Stud
"... We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the p ..."
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Cited by 3 (1 self)
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We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the pLaplacian equation. The results will be used in a companion work in combination with a detailed knowledge of special solutions to obtain sharp a priori bounds and decay estimates for wide classes of solutions of those equations. 1991 Mathematics Subject Classification. 35B05, 35B40, 35J60, 35K55, 35K65, 47H20. Key words. Nonlinear elliptic and parabolic equations, symmetrization, mass concentration, comparison.
On the discretization of the coupled heat and electrical diffusion problems
 Numerical Methods and Applications: 6th International Conference, NMA 2006, Borovets, Bulgaria, August 2024, 2006, Revised Papers
, 2007
"... Abstract. We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which couples both equations. A f ..."
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Cited by 2 (0 self)
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Abstract. We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which couples both equations. A finite element scheme and a finite volume scheme are considered for the discretization of the system; in both cases, we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system. 1.