(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 Lipschitz-continuous boundary ∂Ω. We are interested in the problem ∂u Du = div(

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.

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Abstract. We consider non-linear 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 Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.

. -- 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 div-curl 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 div-curl inequality 15 6 Compactne...

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, F-73376 Le Bourge...

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.

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.

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...

Abstract. We present pointwise gradient bounds for solutions to p-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic equations. 1. Introduction and