On Lebesgue points of entropy solutions to the eikonal equation

Journal of the European Mathematical Society, 2003

In this paper, we establish rectifiability of the jump set of an S 1 -valued conservation law in two space-dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow-ups.

Nonlinear Analysis: Theory, Methods & Applications, 1998

2016

We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C0-sense up to the degeneracy due to the segments where f ′′ = 0. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp. Preprint SISSA 43/2016/MATE

Complex Variables and Elliptic Equations, 2008

In this article, we prove the existence of entropy solutions for the Dirichlet problem (P) − div[ω(x)A(x, u, ∇u)] = f (x) − div(G), in Ω u(x) = 0, on ∂Ω where Ω is a bounded open set of R N , N ≥ 2, f ∈ L 1 (Ω) and G/ω ∈ [L p (Ω, ω)] N .

Communications in Mathematics, 2014

In this article, we prove the existence of entropy solutions for the Dirichlet problem (P) − div[ω(x)A(x, u, ∇u)] = f (x) − div(G), in Ω u(x) = 0, on ∂Ω where Ω is a bounded open set of R N , N ≥ 2, f ∈ L 1 (Ω) and G/ω ∈ [L p (Ω, ω)] N .

Boletim da Sociedade Paranaense de Matemática, 2018

In this article, we study the following degenerate unilateral problems: − div(a(x, ∇u)) + H(x, u, ∇u) = f, which is subject to the weighted Sobolev spaces with variable exponent W 1,p(x) 0 (Ω, ω), where ω is a weight function on Ω, (ω is a measurable, a.e. strictly positive function on Ω and satisfying some integrability conditions). The function H(x, s, ξ) is a nonlinear term satisfying some growth condition but no sign condition and the right hand side f ∈ L 1 (Ω).

Archive for Rational Mechanics and Analysis, 2017

We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C 0-sense up to the degeneracy due to the segments where f = 0. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.

Zeitschrift f�r angewandte Mathematik und Physik, 2004

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.

Boletim da Sociedade Paranaense de Matemática, 2018

We prove existence of solutions for strongly nonlinear elliptic equations of the form $$ \left\{\begin{array}{c} A(u)+g(x,u,\nabla u)=f+\mbox {div}(\phi(u))\quad \textrm{in }\Omega, \\ u\equiv0\quad \partial \Omega. \end{array} \right.$$ Where $A(u)=-\mbox {div}(a(x,u,\nabla u))$ be a Leray-Lions operator defined in $D(A)\subset W^{1}_{0}L_\varphi(\Omega) \rightarrow W^{-1}_{0}L_\psi(\Omega)$, the right hand side belongs in $ L^{1}(\Omega)$, and $\phi\in C^{0}(\mathbb{R},\mathbb{R}^N)$, without assuming the $\Delta_{2}$-condition on the Musielak function.

Comptes Rendus Mathematique, 2015

In this Note, we consider the eikonal equation in one dimensional space describing the evolution of interfaces moving with non-signed velocity. We prove a global existence result of discontinuous viscosity solutions in a weak sense by considering BV intial data. Résumé Résultat d'existence pour l'équation eikonale unidimensionnelle. Dans cette Note, nous considérons l'équation eikonale en une dimension d'espace décrivant le mouvement d'interfaces avec une vitesse non-signée. Nous prouvons un résultat d'existence globale de solutions de viscosité discontinues dans un sens faible en considérant des données initiales BV. Version française abrégée Dans cette Note, nous considérons l'équation eikonale unidimensionnelle, donnée par    ∂ t u(x, t) = c(x, t)|∂ x u(x, t)| dans R × (0, T) u(x, 0) = u 0 (x) dans R, (1) avec les hypothèses (H 1) u 0 ∈ L ∞ (R) ∩ BV (R) (H 2) c ∈ L ∞ (R × (0, T)) ∩ L ∞ ((0, T); BV (R)), où BV (R) est l'espace des fonctions L 1 loc (R) à variations bornées. L'objet de cette Note, est d'établir un résultat d'existence globale en temps d'une solution de viscosité discontinue de (1) dans un sens assez faible, en supposant les hypothèses (H 1) et (H 2).