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A Hamilton-Jacobi approach to junction problems and application to traffic flows

Cyril Imbert, Régis Monneau, Hasnaa Zidani (2013)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They...

A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces

Akihito Unai (2018)

Applications of Mathematics

We shall prove a weak comparison principle for quasilinear elliptic operators - div ( a ( x , u ) ) that includes the negative p -Laplace operator, where a : Ω × N N satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

Picone’s identity for a Finsler p -Laplacian and comparison of nonlinear elliptic equations

Jaroslav Jaroš (2014)

Mathematica Bohemica

In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm H in n which is continuously differentiable for x 0 and such that H p is strictly convex for some p > 1 . Two important special cases are the p -Laplacian and the so-called pseudo p -Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria...

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u > 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 < p < , p - 1 < q p * - 1 , λ > 0 , and 0 < δ < 1 . As usual, p * = N p N - p if 1 < p < N , p * ( p , ) is arbitrarily large if p = N , and p * = if p > N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions...

Uniqueness of solutions for some elliptic equations with a quadratic gradient term

David Arcoya, Sergio Segura de León (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by - Δ u + λ | u | 2 u r = f ( x ) , λ , r > 0 . The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...

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