A variational approach to the local character of -closure : the convex case
A variational inequality for discontinuous solutions of degenerate parabolic equations.
The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. We study in the framework for functions of bounded variation, the well-posedness of the Beltrami flow in the one-dimensional case. We prove existence and uniqueness of the weak solution using lower semi-continuity results for convex functions of measures. The solution is defined via a variational inequality, following Temam?s technique for the evolution problem...
A variational treatment for general elliptic equations of the flame propagation type : regularity of the free boundary
A very singular solution for the dual porous medium equation and the asymptotic behaviour of general solutions.
A W 2, 2 Bound for a Class of Elliptic Equations in Nondivergence Form with Rough Coefficients.
A W1,p-Estimate for Solutions to Mixed Boundary Value Problems for Second Order Elliptic Differential Equtions.
A wave equation associated with mixed nonhomogeneous conditions: The compactness and connectivity of weak solution set.
A wave equation with discontinuous time delay.
A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces
We shall prove a weak comparison principle for quasilinear elliptic operators that includes the negative -Laplace operator, where 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.
A weak maximum principle and estimates of for nonlinear degenerate elliptic equations
A weighted isoperimetric inequality and applications to symmetrization.
A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media
In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution of an elliptic equation, that we write , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to...
About a Variant of the Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"
This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.
About asymptotic and oscillation properties of the Dirichlet problem for delay partial differential equations.
About global existence and asymptotic behavior for two dimensional gravity water waves
The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds.The proof is based on a bootstrap argument involving and estimates. The bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation...
About some types of boundary value problems with interfaces
About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains
This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces...
About steady transport equation. II: Schauder estimates in domains with smooth boundaries.
About the influence of oscillations on Strichartz-type decay estimates.
About the Morse Theory for Certain Variational Problems.