The Stefan problem in heterogeneous media
The Stefan problem with kinetic condition at the free boundary
The support shrinking properties for solutions of quasilinear parabolic equations with strong absorption terms
The upper bound of the number of eigenvalues for a class of perturbed Dirichlet forms
The theory of Markov processes and the analysis on Lie groups are used to study the eigenvalue asymptotics of Dirichlet forms perturbed by scalar potentials.
The use of linear approximation scheme for solving the Stefan problem
This paper deals with the linear approximation scheme to approximate a singular parabolic problem: the two-phase Stefan problem on a domain consisting of two components with imperfect contact. The results of some numerical experiments and comparisons are presented. The method was used to determine the temperature of steel in the process of continuous casting.
The vanishing viscosity method in infinite dimensions
The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.
The verification of the Nirenberg-Treves conjecture
In a series of recent papers, Nils Dencker proves that condition implies the local solvability of principal type pseudodifferential operators (with loss of derivatives for all positive ), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of derivatives.
The waiting time property for parabolic problems trough the nondiffusion of support for the stationary problems.
In this note we study the waiting time phenomenon for local solutions of the nonlinear diffusion equation through its connection with the nondiffusion of the support property for local solutions of the family of discretized elliptic problems. We show that, under a suitable growth condition on the initial datum near the boundary of its support, a finite part of the family of solutions of the discretized problem maintain unchanged its support.
The wave equation approach to an inverse problem for a general convex domain with impedance boundary conditions
The well-posedness of the Dirichlet problem in the cylindric domain for the multidimensional wave equation.
The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces
In this paper we are interested in constructing WKB approximations for the nonlinear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.
The Wolff gradient bound for degenerate parabolic equations
The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.
Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations
We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory. Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate,...
Theoretical and numerical study of a free boundary problem by boundary integral methods
In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.
Theoretical and numerical study of a free boundary problem by boundary integral methods
In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.
Third boundary value problem for the heat equation. I.
Time asymptotic description of an abstract Cauchy problem solution and application to transport equation
In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in -space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.
Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations.
Time-dependent Stokes equations with measure data.
Topological properties of the solution set of a class of nonlinear evolutions inclusions
In the paper we study the topological structure of the solution set of a class of nonlinear evolution inclusions. First we show that it is nonempty and compact in certain function spaces and that it depends in an upper semicontinuous way on the initial condition. Then by strengthening the hypothesis on the orientor field , we are able to show that the solution set is in fact an -set. Finally some applications to infinite dimensional control systems are also presented.