The Method of Lines for Poisson's Equation with Nonlinear or Free Boundary Conditions.
The method of normal splines for linear implicit differential equations of second order.
The method of Rothe and two-scale convergence in nonlinear problems
Modelling of macroscopic behaviour of materials, consisting of several layers or components, cannot avoid their microstructural properties. This article demonstrates how the method of Rothe, described in the book of K. Rektorys The Method of Discretization in Time, together with the two-scale homogenization technique can be applied to the existence and convergence analysis of some strongly nonlinear time-dependent problems of this type.
The microlocal Landau-Zener formula
The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation
We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation on . For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at . We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us...
The micro-support of the complex defined by a convolution operator in tube domains
The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section
We consider the Euler equations for compressible fluids in a nozzle whose cross-section is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl.74 (1995) 483–548]. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class...
The mixed problem for an infinite system of first order functional differential equations.
The mixed problem for the Laplace equation on the rectangle
The mixed regularity of electronic wave functions multiplied by explicit correlation factors
The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics 2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...
The mixed regularity of electronic wave functions multiplied by explicit correlation factors***
The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...
The modulational regime of three-dimensional water waves and the Davey-Stewartson system
The monotone method on the sub-strongly coupled reaction-diffusion systems
The mortar element method for three dimensional finite elements
The Mortar method in the wavelet context
This paper deals with the use of wavelets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method,...
The Mortar Method in the Wavelet Context
This paper deals with the use of wavelets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method,...
The motion of a fluid in an open channel
We consider a free boundary value problem for a viscous, incompressible fluid contained in an uncovered three-dimensional rectangular channel, with gravity and surface tension, governed by the Navier-Stokes equations. We obtain existence results for the linear and nonlinear time-dependent problem. We analyse the qualitative behavior of the flow using tools of bifurcation theory. The main result is a Hopf bifurcation theorem with -symmetry.
The motion of the free surface of a liquid
The Mourre estimate for regular dispersive systems
The multi-dimensional super-replication problem under gamma constraints