Michel Chipot

Finite Element Methods for Elliptic Problems

We consider the framework introduced Chapter 3 for the Lax-Milgram Theorem. In other words let H denote a real Hilbert space and a(u, v) a bilinear form on H satisfying (3.5), (3.6). For f ∈ H’, the dual space of H, we would like to approximate the solution u of the problem

Variational inequalities and flow in porous media

\left\{ {\begin{array}{*{20}{c}} {u \in H,\quad \quad \quad \quad \quad \quad \quad } \\ {a(u,v) = \left\langle {f,v} \right\rangle \quad \forall v \in H} \\ \end{array} } \right.

Numerical analysis of oscillations in nonconvex problems

(8.1)

Numerical approximations in variational problems with potential wells

Unspecified

On the Asymptotic Behaviour of Some Nonlocal Problems

1. Abstract Existence and Uniqueness Results for Solutions of Variational Inequalities.- 2. Examples and Applications.- 3. The Obstacle Problems: A Regularity Theory.- 4. The Dam Problem.- References.

Smoothness of Linear Laminates

Preface.- I. Basic techniques.- 1. Hilbert space techniques.- 2. A survey of essential analysis.- 3. Weak formulation of elliptic problems.- 4. Elliptic problems in divergence form.- 5. Singular perturbation problems.- 6. Problems in large cylinders.- 7. Periodic problems.- 8. Homogenization.- 9. Eigenvalues.- 10. Numerical computations.- II. More advanced theory.- 11. Nonlinear problems.- 12. L(infinity)-estimates.- 13. Linear elliptic systems.- 14. The stationary Navier-Stokes system.- 15. Some more spaces.- 16. Regularity theory.- 17. The p-Laplace equation.- 18. The strong maximum principle.- 19. Problems in the whole space.- A. Fixed point theorems.- Bibliography.- Index.

Approximated convex envelope of a function

SummaryWe study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer.

Existence and uniqueness of solutions to the compressible Reynolds lubrication equation

In this paper, some numerical aspects of variational problems which fail to be convex are studied. It is well known that for such a problem, in general, the infimum of the energy (the functional that has to be minimized) fails to be attained. Instead, minimizing sequences develop oscillations which allow them to decrease the energy.It is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed. It is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant. An estimate of the approximate deformation as it approximates a measure and some numerical results are also given.