The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the Hamiltonian; next the solution to its stationary Hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the Hamiltonian function can be explicitly formulated and when the Jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems.

This paper is concerned with optimal design problems with aspecial assumption on the coefficients of the state equation.Namely we assume that the variations of these coefficientshave a small amplitude. Then, making an asymptotic expansionup to second order with respect to the aspect ratio of thecoefficients allows us to greatly simplify the optimal designproblem. By using the notion of H-measures we are able toprove general existence theorems for small amplitudeoptimal design and to provide simple and efficient numericalalgorithms for their computation. A key feature of thistype of problems is that the optimal microstructures arealways simple laminates.

We define and characterize weak and strong two-scale convergence in Lp ,C 0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness;in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions.We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.

Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the Γ-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are ε-periodic and of size ε. We show that, as ε → 0, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion of material in the perforated domain and is equal to 1 when there are no holes. We also prove that the solution of the approximate controllability problem in the perforated domain behaves, as ε → 0, as that of the problem posed in the perforated domain having as rigth-hand side the (fixed) control of the limit problem.

Integral representation of relaxed energies and of Γ-limits of functionals $$ (u,v)\mapsto \int_\Omega f( x,u(x),v(x))\,dx $$ are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered.

Nous écrivons et nous justifions des conditions aux limites approchéespour descouches minces périodiques recouvrant un objetparfaitement conducteur en polarisation transverse électrique ettransverse magnétique.

This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. They mainly have two effects (at first order with respect to the period ε): first, they add a dispersive term to the homogenized equation, and second, they yield an effective Fourier boundary condition.