Objective
The purpose of the project is to study some open questions related to homogenization of problems of mathematical physics. In particular the following problems will be considered:
homogenization of boundary value problems for nonlinear elliptic and
parabolic higher order equations in perforated domains of general structure;
pointwise estimates of the solutions of nonlinear parabolic higher order
equations in domains with thin holes;
construction of correctors for energetic and uniform convergence for
nonlinear elliptic and parabolic higher order equations in perforated
domains;
asymptotic behaviour and correctors for energetic and uniform convergence
for boundary value problems for Navier-Stokes equations in perforated
domains of general structure;
homogenization of some problems of mathematical physics: elastic composite
materials, motion of the ideal liquid with vortex threads;
homogenization of the attractors for semilinear parabolic equations in
weakly connected domains;
homogenization of nonlinear elliptic and parabolic operators defined on
weakly connected spaces;
homogenization of boundary value problems for nonlinear elliptic higher
order equations in domains with accumulators;
homogenization of nonlinear elliptic and parabolic problems with mixed
boundary conditions; homogenization of free boundary problems and spectral
problems in domains perforated along manifolds;
homogenization of harmonic differential forms on manifolds with complicated
microstructure;
homogenization of linear and nonlinear problems in p-connected domains;
criterion of p-connectedness for two Lipschitz domains in terms of capacity
and Hausdorff dimension of their common boundary;
phenomenon of diffusion through fractal walls;
homogenization on fractals of type of the Cantor networks, Sobolev spaces
and elliptic equations on the Euclidian space with a measure;
variational problems for integrands with non-standard growth conditions:
Lavrentiev phenomenon, Meyers type estimates, Holder continuity of
solutions;
homogenization problems for perforated domains with nonlinear conditions on
the boundary of cavities;
asymptotic behaviour of solutions of nonlinear variational inequalities
connected with domains of composite structure;
homogenization of nonlinear hyperbolic boundary value problems in perforated
domains and in domains with oscillating boundary;
homogenization problems for media with periods of different orders in
different directions;
homogenization problems for media reinforced by irregular systems of
curvilinear fibres;
homogenization problems for media reinforced or weakened by
three-dimensional inclusions;
homogenization for the systems of quasilinear equations of viscous
compressible media dynamics. At the end of the project it is expected to establish rigorous results on the asymptotic behaviour of solutions of boundary value problems for partial differential equations in perforated domains with complicated microstructure, to construct special approximations of these solutions, to describe the corresponding homogenized problems and to apply the results obtained to the study of mathematical models of composite materials.
Topic(s)
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34013 Trieste
Italy