Seminário Análise Funcional e Aplicações 

14/11/2008, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Ioannis Stratis, National and Kapodistrian University of Athens, Greece

Transmission and boundary value problems in the theory of elasticity of hemitropic materials

A material is called hemitropic (acentric, noncentrosymmetric, chiral) if it is isotropic with respect to orthogonal transformations, but not with respect to mirror reflections. Typical examples are quartz crystals, DNA, bones and nanotubes. Although engineers have studied hemitropic materials since the early 1960s, the theory of elasticity for hemitropic materials has only very recently become the object of rigorous mathematical analysis. The governing equations describing time harmonic elastic fields in hemitropic materials, constitute a system of two vector elliptic PDEs of second order, stated in terms of the displacement and the microrotation vectors. When all the “hemitropicity” parameters become zero, this system reduces to the well-known reduced Navier equation of classical (isotropic) linear elasticity. In this talk, we present the basic boundary value and transmission problems arising in 3-dimensional linear hemitropic elasticity. We construct the fundamental solution that satisfies a Sommerfeld-Kupradze type radiation condition. We establish Green’s formulae, based on which we obtain representations of solutions (in bounded and unbounded domains) in terms of appropriate potentials (single-layer, double-layer, Newtonian). We study the jump and mapping properties of these potentials, and of the corresponding boundary integral (pseudodifferential) operators. Using Potential Theory and the Theory of Pseudodifferential Operators we establish the uniqueness and the existence of solutions (in Hölder, Sobolev and Besov spaces) to the Dirichlet, Neumann, and mixed type boundary value problems and to transmission problems. Further, we study the regularity properties of the solutions to these problems. The solvability is treated in both the cases of smooth and of Lipschitz domains.