Influence de l’anisotropie sur l’apparition de singularités de bord dans les problèmes aux limites relatifs aux matériaux composites. (Influence of anisotropy on the appearance of singularities in the boundary value problems for composite materials). (French) Zbl 0608.73016

Sanchez-Palencia, Enrique

C. R. Acad. Sci., Paris, Sér. I 300, 27-30 (1985).

We consider the Neumann problem for an elliptic equation in the variables \(x_ 1\), \(x_ 2\) with coefficients \(a_{ij}(x)\) which are piecewise constant. Taking the origin at a point of the boundary, and r, \(\theta\) the corresponding polar coordinates, it is known that the solution will be singular in general if there exist nonzero solutions of the form \(U(x_ 1,x_ 2)=r^ 2u(\theta)\) with \(0<Re \alpha <1\) (then the gradient will tend to infinity as \(r^{\alpha -1})\). Let \(x_ 2=0\) be the boundary of the domain; we prove that if \(x_ 1\lessgtr 0\) are two regions with different coefficients, there exist values of these coefficients (with \(a_{12}\) discontinuous on \(x_ 1=0)\) such that there is a singularity at the origin. We state a conjecture (without proof) giving the existence (or nonexistence) of singularities in the general case. An application to elasticity is given.

MSC:

74E10	Anisotropy in solid mechanics
35E10	Convexity properties of solutions to PDEs with constant coefficients
35J25	Boundary value problems for second-order elliptic equations

Keywords:

Neumann problem; singularity at the origin; existence (or nonexistence) of singularities