Isotropic realizability of a strain field for the two-dimensional incompressible elasticity system

In the paper we study the problem of the isotropic realizability in R2 of a regular strain field e(U) = 12 ( DU +DU ) for the incompressible Stokes equation, namely the existence of a positive viscosity μ > 0 solving the Stokes equation in R2 with the prescribed field e(U). We show that if e(U) does not vanish at some point, then the isotropic realizability holds in the neighborhood of that point. The global realizability in R2 or in the torus is much more delicate, since it involves the global existence of a regular solution to a semilinear wave equation the coefficients of which depend on the derivatives of U . Using the semilinear wave equation we prove a small perturbation result: If DU is periodic and close enough to its average for the C4-norm, then the strain field is isotropically realizable in a given disk centered at the origin. On the other hand, a counter-example shows that the global realizability in R2 may hold without the realizability in the torus, and it is discussed in connection with the associated semilinear wave equation. The case where the strain field vanishes is illustrated by an example. The singular case of a rank-one laminate field is also investigated.