On the principal eigenvalue of the Stokes operator in cylindrical domains

Abstract

AbstractWe consider the Stokes operator in periodic, three dimensional cylinders supplemented with four types of the boundary conditions: the zero Dirichlet b.c., the Navier b.c., the slip b.c. and the generalized impermeability b.c. We analyze the relation between the principal eigenvalue of the Stokes operator and the diameter of the base of the cylinder. Since the direct computation of the eigenvalues for the Stokes system is very difficult, we examine the vector‐valued Laplacian and then draw some conclusions for the Stokes operator.