Exponential decay estimates for solutions of the polyharmonic equation in a semi-infinite cylinder (Q1323063)

We derive explicit Saint-Venant type decay estimates for solutions of the Dirichlet problem for the polyharmonic equation defined in a semi- infinite cylinder with homogeneous Dirichlet data on the lateral surface of the cylinder. The method we use is a weighted energy technique. The advantage of this weighted energy approach is that it allows us to treat simultaneously polyharmonic problems of any order. However, it achieves the optimal decay rate only in the harmonic case. For the sake of simplicity we consider only the case of three dimensions. In order to make our decay estimates explicit we require a bound for the total weighted energy in the half cylinder. We bound this total weighted energy in terms of ordinary total energy, and then bound the total energy in terms of the Dirichlet data on the finite end of the half cylinder.