Energy estimates for the biharmonic equation in three dimensions
Chang Hao Lin · Quarterly of Applied Mathematics · 1994
1. Introduction.In this paper, we are concerned with energy decay estimates for solutions of the biharmonic equation in a semi-infinite cylinder with nonzero boundary conditions on the finite end.Decay estimates for various energies are derived and compared.Saint-Venant's principle in the context of plane elastostatics has been investigated by many authors [1][2][3][4][5].Various types of arguments are employed to establish energy decay estimates for the resulting biharmonic equation in a semifinite strip.It is, however, not yet known how to employ a simple technique to yield a sharp decay rate-one which approximates the exact decay rate and makes effective application of Saint-Venant's principle possible.Although the analogous three-dimensional biharmonic problem does not have the same elasticity interpretation as its counterpart in the plane, it is nevertheless of mathematical interest to try to extend the decay results in R2 to analogous results in R3.In this paper, we derive and compare decay estimates using both direct analogues of known arguments for the two-dimensional case and new techniques.In §3 we derive exponential decay estimates for the first-order energy and the second-order energy.Then we establish an alternate energy decay estimate in §4.An equivalence between the first-order energy and the second-order energy is derived in §5.Finally, we establish upper bounds for the total energies in terms of the given boundary data on the finite end.As we shall see, although there is an obvious similarity between the plane biharmonic equation and the spatial biharmonic equation, our arguments in this paper are quite different from those of [1-3], Meanwhile, the techniques and results of this paper could be extended to the case of arbitrary dimension.2. Preliminary results.Let R be the interior of a semi-infinite cylinder whose cross section D is bounded by one or more piecewise smooth simple closed curves.Choose Cartesian coordinates x{, x2, and x} with the origin as one end of the cylinder and the x3-axis parallel to the generator.Let 0 be a smooth solution of the biharmonic equation AA0 = 0 on R = D x [0, oo) (2.1)