MAXIMUM-PRINCIPLES FOR THE POLYHARMONIC EQUATION ON LIPSCHITZ-DOMAINS

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in R(3). In R(n), n greater than or equal to 4, the Agmon-Miranda maximum principle and L(p)-Dirichlet estimates for certain p > 2 are shown to fail in Lipschitz domains for these equations. In particular if 4 less than or equal to n less than or equal to 2m + 1 the L(p) Dirichlet problem for Delta(m) fails to be solvable for p > 2(n - 1)/(n - 3).