# GREEN'S FUNCTION METHODS FOR THE POLYHARMONIC EQUATION

The solution of the polyharmonic equation $\Delta\sp{\rm m}$u = 0 in a domain D, with conditions u = f$\sb0$ and normal derivatives ${\partial{\rm u}\over\partial

u}$ = ${\rm f}\sb1,\dots,{\partial\sp{\rm m-1}{\rm u}\over\partial

u\sp{\rm m-1}}$ = ${\rm f\sb{m-1}}$ on the boundary $\partial$D, is considered. The solution for two-dimensional domains is first obtained by the Green's function method. Green's second and third identities for $\Delta\sp{\rm m}$u are derived and used to show uniqueness of the solution. A polyharmonic Green's function G$\sb{\rm m}$(Q,P) is defined which eliminates nonprescribed boundary terms from Green's third identity for $\Delta\sp{\rm m}$u. This allows the value of a polyharmonic function at a point P of D to be represented by a line integral in terms of the given boundary data on $\partial$D. The symmetry of G$\sb{\rm m}$(Q,P) in Q and P is demonstrated. The polyharmonic Green's functions for the half-plane and for the disk are constructed for arbitrary m, and the invariance of their sign is proved. Solutions of the polyharmonic boundary value problem for the right half-plane and for the disk for m = 1, 2, 3, 4 are exhibited. The solutions of the polyharmonic boundary value problem for the right half-plane and for the disk are then also derived by a direct method, which avoids the use of the Green's functions. This method is based on a generalization, in complex form, of formulas first given by E. Goursat. For the right half-plane, a solution is obtained for arbitrary m; for the disk, solutions for m = 1, 2, 3 are displayed. The Green's functions and solution formulas for the biharmonic equation are also constructed for the half-hyperspace x$\sb{\rm n} >$ 0 and for hypersphere. A kernel function for $\Delta\sp{\rm m}$u = 0 is defined for m even and m = 1, and calculated for the disk for m = 1, 2, 4. For m = 1 and m = 2, the kernel function for the disk is expanded in a convergent bilinear series of harmonic functions which are orthonormal and complete with respect to an appropriately prescribed scalar product.