Biharmonic Green functions

  • Heinrich Begehr
Keywords: biharmonic Green, Neumann, Robin, hybrid Green functions, boundary value problems, inhomogeneous biharmonic equation, unit disc in complex plane


The harmonic Green and Neumann function and a particular Robin function are used to construct bi-harmonic Green, Neumann and particular Robin functions. Moreover hybrid bi-harmonic Green functions are given. They all are constructed via a convolution of the mentioned harmonic particular fundamental solutions. In case of the unit disc they are explicitly expressed.
Besides these 9 bi-harmonic Green functions there is another bi-harmonic Green function in explicit form for the unit disc not defined by convolution. Related boundary value problems are not all well posed. In case they are, the unique solutions are given. For the other cases solvability conditions are determined and the unique solutions found. There are all together 12 Dirichlet kind boundary value problems for the inhomogeneous bi-harmonic equation treated. The investigation is restricted to the two dimensional case and complex notation is used.