Polyharmonic Boundary Value Problems

This monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. Underlying models and, in particular, the role of di erent boundary conditions are explained in some detail. §As for linear problems, after a brief summary of the existence theory and Lp- and Schauder-estimates, the focus is on positivity or -since, in contrast to second order equations, a general form of a comparison principle does not exist -on "almost positivity". The required kernel estimates are also presented in some detail. §As for nonlinear problems, several techniques well-known from second order equations are no longer available and have to be replaced by new and di erent methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the rst part is emphasised and, moreover, a far reaching Gidas-Ni-Nirenberg-type symmetry result is included. Finally, some recent progress on the Dirichlet problem for Willmore surfaces under symmetry assumptions is reported.