Hans-Christoph Grunau

Polyharmonic Boundary Value Problems

Page and line numbers refer to the final version which appeared at Springer-Verlag. The preprint version, which can be found on our personal web pages, has different page and line numbers.

Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ R can be refined by adding remainder terms which involve L p norms. In the higher-order case further L p norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W 1,p 0.

A Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in (P.-L. Lions, SIAM Rev., 24 (1982), pp. 441-467), we partially extend results known for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list those of particular interest in the final section.

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

AbstractWe are concerned with the semilinear polyharmonic model problem (−Δ)K v = λv +v|v|s−1 inB,Dαv|∂B = 0 for ¦α|<-K − 1. HereK ε ℕ,B is the unit ball in ℓn,n >2K,

Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data

s = \frac{{n + 2K}}{{n - 2K}}

Classical solutions for some higher order semilinear elliptic equations under weak growth conditions

is the critical Sobolev exponent. Let λ1 denote the first Dirichlet eigenvalue of (-Δ)K inB. The existence of a positive radial solutionv is shown for

Stability and Symmetry in the Navier Problem for the One-Dimensional Willmore Equation

\begin{array}{*{20}c} { - \lambda \in (0,\lambda _1 ), if n \geqslant 4K,} \\ { - \lambda \in (\bar \lambda ,\lambda _1 ) for some \bar \lambda = \bar \lambda (n,K) \in (0,\lambda _1 ), if 2K + 1 \leqslant n \leqslant 4K - 1.} \\ \end{array}