Guido Sweers

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.

Sharp estimates for iterated Green functions

Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second order elliptic operators these estimates hold true on bounded C 1,1 domains. For higher order elliptic operators we have to restrict ourselves to the polyharmonic operator on balls. We will also consider applications to noncooperatively coupled elliptic systems and to the lifetime of conditioned Brownian motion.

On the bifurcation curve for an elliptic system of FitzHugh–Nagumo type

Abstract We study a system of partial differential equations derived from the FitzHugh–Nagumo model. In one dimension solutions are required to satisfy zero Dirichlet boundary conditions on the interval Ω=(−1,1). Estimates are given to describe bounds on the range of parameters over which solutions exist; numerical computations provide the global bifurcation diagram for families of symmetric and asymmetric solutions. In the two-dimensional case we use numerical methods for zero Dirichlet boundary conditions on the square domain Ω=(−1,1)×(−1,1) . Numerical computations are given both for symmetric and asymmetric, and for stable and unstable solutions.

A survey on boundary conditions for the biharmonic

For second-order elliptic equations one usually extensively studies the case of Dirichlet boundary conditions and other boundary conditions are left to the reader. For the biharmonic equation Δ2 u = f on a bounded domain in ℝ n it is not that obvious which boundary condition would serve as a role model. Usually a good approach is to focus on some boundary conditions which describe physically relevant situations. We will discuss boundary conditions for a slender beam and for the thin elastic plate.

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

We want to explain the crucial difference between the second order (m = 1) and higher order (m > 1) case. If m = 1 a very satisfactory result is known. Indeed, the sign condition (2) alone is sufficient to ensure classical solvability of the Dirichlet problem (1). There are two very strong devices in the theory of second order elliptic equations, which make the proof of the needed a-priori maximum estimate an easy exercise:

A strong maximum principle for a noncooperative elliptic system

In this note it is shown that on a ball in

Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains

\mathbb{R}^N

Regions of positivity for polyharmonic Green functions in arbitrary domains

, with

On a conditioned Brownian motion and a maximum principle on the disk

N > 2

No Gidas–Ni–Nirenberg Type Result for Semilinear Biharmonic Problems

, a maximum principle holds for a special elliptic system. This system is such that the classical maximum principle is not applicable.