Polyharmonic boundary value problems

1.2 The Boggio-Hadamard conjecture for a clamped plate 11
sign change occurs already in ellipses with ratio of half axes ≈ 1.2. Nakai and Sario
[317] give a construction how to extend Garabedian’s example also to higher dimensions.
Sign change is also proven by Coffman-Duffin [108] in any bounded domain
containing a corner, the angle of which is not too large. Their arguments are based
on previous results by Osher and Seif [326, 367] and cover, in particular, squares.
This means that neither in arbitrarily smooth uniformly convex nor in rather symmetric
domains the Green function needs to be positive. Moreover, in [120] it has
been proved that Hadamard’s claim for the limaçons is not correct. Limaçons are a
one-parameter family with circle and cardioid as extreme cases. For domains close
enough to the cardioid, the Green function is no longer positive. Surprisingly, the
extreme case for positivity is not convex. Hence convexity is neither sufficient nor
necessary for a positive Green function. One should observe that in one dimension
any bounded interval is a ball and so, one always has positivity there thanks to Boggio’s
formula.
For the history of the Boggio-Hadamard conjecture one may also see Maz’ya’s
and Shaposhnikova’s biography [294] of Hadamard.
Fig. 1.2 Limaçons vary from circle to cardioid. The fifth limaçon from the left is critical for a
positive Green function.
Despite the fact that the Green function is usually sign changing, it is very hard
to find real world experiments where loss of positivity preserving can be observed.
Moreover, in all numerical experiments in smooth domains, it is very difficult to
display the negative part and heuristically, one feels that the negative part of G ∆ 2 ,Ω
– if present at all – is small in a suitable sense compared with the “dominating”
positive part. We refine the Boggio-Hadamard conjecture as follows:
In arbitrary domains Ω ⊂ R n , the negative part of the biharmonic Green’s function G ∆ 2 ,Ω
is small relative to the singular positive part. In the investigation of nonlinear problems,
the negative part is technically disturbing but it does not give rise to any substantial additional
assumption in order to have existence, regularity, etc. when compared with analogous
second order problems.
The present book may be considered as a first contribution to the discussion
of this conjecture and Chapters 5 and 6 are devoted to it. Chapter 4 provides the
necessary kernel estimates. Let us mention some of those results which we have
obtained so far to give support to this conjecture. For any smooth domain Ω ⊂ R n
(n ≥ 2) we show that there exists a constant C = C(Ω) such that for the biharmonic
Green’s function G ∆ 2 ,Ω under Dirichlet boundary conditions one has the following
estimate from below:
G ∆ 2 ,Ω (x,y) ≥ −C dist(x,∂Ω) 2 dist(y,∂Ω) 2 .