Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 47, pp. 54.
Title: Regularity for a clamped grid equation $u_{xxxx}+u_{yyyy}=f$
on a domain with a corner
Authors: Tymofiy Gerasimov (Delft Univ. of Technology, The Netherlands)
Guido Sweers (Univ. zu Koln, Cologne, Germany)
Abstract:
The operator $L=\frac{\partial ^{4}}{\partial x^{4}}
+\frac{\partial ^{4}}{\partial y^{4}}$ appears in a model for the
vertical displacement of a two-dimensional grid that consists of
two perpendicular sets of elastic fibers or rods. We are interested
in the behaviour of such a grid that is clamped at the boundary and
more specifically near a corner of the domain.
Kondratiev supplied the appropriate setting in the sense of Sobolev
type spaces tailored to find the optimal regularity. Inspired by
the Laplacian and the Bilaplacian models one expect, except maybe for
some special angles that the optimal regularity improves when angle
decreases. For the homogeneous Dirichlet problem with this special
non-isotropic fourth order operator such a result does not hold true.
We will show the existence of an interval
$( \frac{1}{2}\pi ,\omega _{\star })$,
$\omega _{\star }/\pi \approx 0.528\dots$
(in degrees $\omega _{\star }\approx 95.1\dots^{\circ} $),
in which the optimal regularity improves with increasing opening angle.
Submitted December 10, 2008. Published April 02, 2009.
Math Subject Classifications: 35J40, 46E35, 35P30.
Key Words: Nonisotropic; fourth order PDE; domain with corner;
clamped grid; weighted Sobolev space; regularity.