Elliptic equations appear for instance as
models describing stationary phenomena. Most of those equations are either of
second or fourth order. Although the general theory for second and higher order
elliptic equations has much in common there is a crucial difference. For second
order elliptic equations a so-called maximum principle exists which implies
that a positive source term yields that the solution is positive. Think of a
membrane fixed at its edges. If one pushes upwards somewhere the membrane will
move upwards everywhere. A similar result for fourth and higher order does not
hold in the same generality. A simple model for a plate which is clamped at its
edges is of fourth order. Pushing a circular plate upwards somewhere inside
results in the plate bending upwards everywhere. However, a plate which has the
shape of a long rectangle behaves differently. Pushing upwards will move the
plate upwards nearby but might result in downwards movement near the other
side. The lack of such a sign preserving property is a main complication in the
study of higher order (non)linear elliptic equations.
The separation of the local sign-preserving behaviour
and the behaviour away from the source could be one
of the themes of the project.
membrane (unit circle), pushed upwards near
(.5,0), from above and side-view
The deflection u of this membrane is
modeled by the
clamped plate (unit circle), pushed upwards
near (.5,0), from above and side-view
A
linear model for the deflection u of a clamped plate by a load f is uxxxx + 2 uxxyy
+ uyyyy = f with u = un
= 0 on the boundary.
a clamped rectangular plate pushed
downwards near one side bends up on the opposite side (red area).
Again the linear model for the clamped
plate: uxxxx + 2 uxxyy
+ uyyyy = f with u = un
= 0 on the boundary. For the rectangle also the first eigenfunction
is known to change sign near a corner.
a clamped rectangular grid bending
down near corners
A clamped square grid (having a square
mesh) is modeled by an orthotropic fourth order operator uxxxx
+ uyyyy = f again with u = un = 0 on the boundary. Although now the first eigenfunction is of one sign, numerical evidence indicates
that pushing down (yellow bar) might result in the grid moving upwards near
some corners (red area).
from disk to cardioid
by limacons; the orange one is critical for the positivity preserving property; for the two reds there are
positive loads that result in a sign-changing deflection.
For more reading: A. Dall'Acqua and G. Sweers,
The
clamped plate equation on the Limaçon, to
appear in Annali di Matematica
Pura ed Applicata. (The
original publication will be available at springerlink.com, © Springer).
(∂x∂x
+∂y∂y)(u ◦ h) = |(h1,x,h1,y)|2
((∂x∂x +∂y∂y)u ) ◦ h .
This is
very helpful for the
These domains are constructed by
applying an inversion to one of the above green limacons
with the center pointed at by the red arrow.
For more reading: A. Dall'Acqua and G. Sweers,
On
domains for which the clamped plate system is positivity
preserving, to appear in: Partial Differential Equations and Inverse
Problems, ed. by Carlos Conca, Raul Manasevich, Gunter Uhlmann and
Michael Vogelius, AMS, 2004.
The slides of a lecture on this subject can
be downloaded through lecture.pdf.