On sign preserving properties of higher order elliptic equations.

RESEARCH THEME

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.

EXAMPLES

1. The circular membrane (second order):

membrane (unit circle), pushed upwards near (.5,0), from above and side-view

The deflection u of this membrane is modeled by the laplace equation -(u_xx+u_yy) = f with boundary condition u = 0.

2. The clamped circular plate:

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 u_xxxx + 2 u_xxyy + u_yyyy = f with u = u_n = 0 on the boundary.

3. A clamped rectangular plate:

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: u_xxxx + 2 u_xxyy + u_yyyy = f with u = u_n = 0 on the boundary. For the rectangle also the first eigenfunction is known to change sign near a corner.

4. A clamped square grid:

a clamped rectangular grid bending down near corners

A clamped square grid (having a square mesh) is modeled by an orthotropic fourth order operator u_xxxx + u_yyyy = f again with u = u_n = 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).

1. Clamped plate equation for other domains:
   Hadamard claimed in 1907 that the clamped plate problem on any ‘Limaçon de Pascal’ is positivity preserving. These limacons describe a one-parameter family of domains between disk and cardioid. Although his claim is wrong in its full generality we have been able to show that at least up to some critical limacon pushing down the surface results in bending down.
   The result is remarkable in the sense that it seems to be the first example of a non-convex domain for which the clamped plate problem is positivity preserving.

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).

2. Biharmonic and conformal mappings:
   A transformation by a conformal mapping and the laplace equation almost commute. Indeed, if h is conformal then:

(∂_x∂_x +∂_y∂_y)(u ◦ h) = |(h_1,x,h_1,y)|² ((∂_x∂_x +∂_y∂_y)u ) ◦ h .

This is very helpful for the laplace equation in 2 dimensions, but in general not for the bilaplace (∂_x∂_x +∂_y∂_y)² u = f. There is one nontrivial exception namely the inversion h(z) = z ^-1. Applying such an inversion (combined with simple transformations like shifts, dilatations and rotations) to a ‘positivity preserving’ limacon one finds that clamped plates with the following shapes are positivity preserving.

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.

LECTURE

The slides of a lecture on this subject can be downloaded through lecture.pdf.

g.h.sweers@its.tudelft.nl.