ANTI-MAXIMUM PRINCIPLES


GUIDO SWEERS

G.H.Sweers@its.tudelft.nl

Dept. of Applied Mathematical Analysis
ITS Faculty
Delft University of Technology
PO Box 5031, 2600 GA Delft, The Netherlands






For second order elliptic boundary value problems such as $ -\Delta
u=\lambda u+f$ in $ \Omega $ and $ u=0$ on $ \partial \Omega $ it is well known that for any reasonable bounded domain in $ \mathbb{R}^{n}$ a first eigenvalue $ \lambda _{1}$ exists and moreover, for any $ \lambda <\lambda _{1}$ a positive source term $ f$ implies that the solution $ u$ is positive. Clément and Peletier showed that for $ \lambda $ in a right neighbourhood of $ \lambda _{1}$ opposite behaviour occurs, an phenomenon which they named the anti-maximum principle: certain $ f>0$ imply $ u<0.$ Such behaviour is roughly explained by the pole of the resolvent at $ \lambda _{1}$ and the positive sign of the corresponding eigenfunction $ \varphi _{1}:$

$\displaystyle u=\frac{1}{\lambda _{1}-\lambda }\left\langle
\varphi _{1},f\right\rangle \varphi _{1}+R_{\lambda }f. $

with $ R_{\lambda }f$ the remaining regular part. A more precise statement reads as: if $ 0<f\in L^{p}\left( \Omega \right) $ with $ p>n,$ then there is $ %%
\delta _{f}>0$ such that for $ \lambda \in \left( \lambda
_{1},\lambda _{1}+\delta _{f}\right) $ the solution satisfies $ u<0.$ For higher order elliptic boundary value problems with appropriate boundary conditions sometimes a stronger uniform anti-maximum principle holds, that is, for $ Lu=\lambda u+f$ in $ \Omega $ with $ Bu=0$ on $ \partial \Omega $ there exists $ \delta >0$ such that for all $ \lambda \in \left(
\lambda _{1},\lambda _{1}+\delta \right) $ and $ f>0$ one obtains $ u<0.$

The anti-maximum principle for higher order elliptic boundary value problems is joint work with Ph.Clément. Sharp conditions for the uniform anti-maximum principle will appear in a joint work with H.-Ch.Grunau.


EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts