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 $\varphi _{1}:$

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

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.