FUŠCSIK SPECTRUM OF THE LAPLACIAN ON SOME SUBDOMAINS OF THE PLANE


JIŠRSI HORÁK

horak@math.unibas.ch



WOLFGANG REICHEL

reichel@math.unibas.ch

Mathematisches Institut
Universität Basel
Rheinsprung 21, CH-4057, Basel, Switzerland






The problem $ \Delta u + \mu u^+ - \nu u^- = 0$ in $ \Omega\subset\mathbb{R}^2$ with $ u=0$ on $ \partial\Omega$ is studied, where $ u^\pm=\max(0,\pm u)$, $ \mu,\nu\in\mathbb{R}$. A pair $ (\mu,\nu)$ is called a Fucík eigenvalue if there exists a function $ u_{\mu,\nu}\neq 0$ that solves the problem. Local existence results are obtained and a numerical method is used to approximate some of the Fucík eigenvalues on various domains in $ \mathbb{R}^2$. Based on a variational formulation in $ W^{1,2}_0(\Omega)$, critical points of the functional $ F(u)=\int_\Omega \vert\nabla u\vert^2$d$ x$ subject to constraints $ \int_\Omega (u^+)^2$d$ x=t$ and $ \int_\Omega (u^-)^2$d$ x=1-t$ with $ t\in(0,1)$ are sought. The information obtained numerically is compared to the known analytical results.


EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts