Lyapunov inequalities on fractals

JUAN PABLO PINASCO; CRISTIAN SCAROLA

Banff

Workshop; Multifractal Analysis: From Theory to Applications and Back; 2014

Banff International Research Station

We analyze the existence of Lyapunov type inequalities for the Laplace-operator $\Delta_{\mu}$ on a bounded open subset $\Omega\subset \R^d$ with a positive finite Borel measure $\mu$ supported on $\overline{\Omega}$. We show that an inequality involving some geometric properties of the domain must hold in order to have a nontrivial solution. For the eigenvalue problem $$-\Delta_{\mu} u = \lambda u,$$ this inequality gives lower bounds of the eigenvalues and enable us to give an upper estimate of the spectral counting function without the use of the renewal equation.