Speaker
Gerd Grubb, University of Copenhagen
Abstract
We consider the Dirichlet problem for a strongly elliptic pseudodifferential operator \(P\) of order \(2a\) \((0< a<1)\) generalizing the fractional Laplacian \((-\Delta)^a\), on a bounded open subset \(\Omega\) of \(\mathbb{R}^n\) with some smoothness. The Dirichlet realization \(P_D \)in \(L_2(\Omega)\) has a domain involving the factor \(d^a\), where \(d(x)\) is the distance to the boundary; similar observations hold in Bessel-potential spaces \(H^s_q\) with \( 1 < q < \infty \). \(P_D\) has discrete spectrum in a sectorial region around \( \mathbb{R}_+ \). The eigenfunctions behave at best like \(d^aC^{2a}\) near the boundary. The eigenvalue asymptotics are described by Weyl-type formulas, when \(\Omega\) is Lipschitz. Resolvent estimates outside the sectorial region are used to obtain solvability of evolution problems for \(P_D\). In particular, there are new results on maximal \(L_q\)-regularity.