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On the shape of the ground state eigenfunction for stable processes. (English) Zbl 1105.31006
For a bounded convex domain in \(\mathbb R^d\) the authors consider the eigenvalue problem \((-\Delta)^{\alpha/2}\phi_n (x) = \lambda_n \phi_n (x)\) for \(x \in D\), \(\phi_n (x)=0\) for \(x \in D^c\), the Dirichlet problem for stable processes, \(0<\alpha<2\), \(0 < \lambda_1 < \lambda_2 < \lambda_3 <\dots \rightarrow \infty\). Their aim is to generalize some results known for the Brownian motion case \(\alpha= 2\) and the Cauchy case \(\alpha = 1\). For a convex domain symmetric with respect to each coordinate axis they define a concept “mid-concavity” first for certain straight segments parallel to a coordinate axis and then for the whole domain \(D\). Then, in a sequence of propositions, lemmata and corollaries, they proceed to the proof of their main result which says that if \(D\) is a rectangle \(Q\) symmetric to the origin and with each edge parallel to a cordinate axis the “ground state eigenfunction” \(\phi_1\) is mid-concave.

MSC:
31C45 Other generalizations (nonlinear potential theory, etc.)
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
60G52 Stable stochastic processes
60J45 Probabilistic potential theory
26A33 Fractional derivatives and integrals
35P99 Spectral theory and eigenvalue problems for partial differential equations
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