Search

In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius $ R $ and outer radius $ R+1 $. We show that for $ R $ sufficiently large any corresponding second eigenfunction of this annulus is nonradial. In the second part, we investigate the second eigenvalue in domains of the form $ B_1(0)\setminus \overline{B_{\tau}(a)} $, where $ a $ is in the unitary ball and $ 0<\tau<1-|a| $. We show that this value is maximized for $ a = 0 $, if the set $ B_1(0)\setminus \overline{B_{\tau}(0)} $ has no radial second eigenfunction. We emphasize that the first part of our paper implies that this assumption is indeed nonempty. [ABSTRACT FROM AUTHOR]

We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary. [ABSTRACT FROM AUTHOR]

Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with F [ABSTRACT FROM AUTHOR]