TY  - THES
A1  - Feulefack, Pierre Aimé
T1  - Spectral characteristics of Dirichlet problems for nonlocal operators
N2  - We present new results on nonlocal Dirichlet problems established by means of suitable spectral theoretic and variational methods, taking care of the nonlocal feature of the operators. We  mainly address: First,  we estimate the Morse index of radially symmetric sign changing bounded weak solutions to a semilinear Dirichlet problem involving the fractional Laplacian.  In particular,  we  derive a conjecture due to Bañuelos and Kulczycki on the geometric structure of the second  Dirichlet eigenfunctions. Secondly, we study a small order asymptotics with respect to the parameter s  of the Dirichlet eigenvalues problem for the fractional Laplacian. Thirdly, we deal with the logarithmic Schrödinger operator. In particular, we provide an alternative to derive the singular integral representation corresponding to the associated Fourier symbol  and introduce tools and functional analytic framework for variational studies. Finaly, we study nonlocal operators of order strictly below one. In particular, we investigate interior regularity properties of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side.
KW  - Mathematik
Y1  - 2022
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/67965
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-679658
IS  - 46
ER  -