Spectral characteristics of Dirichlet problems for nonlocal operators

  • 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.

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Author:Pierre Aimé FeulefackORCiDGND
Referee:Tobias WethORCiDGND, Mouhamed Moustapha Fall
Document Type:Doctoral Thesis
Date of Publication (online):2022/05/12
Year of first Publication:2022
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Granting Institution:Johann Wolfgang Goethe-Universität
Date of final exam:2022/04/21
Release Date:2022/05/12
Page Number:163
Institutes:Informatik und Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoDeutsches Urheberrecht