h1

% IMPORTANT: The following is UTF-8 encoded.  This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.

@PHDTHESIS{Bundrock:973341,
      author       = {Bundrock, Lukas},
      othercontributors = {Wagner, Alfred and von der Mosel, Heiko and Henrot,
                          Antoine},
      title        = {{T}he {R}obin eigenvalue in exterior domains},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {RWTH Aachen University},
      reportid     = {RWTH-2023-10726},
      pages        = {1 Online-Ressource : Illustrationen},
      year         = {2023},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; Dissertation, RWTH Aachen University, 2023},
      abstract     = {In the present thesis we consider eigenvalues of the Robin
                      Laplacian, the Laplace operator with Robin boundary
                      conditions, in the exterior of a compact set. In contrast to
                      the Robin Laplacian in bounded domains, the essential
                      spectrum is not empty. D. Krejcirik and V. Lotoreichik show
                      that there is a discrete eigenvalue below the essential
                      spectrum if and only if $\alpha$, the parameter of the
                      boundary condition, is smaller than a constant $\alpha^*$
                      which depends on the domain.Using the results of G. Auchmuty
                      and Q. Han, we show that $\alpha^*$ coincides with the first
                      eigenvalue of an appropriate Steklov eigenvalue problem. In
                      addition, we characterize the existence of exactly $k$
                      discrete eigenvalues of the Robin Laplacian depending on
                      $\alpha$. If the first eigenvalue of the Robin Laplacian
                      $\lambda_1$ is discrete, we consider a geometric
                      optimization problem. We follow the ideas of C. Bandle and
                      A. Wagner and show that the ball is a local maximizer of
                      $\lambda_1(\mathbb{R}^n \setminus \overline{\Omega})$ among
                      all smooth bounded domains $\Omega \subseteq \mathbb{R}^n $
                      with fixed volume. If we exclude translations, the ball is a
                      strict local maximizer. In addition, we show that the
                      exterior problem can be approximated by spherical shells and
                      illustrate the transition from a discrete spectrum to an
                      essential spectrum. Furthermore, we show that the ball is a
                      local maximizer of the first Steklov eigenvalue in exterior
                      domains.},
      cin          = {111810 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)111810_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2023-10726},
      url          = {https://publications.rwth-aachen.de/record/973341},
}