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@PHDTHESIS{Deipenbrock:480364,
      author       = {Deipenbrock, Matthias},
      othercontributors = {Wagner, Alfred and von der Mosel, Heiko},
      title        = {{R}obin boundary conditions in shape optimization},
      school       = {Aachen, Techn. Hochsch.},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {Publikationsserver der RWTH Aachen University},
      reportid     = {RWTH-2015-03630},
      pages        = {VII, 79 S.},
      year         = {2015},
      note         = {Aachen, Techn. Hochsch., Diss., 2015},
      abstract     = {The present thesis is concerned with the problem of proving
                      the existence of optimal domains for functionals subjected
                      to Robin Boundary conditions. We treat both cases of
                      positive and negative Robin parameters. In the case of
                      positive Robin parameters we prove the existence of a
                      minimizing domain in a class of Lipschitz domains of given
                      measure, that are uniform extension domains. In addition to
                      the linear case, i.e. the case of the first eigenvalue, we
                      consider Rayleigh quotients corresponding to the Sobolev
                      embedding theorem, up to the critical exponent.
                      Subsequently, we show that the volume constraint can be
                      replaced by a surface area constraint.For negative Robin
                      parameters we restrict the class of domains. We consider
                      domains that are starshaped with respect to a fixed ball,
                      thus fixing the topology of the domains. This exludes recent
                      counter examples to the reverse Faber-Krahn inequality.
                      Using a uniform trace inequality, we prove the existence of
                      a maximizing domain for the first eigenvalue of the Robin
                      Laplacian. Subsequently, we present an additional existence
                      result in a class resembling spherical shells. Moreover, we
                      prove the existence of optimal domains in a smoother
                      setting, using a constraint on the mean curvature to obtain
                      the compactness of the class of domains with respect to the
                      stronger topology. As a consequence of the smoother setting,
                      we are able to discuss further regularity properties of
                      optimal domains.},
      cin          = {111810 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)111810_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      urn          = {urn:nbn:de:hbz:82-rwth-2015-036301},
      url          = {https://publications.rwth-aachen.de/record/480364},
}