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@ARTICLE{Hoeltgen:862631,
      author       = {Hoeltgen, Laurent and Kleefeld, Andreas and Harris, Isaac
                      and Breuss, Michael},
      title        = {{T}heoretical foundation of the weighted {L}aplace
                      inpainting problem},
      journal      = {Applications of mathematics},
      volume       = {64},
      number       = {3},
      issn         = {0862-7940},
      address      = {Dordrecht [u.a.]},
      publisher    = {Springer Science + Business Media B.V},
      reportid     = {FZJ-2019-02893},
      pages        = {281-300},
      year         = {2019},
      abstract     = {Laplace interpolation is a popular approach in image
                      inpainting using partial differential equations. The classic
                      approach considers the Laplace equation with mixed boundary
                      conditions. Recently a more general formulation has been
                      proposed, where the differential operator consists of a
                      point-wise convex combination of the Laplacian and the known
                      image data. We provide the first detailed analysis on
                      existence and uniqueness of solutions for the arising mixed
                      boundary value problem. Our approach considers the
                      corresponding weak formulation and aims at using the Theorem
                      of Lax-Milgram to assert the existence of a solution. To
                      this end we have to resort to weighted Sobolev spaces. Our
                      analysis shows that solutions do not exist unconditionally.
                      The weights need some regularity and must fulfil certain
                      growth conditions. The results from this work complement
                      findings which were previously only available for a discrete
                      setup.},
      cin          = {JSC},
      ddc          = {510},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {511 - Computational Science and Mathematical Methods
                      (POF3-511)},
      pid          = {G:(DE-HGF)POF3-511},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000469441500002},
      doi          = {10.21136/AM.2019.0206-18},
      url          = {https://juser.fz-juelich.de/record/862631},
}