Existence of variational solutions to a Cauchy–Dirichlet problem with time-dependent boundary data on metric measure spaces

• Collins, Michael ^[1]
  1. [1] Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11, 91058, Erlangen, Germany
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 2, 2021, págs. 281-306
• Idioma: inglés
• DOI: 10.1007/s13348-020-00288-0
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• Resumen

  • The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space ({\mathcal {X}}, d, \mu ). For such parabolic minimizers that coincide with Cauchy-Dirichlet data \eta on the parabolic boundary of a space-time-cylinder \varOmega \times (0, T) with an open subset \varOmega \subset {\mathcal {X}} and T > 0, we prove existence in the parabolic Newtonian space L^p(0, T; {\mathcal {N}}^{1,p}(\varOmega )). In this paper we generalize results from Collins and Herán (Nonlinear Anal 176:56–83, 2018) where only time-independent Cauchy–Dirichlet data have been considered. We argue completely on a variational level.