JournalsrmiVol. 31, No. 1pp. 161–214

Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology

  • Anders Björn

    Linköping University, Sweden
  • Jana Björn

    Linköping University, Sweden
Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology cover
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Abstract

We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain the Adams criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal pp-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Many of the results are new even for open EE (apart from those which are trivial in this case) and also on Rn\mathbb R^n.

Cite this article

Anders Björn, Jana Björn, Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology. Rev. Mat. Iberoam. 31 (2015), no. 1, pp. 161–214

DOI 10.4171/RMI/830