Annales Academiæ Scientiarum Fennicæ
Volumen 40, 2015, 89-108
Linköping University, Department of Mathematics
SE-581 83 Linköping, Sweden; daniel.hansevi 'at' liu.se
Abstract. The obstacle problem associated with p-harmonic functions is extended to unbounded open sets, whose complement has positive capacity, in the setting of a proper metric measure space supporting a (p,p)-Poincaré inequality, 1 < p < ∞, and the existence of a unique solution is proved. Furthermore, if the measure is doubling, then it is shown that a continuous obstacle implies that the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.
2010 Mathematics Subject Classification: Primary 31E05; Secondary 31C45, 35D30, 35J20, 35J25, 35J60, 47J20, 49J40, 49J52, 49Q20, 58J05, 58J32.
Key words: Dirichlet problem, Dirichlet space, doubling measure, metric space, minimal p-weak upper gradient, Newtonian space, nonlinear, obstacle problem, p-harmonic, Poincaré inequality, potential theory, upper gradient.
Reference to this article: D. Hansevi: The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in Rn and metric spaces. Ann. Acad. Sci. Fenn. Math. 40 (2015), 89-108.
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