Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 40, 2015, 89-108

# THE OBSTACLE AND DIRICHLET PROBLEMS
ASSOCIATED WITH *p*-HARMONIC FUNCTIONS
IN UNBOUNDED SETS IN **R**^{n}
AND METRIC SPACES

## Daniel Hansevi

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 **R**^{n} and metric spaces.
Ann. Acad. Sci. Fenn. Math. 40 (2015), 89-108.

Full document as PDF file

doi:10.5186/aasfm.2015.4005

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