Annales Academiĉ Scientiarum Fennicĉ

Mathematica

Volumen 34, 2009, 261-277

# THE DOUBLE OBSTACLE PROBLEM ON METRIC SPACES

## Zohra Farnana

Linköping University, Department of Mathematics

SE-581 83 Linköping, Sweden; zofar 'at' mai.liu.se

**Abstract.**
We study the double obstacle
problem on a metric measure space equipped
with a doubling measure and supporting a *p*-Poincaré inequality.
We prove existence and uniqueness. We also prove the continuity of
the solution of the double obstacle problem with continuous obstacles
and show that the continuous solution is a minimizer in the open set
where it does not touch the two obstacles. Moreover we consider the
regular boundary points and show that the solution of the double
obstacle problem on a regular open set with continuous obstacles
is continuous up to the boundary.
Regularity of boundary points is further characterized in some other
ways using the solution of the double obstacle problem.

**2000 Mathematics Subject Classification:**
Primary 49J40; Secondary 31C45.

**Key words:**
Double obstacle problem, doubling measure,
metric space, nonlinear, *p*-harmonic, Poincaré inequality,
potential theory, regularity.

**Reference to this article:** Z. Farnana:
The double obstacle problem on metric spaces.
Ann. Acad. Sci. Fenn. Math. 34 (2009), 261-277.

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