Let $ O$ be an open bounded connected set in $ R^n$ and K its boundary. Given a continuous real function $ f$ defined on $ K$ , I would like to extend $ f$ to a continuous real function $ g$ (i.e. $ g$ restricted to $ K$ equals $ f$ ) defined in the closure of $ O$ , in such a way that g is also monotone (e.g. the min and max of $ g$ on each closed ball contained in $ O$ is attained on the boundary of the ball). In case $ K$ has the extra property of being Holder-continuous, there exists a harmonic function $ g$ as intended; in particular $ g$ is real-analytic in $ O$ and is monotonic by the min and max principles. What about if no extra property of $ K$ is known, can one still find a monotonic $ g$ as intended?
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