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# Mathematical Research Letters

## Volume 19 (2012)

### Number 6

### $L^∞$-variational problem associated to Dirichlet forms

Pages: 1263 – 1275

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n6.a7

#### Authors

#### Abstract

We study the $L^∞$-variational problem associated to a general regular, strongly local Dirichlet form.We show that the intrinsic distance determines the absolute minimizer (infinite harmonic function) of the corresponding $L^∞$-functional. This leads to the existence and uniqueness of the absolute minimizer on a bounded domain, given a continuous boundary data. Applying this, we also obtain that an infinity harmonic function on $\mathbb{R}^n$ may be the minimizer for several different variational problems. Finally, we apply our results to Carnot–Carathéodory spaces.

#### Keywords

Dirichlet form, intrinsic distance, differential structure, $L^∞$-variational problem, absolute minimizer, metric measure space, Carnot–Carathéodory space