Mathematical Research Letters

Volume 19 (2012)

Number 6

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

Pages: 1263 – 1275



Pekka Koskela (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Nageswari Shanmugalingam (Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio, U.S.A.)

Yuan Zhou (Department of Mathematics, Beijing University of Aeronautics and Astronautics, Beijing China)


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.


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

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