On fractal minimizers and potentials of occupation measures

Abstract

We consider four prototypes of variational problems and prove the existence of fractal minimizers through the direct method in the calculus of variations. By design these minimizers are H\"older curves or H\"older parametrizations of hypersurfaces whose images generally have a non-integer Hausdorff dimension. Although their origin is deterministic, their regularity properties are roughly similar to those of typical realizations of stochastic processes. As a key tool, we prove novel continuity and boundedness results for potentials of occupation measures of Gaussian random fields. These results complement well-known results for local times, but hold under much less restrictive assumptions. In an auxiliary section, we generalize earlier results on non-linear compositions of fractional Sobolev functions with $BV$-functions to higher dimensions.