Local boundedness for minimizers of convex integral functionals in metric measure spaces

  • Huiju Wang
  • Pengcheng Niu

Abstract

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

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Published
2020-05-06
How to Cite
Wang, H., & Niu, P. (2020). Local boundedness for minimizers of convex integral functionals in metric measure spaces. MATHEMATICA SCANDINAVICA, 126(2), 259-275. https://doi.org/10.7146/math.scand.a-116244
Section
Articles