Existence of parabolic minimizers on metric measure spaces. (November 2018)

• Record Type: Journal Article
• Authors: Collins, Michael; Herán, Andreas
• Is Part Of: Nonlinear analysis. Volume 176(2018)
• Page Start: 56
• Page End: 83
• Publication Date: 2018-11
• Subjects: 49J27 -- 49J40 -- 30L99 -- 35A15; Existence -- Parabolic minimizers -- Metric measure spaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.06.005
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The objects of our studies are vector valued parabolic minimizers u associated to a convex Carathéodory integrand f obeying a p -growth assumption from below and a certain monotonicity condition in the gradient variable. Here, the functions being considered are defined on a metric measure space ( X, d, μ ) . For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u 0 on the parabolic boundary of a space–time-cylinder Ω × ( 0, T ) with an open subset Ω ⊂ X and T > 0, we prove existence in the parabolic Newtonian space L p ( 0, T ; N 1, p ( Ω ; R N ) ) . In this paper we generalize results from Bögelein et al. (2014, 2015) to the metric setting and argue completely on a variational level.