2018/09/19 by Anders Björn, Jana Björn, Nageswari Shanmugalingam
Mathematics · #Discrete measure #Finite set #Geometric Analysis and Curvature Flows #Measure (data warehouse) #Metric (unit) #Metric space #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Quasiconvex function #Real line #Space (punctuation) #math.AP #math.MG #msc:30L99 #msc:31C45 #msc:31E05 #msc:35J20 #msc:35J92 #msc:49Q20
paper · pdf · doi:10.1007/s00209-020-02536-2
published as Math. Z. 297 ( 2021), 827-854 · 27 pages
arxiv created 2018/09/19 · openalex created_date 2018/09/27 · openalex publication_date 2020/06/24 · arxiv updated 2021/01/28 · openalex updated_date 2026/08/06
Abstract We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite p th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line \mathbf R <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , we characterize all locally doubling measures, supporting a local p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -Poincaré inequality, for which there exist nonconstant quasiminimizers of finite p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -energy, and show that a quasiminimizer is of finite p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -energy if and only if it is bounded. As p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -harmonic functions are quasiminimizers they are covered by these results.