We prove the nonlinear fundamental convergence theorem for
superharmonic functions on metric measure spaces. Our proof seems to
be new even in the Euclidean setting. The proof uses direct methods
in the calculus of variations and, in particular, avoids advanced
tools from potential theory. We also provide a new proof for the
fact that a Newtonian function has Lebesgue points outside a set of
capacity zero, and give a sharp result on when superharmonic
functions have $L^q$-Lebesgue points everywhere.
Publié le : 2010-03-15
Classification:
$mathcal{A}$-harmonic,
fundamental convergence theorem,
Lebesgue point,
metric space,
Newtonian function,
nonlinear,
$p$-harmonic,
quasicontinuous,
Sobolev function,
superharmonic,
superminimizer,
supersolution,
weak upper gradient,
31C45,
31C05,
35J60
@article{1266330121,
author = {Bj\"orn
,
Anders and Bj\"orn
,
Jana and Parviainen
,
Mikko},
title = {Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces},
journal = {Rev. Mat. Iberoamericana},
volume = {26},
number = {1},
year = {2010},
pages = { 147-174},
language = {en},
url = {http://dml.mathdoc.fr/item/1266330121}
}
Björn
,
Anders; Björn
,
Jana; Parviainen
,
Mikko. Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces. Rev. Mat. Iberoamericana, Tome 26 (2010) no. 1, pp. 147-174. http://gdmltest.u-ga.fr/item/1266330121/