# The mean curvature measure

Quiyi Dai; Neil S. Trudinger; Xu-Jia Wang

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 3, page 779-800
- ISSN: 1435-9855

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topDai, Quiyi, Trudinger, Neil S., and Wang, Xu-Jia. "The mean curvature measure." Journal of the European Mathematical Society 014.3 (2012): 779-800. <http://eudml.org/doc/277659>.

@article{Dai2012,

abstract = {We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the corresponding Dirichlet problem when the inhomogeneous term is a measure.},

author = {Dai, Quiyi, Trudinger, Neil S., Wang, Xu-Jia},

journal = {Journal of the European Mathematical Society},

keywords = {mean curvature measure; Harnack inequality; weak continuity of mean curvature operator; weak solution; mean curvature measure; Harnack inequality; weak continuity of mean curvature operator; weak solution},

language = {eng},

number = {3},

pages = {779-800},

publisher = {European Mathematical Society Publishing House},

title = {The mean curvature measure},

url = {http://eudml.org/doc/277659},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Dai, Quiyi

AU - Trudinger, Neil S.

AU - Wang, Xu-Jia

TI - The mean curvature measure

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 3

SP - 779

EP - 800

AB - We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the corresponding Dirichlet problem when the inhomogeneous term is a measure.

LA - eng

KW - mean curvature measure; Harnack inequality; weak continuity of mean curvature operator; weak solution; mean curvature measure; Harnack inequality; weak continuity of mean curvature operator; weak solution

UR - http://eudml.org/doc/277659

ER -

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