Energy quantization and mean value inequalities for nonlinear boundary value problems

Katrin Wehrheim

Journal of the European Mathematical Society (2005)

  • Volume: 007, Issue: 3, page 305-318
  • ISSN: 1435-9855

Abstract

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We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal derivative: One obtains local uniform bounds on the complement of finitely many points, where some minimum quantum of energy concentrates.

How to cite

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Wehrheim, Katrin. "Energy quantization and mean value inequalities for nonlinear boundary value problems." Journal of the European Mathematical Society 007.3 (2005): 305-318. <http://eudml.org/doc/277664>.

@article{Wehrheim2005,
abstract = {We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal derivative: One obtains local uniform bounds on the complement of finitely many points, where some minimum quantum of energy concentrates.},
author = {Wehrheim, Katrin},
journal = {Journal of the European Mathematical Society},
keywords = {variational methods; extremal problems; energy quantization principle},
language = {eng},
number = {3},
pages = {305-318},
publisher = {European Mathematical Society Publishing House},
title = {Energy quantization and mean value inequalities for nonlinear boundary value problems},
url = {http://eudml.org/doc/277664},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Wehrheim, Katrin
TI - Energy quantization and mean value inequalities for nonlinear boundary value problems
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 3
SP - 305
EP - 318
AB - We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal derivative: One obtains local uniform bounds on the complement of finitely many points, where some minimum quantum of energy concentrates.
LA - eng
KW - variational methods; extremal problems; energy quantization principle
UR - http://eudml.org/doc/277664
ER -

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