# The restriction theorem for fully nonlinear subequations

• [1] Rice University Department of Mathematics P.O. Box 1892 MS-172 Houston, 77251(Texas)
• [2] Stony Brook University Department of Mathematics Stony Brook NY, 11794-3651 (USA)
• Volume: 64, Issue: 1, page 217-265
• ISSN: 0373-0956

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## Abstract

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Let $X$ be a submanifold of a manifold $Z$. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$, restrict to be viscosity subsolutions of the restricted subequation on $X$? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed to a constant coefficient (i.e., euclidean) model. This provides a long list of geometrically and analytically interesting cases where restriction holds.

## How to cite

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Harvey, F. Reese, and Lawson, H. Blaine. "The restriction theorem for fully nonlinear subequations." Annales de l’institut Fourier 64.1 (2014): 217-265. <http://eudml.org/doc/275604>.

@article{Harvey2014,
abstract = {Let $X$ be a submanifold of a manifold $Z$. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$, restrict to be viscosity subsolutions of the restricted subequation on $X$? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed to a constant coefficient (i.e., euclidean) model. This provides a long list of geometrically and analytically interesting cases where restriction holds.},
affiliation = {Rice University Department of Mathematics P.O. Box 1892 MS-172 Houston, 77251(Texas); Stony Brook University Department of Mathematics Stony Brook NY, 11794-3651 (USA)},
author = {Harvey, F. Reese, Lawson, H. Blaine},
journal = {Annales de l’institut Fourier},
keywords = {Viscosity solution; viscosity subsolution; nonlinear second-order elliptic equations; restriction; submanifold; pluripotential theory; viscosity solution},
language = {eng},
number = {1},
pages = {217-265},
publisher = {Association des Annales de l’institut Fourier},
title = {The restriction theorem for fully nonlinear subequations},
url = {http://eudml.org/doc/275604},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Harvey, F. Reese
AU - Lawson, H. Blaine
TI - The restriction theorem for fully nonlinear subequations
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 1
SP - 217
EP - 265
AB - Let $X$ be a submanifold of a manifold $Z$. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$, restrict to be viscosity subsolutions of the restricted subequation on $X$? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed to a constant coefficient (i.e., euclidean) model. This provides a long list of geometrically and analytically interesting cases where restriction holds.
LA - eng
KW - Viscosity solution; viscosity subsolution; nonlinear second-order elliptic equations; restriction; submanifold; pluripotential theory; viscosity solution
UR - http://eudml.org/doc/275604
ER -

## References

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