Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

Andrzej Świȩch

doi:10.3934/dcds.2020156

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2020, Volume 40, Issue 5: 2945-2962. Doi: 10.3934/dcds.2020156

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Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

Andrzej Świȩch^,

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

^* Corresponding author: Andrzej Świȩch
^* Corresponding author: Andrzej Świȩch

Received: August 31, 2019

Published: March 2020

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Abstract

We prove various pointwise properties of $ L^p $-viscosity solutions of fully nonlinear uniformly elliptic equations with unbounded measurable terms and possibly quadratically growing gradient terms. These include differentiability properties and "pointwise maximum principle". In particular we discuss an equivalent pointwise definition of $ L^p $-viscosity solution which, together with pointwise properties, allows to operate on $ L^p $-viscosity solutions almost as if they were strong solutions and move them freely from one equation to another. Such results were proved before in [6,8,30] for equations with linearly growing gradient terms, however it appears that they have not been widely used. Here we generalize pointwise properties of $ L^p $-viscosity solutions to a larger class of equations and show that they create a very powerful set of tools which can be used for instance to prove regularity results for equations and differential inequalities.

Keywords:

Fully nonlinear uniformly elliptic equations,
L^p-viscosity solutions,
maximum principle,
superlinear gradient terms,
differential inequalities.

Mathematics Subject Classification: Primary: 35D40, 35B50, 35B65, 35J15, 35J60.

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